Research ArticleMathematical Modeling of Transmission Dynamics and OptimalControl of Vaccination and Treatment for Hepatitis B Virus

Ali Vahidian Kamyad1 Reza Akbari2 Ali Akbar Heydari3 and Aghileh Heydari4

1 Department of Mathematics Sciences Ferdowsi University of Mashhad Mashhad Iran2Department of Mathematical Sciences Payame Noor University Iran3 Research Center for Infection Control and Hand Hygiene Mashhad University of Medical Sciences Mashhad Iran4 Payame Noor University Khorasan Razavi Mashhad Iran

Correspondence should be addressed to Reza Akbari r9rezayahoocom

Received 30 December 2013 Accepted 26 February 2014 Published 9 April 2014

Academic Editor Chris Bauch

Copyright copy 2014 Ali Vahidian Kamyad et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Hepatitis B virus (HBV) infection is a worldwide public health problem In this paper we study the dynamics of hepatitis B virus(HBV) infection which can be controlled by vaccination as well as treatment Initially we consider constant controls for bothvaccination and treatment In the constant controls case by determining the basic reproduction number we study the existence andstability of the disease-free and endemic steady-state solutions of the model Next we take the controls as time and formulate theappropriate optimal control problemandobtain the optimal control strategy tominimize both the number of infectious humans andthe associated costs Finally at the end numerical simulation results show that optimal combination of vaccination and treatmentis the most effective way to control hepatitis B virus infection

1 Introduction

Hepatitis B is a potentially life-threatening liver infectioncaused by the hepatitis B virus It is a major global healthproblem It can cause chronic liver disease and chronicinfection and puts people at high risk of death from cirrhosisof the liver and liver cancer [1] Infections of hepatitis B occuronly if the virus is able to enter the blood stream and reach theliver Once in the liver the virus reproduces and releases largenumbers of new viruses into the blood stream [2]

This infection has two possible phases (1) acute and (2)chronic Acute hepatitis B infection lasts less than sixmonthsIf the disease is acute your immune system is usually ableto clear the virus from your body and you should recovercompletely within a few months Most people who acquirehepatitis B as adults have an acute infection Chronic hepatitisB infection lasts six months or longer Most infants infectedwith HBV at birth and many children infected between 1and 6 years of age become chronically infected [1] Abouttwo-thirds of people with chronic HBV infection are chroniccarriersThese people do not develop symptoms even though

they harbor the virus and can transmit it to other peopleTheremaining one-third develop active hepatitis a disease of theliver that can be very serious More than 240 million peoplehave chronic liver infections About 600 000 people die everyyear due to the acute or chronic consequences of hepatitis B[1]

Transmission of hepatitis B virus results from exposureto infectious blood or body fluids containing blood Possibleforms of transmission include sexual contact blood trans-fusions and transfusion with other human blood products(horizontal transmission) and possibly from mother tochild during childbirth (vertical transmission) [3] The mostimportant influence on the probability of developing carriageafter infection is age [4] Children less than 6 years of age whobecome infected with the hepatitis B virus are the most likelyto develop chronic infections

80ndash90 of infants infected during the first year of lifedevelop chronic infections30ndash50 of children infected before the age of 6 yearsdevelop chronic infections

Hindawi Publishing CorporationComputational and Mathematical Methods in MedicineVolume 2014 Article ID 475451 15 pageshttpdxdoiorg1011552014475451

2 Computational and Mathematical Methods in Medicine

In adults

lt5 of otherwise healthy adults who areinfected will develop chronic infection15ndash25 of adults who become chronicallyinfected during childhood die from hepatitis Brelated liver cancer or cirrhosis [1 5]

Patients with chronic carrier often have no history ofacute illness but may develop cirrhosis (liver scarring) thatcan lead to liver failure and may also develop liver cancer[5] A small portion (1ndash6) of chronic carries clear thevirus naturally [5] Someone who is infected with hepatitisB may have symptoms similar to those caused by other viralinfections However many people infected with hepatitis Bdo not have any symptoms until when more serious sideeffects such as liver damage can occur Someone who hasbeen exposed to hepatitis B may have signs 2 to 5 monthslater Some people with hepatitis B do not notice warningsigns until they become very intense Some have few orno symptoms but even someone who does not notice anysymptoms can still transmit the disease to others and canstill develop complications later in life Some people carry thevirus in their bodies and are contagious for the rest of theirlives [6]

The risk of transmission from mother to newborn canbe reduced from 20ndash90 to 5ndash10 by administering to thenewborn hepatitis B vaccine (HBV) and hepatitis B immuneglobulin (HBIG) within 12 hours of birth followed by asecond dose of hepatitis B vaccine (HBV) at 1-2 months and athird dose at and not earlier than 6 months (24 weeks) [7 8]The hepatitis B infection does not usually require treatmentbecause most adults clear the infection spontaneously [9]Early antiviral treatment may be required in less than 1of people whose infection takes a very aggressive course(fulminant hepatitis) or who are immunocompromised Onthe other hand treatment of chronic infection may benecessary to reduce the risk of cirrhosis and liver cancerTreatment lasts from six months to a year depending onmedication and genotype [10]

One of the primary reasons for studying hepatitis Bvirus (HBV) infection is to improve control and finally toput down the infection from the population Mathematicalmodels can be a useful tool in this approach which helps usto optimize the use of finite sources or simply to goal (theincidence of infection) control measures more impressivelyAnderson and May [11] used a simple mathematical modelto illustrate the effects of carriers on the transmission ofHBV A hepatitis B mathematical model (Medley et al[4]) was used to develop a strategy for eliminating HBVin New Zealand [5 12] Zhao et al [13] proposed an agestructuremodel to predict the dynamics ofHBV transmissionand evaluate the long-term effectiveness of the vaccinationprogramme in ChinaWang et al [14] proposed and analyzedthe hepatitis B virus infection in a diffusion model confinedto a finite domain Xu and Ma [15] proposed a hepatitisB virus (HBV) model with spatial diffusion and saturationresponse of the infection rate was investigated Zou et al[16] also proposed a mathematical model to understand the

transmission dynamics and prevalence of HBV in mainlandChina Pang et al [17] developed a model to explore theimpact of vaccination and other controlling measures ofHBV infection Zhang and Zhou [18] proposed analysis andapplication of an HBV model Bhattacharyya and Ghosh[19] Kar and Batabyal [20] and Kar and Jana [21] proposedoptimal control of infectious diseases

In this paper we study the dynamics of hepatitis B virus(HBV) infection under administration of vaccination andtreatment where HBV infection is transmitted in two waysthrough vertical transmission and horizontal transmissionWhile the horizontal transmission is reduced through theadministration of vaccination to those susceptible individ-uals the vertical transmission gets reduced through theadministration of treatment to infected individuals thereforethe vaccine and the treatment play different roles in control-ling the HBV [19] In this study we analyze and apply optimalcontrol to determine the possible impacts of vaccination tosusceptible individuals and treatment to infected individualsSome numerical simulations of the model are also givento illustrate the results and to find optimal strategies incontrolling HBV infection

The paper is organized as follows In Section 2 weproposed an HBV infection model with vaccination andtreatment In Section 3 we analyzed the qualitative propertyof themodel In Section 4 we considered the optimal analysisof the model and in Section 5 we considered some numericalexperiments under special choice of parameter values Thepaper will be finished with a brief discussion and conclusion

2 Model and Preliminaries

To analyze and control hepatitis B virus (HBV) infection inthe present paper we consider a model with two controlsvaccination and treatment The model is constructed basedon the characteristics of HBV transmission and the modelof Medley et al [4] According to the natural history ofHBV this study has a unique characteristic that distinguishesit from Medleyrsquos model as follows Firstly in this studytwo controlling variables are considered (vaccination andtreatment) in order to prevent the spread of the HBVand finally to put down the infection from the populationwhereas in Medleyrsquos model only one controlling variablethe vaccination was employed Secondly in our model wehave considered two different ways of the transmission ofhepatitis B virus infection namely vertical transmission(hepatitis B virus infection transmits directly from the par-ents to the offspring) and horizontal transmission (hepatitisB virus infection transmits through contact with infectiveindividuals) [19]Thirdly we have considered a proportion ofthe vaccination in susceptible individuals to all age groupswhereas Medley et al [4] have utilized only the newbornsvaccination In fact the adults especially the teenagers areencouraged to hepatitis B vaccine [17] Fourthly ldquoimmunityrdquohas been considered as ldquopermanentrdquo in Medleyrsquos modelwhereas the effect of the vaccination may be decreased aftersometime and the person may be affected and be susceptible[22] Consequently we have considered a parameter 120582

4to

Computational and Mathematical Methods in Medicine 3

SS E I C R

minus p1C minus p2R u1S

120588(I + 120579C)S

E

1205821E

I

p31205822I

p1C C

(1 minus p3)1205822I

u2C

1205823C

R

p2R

1205824R

Figure 1 FlowdiagramofHBVdynamics under application of vaccine and treatment 119878119864 119868119862 and119877denote five compartment of susceptibleexposed acute infection chronic carriers and immune or recovered class respectively The 120588(119868 + 120579119862)119878 indicates the horizontal transmissionfrom compartment 119878 to 119864 whereas ]119901

1119862 denotes the vertical transmission from 119862 to 119862 by birth of offspring from a carrier individual

Similarly ]1199012119877 represents the proportion of immune newborn from recovered class 120582

3119862 shows individualsrsquo spontaneous recovery and move

from compartment 119862 to 119877 and 1199062119862 denotes proportions of carriers move to recovered class by treatment 120582

4119877 denotes a portion that moves

from compartment 119877 to 119878 due to loss of immunity 11990131205822119868 shows proportion of acute infection individuals who become carries and (1minus119901

3)1205822119868

shows proportion of acute infection individuals clear HBV and move from 119868 to 119877 1199061119878 denotes proportions of susceptible move to recovered

class by vaccination and each compartment has its own death rate

account for the turning back recovered people to susceptibleones [16 17]

We consider an 119878-119864-119868-119862-119877 epidemic model by dividingtotal population into five time-dependent classes namely thesusceptible individuals 119878(119905) infected but not yet infectiousindividuals (exposed) 119864(119905) acute infection individuals 119868(119905)chronic HBV carriers 119862(119905) and recovered 119877(119905) for hepatitisB virus (HBV) infection that propagates through contactbetween the infected and the susceptible individuals and alsothrough the infected parents to the offspring A flow chartof this compartmental model is shown in Figure 1 Theseassumptions however lead to the following dynamic model

119878 (119905) = ] minus ]1199011119862 minus ]119901

2119877 minus 120588 (119868 + 120579119862) 119878 minus ]119878 minus 119906

1119878 + 1205824119877

(119905) = 120588 (119868 + 120579119862) 119878 minus (] + 1205821) 119864

119868 (119905) = 1205821119864 minus (] + 120582

2) 119868

(119905) = ]1199011119862 + 119901

31205822119868 minus (] + 120582

3) 119862 minus 119906

2119862

(119905) = ]p2119877 + (1 minus 119901

3) 1205822119868 + 1205823119862 minus ]119877 minus 120582

4119877 + 1199061119878 + 1199062119862

(1)

In these equations all the parameters are nonnegativeWe assume stable population with equal per capita birth anddeath rate ] (as disease-induced death rate is not consideredin system) and 120582

1is the rate of exposed individuals becoming

infections 1205822is the rate at which individuals leave the

acute infection class 1205823is the rate of carrier individuals

who recover from the disease by natural way (spontaneousrecovery) and move from carrier to recovered [16 21] and 120579is the infectiousness of carriers relative to acute infections Aproportion 119901

3of acute infection individuals become carriers

and another clear HBV moves directly to immunity class[17] The horizontal transmission of disease propagation isdenoted by the mass action term 120588(119868 + 120579119862)119878 where 120588represents the contact rate For vertical transmission weassume that a fraction 119901

1of newborns from infected class

are infected and it is denoted by the term ]1199011119862 (1199011lt 1)

Similarly a fraction 1199012of newborns from recovered class

are immune and it is denoted by ]1199012119877 (1199012lt 1) [23]

Consequently the birth flux into the susceptible class is givenby ] minus ]119901

1119862 minus ]119901

2119877

For simplicity we normalize the population size to 1 thatis now 119878 119864 119868 119862 and 119877 are respectively the fractions of thesusceptible the exposed the acute infective the carriers andthe recovered individuals in the population and 119878+119864+119868+119862+119877 = 1 holds [17] Hence the fifth equation may be omittedand (1) becomes

119878 (119905) = ] minus ]1199011119862 minus 120588 (119868 + 120579119862) 119878 minus ]119878 minus 119906

1119878

+ (1205824minus ]1199012) (1 minus 119878 minus 119864 minus 119868 minus 119862)

(119905) = 120588 (119868 + 120579119862) 119878 minus (] + 1205821) 119864

119868 (119905) = 1205821119864 minus (] + 120582

2) 119868

(119905) = ]1199011119862 + 119901

31205822119868 minus (] + 120582

3) 119862 minus 119906

2119862

(2)

Among them 1199061is the proportion of the susceptible

that is vaccinated per unit time further we assume thatthe vaccination is not perfect that is although vaccinationoffers a very powerful method of disease control there aremany associated difficulties Generally vaccines are not 100effective and therefore only a proportion of vaccinated indi-viduals are protected then some proportion of the vaccinatedindividuals may be susceptible again to that disease [22]Similarly 119906

2is the proportion of the chronic HBV carriers

that is treated per unit time The transfer rate from therecovered class to the susceptible class is taken as 120582

4(1205824ge 0)

We assume that the population of newborn carriers born tocarriers is less than the sum of the death carriers and thepopulation moving from carrier to recovery state [16] In thiscase we have ]119901

1lt ] + 119906

2+ 1205823 otherwise carriers would

keep increasing rapidly as long as there is infection that is119889119862119889119905 gt 0 for 119862 = 0 or 119868 = 0 and 119905 ge 0

Let119883(119905) = 119878(119905) + 119862(119905) + 119868(119905) + 119862(119905) then we have

d119883119889119905

=

119889119878

119889119905

+

119889119864

119889119905

+

119889119868

119889119905

+

119889119862

119889119905

(3)

4 Computational and Mathematical Methods in Medicine

that is

119889119883

119889119905

+ (] + 1205824minus ]1199012)119883 le ] + 120582

4 (4)

Now integrating both sides of the above inequality and usingthe theory of differential inequality [21 24] we get

119883(119905) le 119890minus(]+120582

4minus]1199012)119905[

] + 1205824

] + 1205824minus ]1199012

119890(]+1205824minus]1199012)119905+ 119896] (5)

where 119896 is a constant and letting 119905 rarr infin we have

119878 + 119864 + 119868 + 119862 le

] + 1205824

] + 1205824minus ]1199012

119878 (119905) = ] minus ]1199011119862 minus 120588 (119868 + 120579119862) 119878 minus ]119878 minus 119906

1119878

+ (1205824minus ]1199012) (1 minus 119878 minus 119864 minus 119868 minus 119862)

(6)

then

119878 le

] + 1205824

] + 1205824+ 1199061minus ]1199012

(7)

then

Π = (119878 119864 119868 119862) isin R4

+| 119878 le

] + 1205824

] + 1199061+ 1205824minus ]1199012

119878 + 119864 + 119868 + 119862 le

] + 1205824

] + 1205824minus ]1199012

(8)

is positively invariant Hence the system is mathematicallywell-posed There for initial starting point 119909 isin R4

+ the

trajectory lies inΠTherefore wewill focus our attention onlyon the region Π

3 Analysis of the System forConstant Controls

In this section we assume that the control parameters 1199061(119905)

and 1199062(119905) are constant functions

31 Equilibrium Points and Basic Reproduction Number Wewill discuss the existence of all the possible equilibriumpointsof the system (2) We see that system (2) has two possiblenonnegative equilibria

(i) Disease-free equilibrium point1198640= (1198780 0 0 0) where

1198780=

] minus ]1199012+ 1205824

1199061+ ] + 120582

4minus ]1199012

(9)

it is always feasible In the absence of vaccination this isreduced to the equilibrium (1 0 0 0)

Definition 1 The basic reproduction number denoted by 1198770

is the expected number of secondary cases produced ina completely susceptible population by a typical infectiveindividual [25]

Using the notation in van den Driessche and Watmough[25] we have

F = [

[

0 12058811987801205881205791198780

0 0 0

0 0 0

]

]

V = [

[

] + 1205821

0 0

minus1205821

] + 1205822

0

0 minus11990131205822] + 1205823+ 1199062minus ]1199011

]

]

(10)

The reproduction number is given by 120588(119865119881minus1) and

1198770

=

1205881205821(] + 120582

3+ 1199062minus 1199011] + 120579119901

31205822) (] minus ]119901

2+ 1205824)

(] + 1205821) (] + 120582

2) (] + 120582

3+ 1199062minus 1199011]) (1199061+ ] + 120582

4minus ]1199012)

=

1205881205821(] + 120582

3+ 1199062minus 1199011] + 120579119901

31205822)

(] + 1205821) (] + 120582

2) (] + 120582

3+ 1199062minus 1199011])1198780

=

1205881205821

(] + 1205821) (] + 120582

2)

[1 +

12057911990131205822

(] + 1205823+ 1199062minus ]1199011)

] 1198780

=

1205881205821

(] + 1205821) (] + 120582

2)

times [1 +

12057911990131205822

(] + 1205823+ 1199062minus ]1199011)

] [

] minus ]1199012+ 1205824

1199061+ ] + 120582

4minus ]1199012

]

(11)

Define

1198771=

12058812058211198780

(] + 1205821) (] + 120582

2)

1198772=

1205881205791199013120582112058221198780

(] + 1205821) (] + 120582

2) (] + 120582

3+ 1199062minus ]1199011)

(12)

then we can see that 1198770= 1198771+ 1198772

Remark 2 We should note from (11) that the use both ofvaccine and treatment control to both reduce the value of 119877

0

and at the same time effects of both intervention strategieson 1198770are not simply the addition of two independent effects

rather they multiply together in order to improve the overalleffects of population level independently (Figure 2) Considerthe following

(ii) endemic equilibria 119864lowastlowast = (119878lowast 119864lowast 119868lowast 119862lowast) which hasfour different cases

Computational and Mathematical Methods in Medicine 5

0

1

2

3

4

5

6

Repr

oduc

tion

num

ber

0 01 02 03 04 05 06 07 08 09 1

Control (u1 u2)

u1 = 0 u2 = 0

u1 = 0 u2 = 1

u1 = 1 u2 = 0

u1 = 1 u2 = 1

Figure 2 Variation of different reproduction numbers by usingof two control variables (vaccination and treatment) Figure 2illustrates the impact of application of two controls (vaccination andtreatment) on different basic reproduction numbers 119877

0= 39810

when there is no vaccination and treatment (1199061= 0 119906

2= 0) The

figure shows that application of vaccination reduces1198770more rapidly

than treatment though mixed intervention strategies always workbetter to reduce the disease burden

(a) No vaccine no treatment control (1199061= 0 119906

2= 0)

119864lowastlowast

1= (119878lowast

1 119864lowast

1 119868lowast

1 119862lowast

1) where

119878lowast

1=

(] + 1205821) (] + 120582

2) (] + 120582

3minus 1199011])

1205881205821(] + 120582

3+ 12057912058221199013minus 1199011])

119864lowast

1=

120579120588 (] + 1205822) 119862lowast

1119878lowast

1

(] + 1205821) (] + 120582

2) minus 120588120582

1119878lowast

1

119868lowast

1=

1205791205881205821119862lowast

1119878lowast

1

(] + 1205821) (] + 120582

2) minus 120588120582

1119878lowast

1

119862lowast

1= (120582112058221199013(] + 120582

4minus ]1199012) 119878lowast

1(1198770minus 1))

times ( (] + 1205823minus 1199011])

times [(] + 1205824minus ]1199012) (] + 120582

2+ 1205821) + 12058211205822]

+120582112058221199013(]1199011minus ]1199012+ 1205824) )minus1

(13)

(b) No vaccine with treatment control (1199061= 0 119906

2= 0)

119864lowastlowast

2= (119878lowast

2 119864lowast

2 119868lowast

2 119862lowast

2) where

119878lowast

2=

(] + 1205821) (] + 120582

2) (] + 120582

3minus 1199011] + 1199062)

1205881205821(] + 120582

3+ 12057912058221199013minus 1199011] + 1199062)

119864lowast

2=

120579120588 (] + 1205822) 119862lowast

2119878lowast

2

(] + 1205821) (] + 120582

2) minus 120588120582

1119878lowast

2

119868lowast

2=

1205791205881205821119862lowast

2119878lowast

2

(] + 1205821) (] + 120582

2) minus 120588120582

1119878lowast

2

119862lowast

2= (120582112058221199013(] + 120582

4minus ]1199012) 119878lowast

2(1198770minus 1))

times ( (] + 1205823minus 1199011] + 1199062)

times [(] + 1205824minus ]1199012) (] + 120582

2+ 1205821) + 12058211205822]

+120582112058221199013(]1199011minus ]1199012+ 1205824) )minus1

(14)

(c) With vaccine no treatment control (1199061= 0 1199062= 0)

119864lowastlowast

3= (119878lowast

3 119864lowast

3 119868lowast

3 119862lowast

3) where

119878lowast

3=

(] + 1205821) (] + 120582

2) (] + 120582

3minus 1199011])

1205881205821(] + 120582

3+ 12057912058221199013minus 1199011])

119864lowast

3=

120579120588 (] + 1205822) 119862lowast

3119878lowast

3

(] + 1205821) (] + 120582

2) minus 120588120582

1119878lowast

3

119868lowast

3=

1205791205881205821119862lowast

3119878lowast

3

(] + 1205821) (] + 120582

2) minus 120588120582

1119878lowast

3

119862lowast

3= (120582112058221199013(] + 120582

4minus ]1199012+ 1199061) 119878lowast

3(1198770minus 1))

times ( (] + 1205823minus 1199011])

times [(] + 1205824minus ]1199012) (] + 120582

2+ 1205821) + 12058211205822]

+120582112058221199013(]1199011minus ]1199012+ 1205824) )minus1

(15)

(d) With vaccine with treatment control (1199061= 0 1199062= 0)

119864lowastlowast

4= (119878lowast

4 119864lowast

4 119868lowast

4 119862lowast

4) where

119878lowast

4=

(] + 1205821) (] + 120582

2) (] + 120582

3minus 1199011] + 1199062)

1205881205821(] + 120582

3+ 12057912058221199013minus 1199011] + 1199062)

119864lowast

4=

120579120573 (] + 1205822) 119862lowast

4119878lowast

4

(] + 1205821) (] + 120582

2) minus 120588120582

1119878lowast

4

119868lowast

4=

1205791205881205821119862lowast

4119878lowast

4

(] + 1205821) (] + 120582

2) minus 120588120582

1119878lowast

4

119862lowast

4= (120582112058221199013(] + 120582

4minus ]1199012+ 1199061) 119878lowast

4(1198770minus 1))

times ( (] + 1205823minus 1199011] + 1199062)

times [(] + 1205824minus ]1199012) (] + 120582

2+ 1205821) + 12058211205822]

+120582112058221199013(]1199011minus ]1199012+ 1205824) )minus1

(16)

Clearly 119864lowastlowast is feasible if 119862lowast gt 0 that is if 1198770gt 1 Also for

1198770= 1 the endemic equilibrium reduces to the disease-free

equilibrium and for 1198770lt 1 it becomes infeasible Hence we

may state the following theorem

Theorem 3 (i) If 1198770lt 1 then the system (2) has only one

equilibrium which is disease-free(ii) If 119877

0gt 1 then the system (2) has two equilibria one is

disease-free and the other is endemic equilibrium(iii) If 119877

0= 1 then the endemic equilibrium reduces to the

disease-free equilibrium

6 Computational and Mathematical Methods in Medicine

32 Stability Analysis In this section we will discuss thestability of different equilibria Firstly we analyse the localstability of the disease-free equilibrium

Theorem 4 If 1198770lt 1 then the disease-free equilibrium is

locally asymptotically stable

Proof The Jacobian matrix of system (2) at the disease-freeequilibrium is

J0 =[[[[

[

minus (] + 1205824minus ]1199012+ 1199061) minus (120582

4minus ]1199012) minus (120588119878

0+ 1205824minus ]1199012) minus (]119901

1+ 1205824minus ]1199012+ 120579120588119878

0)

0 minus (] + 1205821) 120588119878

01205791205881198780

0 1205821

minus (] + 1205822) 0

0 0 11990131205822

]1199011minus ] minus 120582

3minus 1199062

]]]]

]

(17)

The characteristic polynomial of 1198690given by

119875 (120582) = (120582 + 1198970) (1205823+ 11989711205822+ 1198972120582 + 1198973) (18)

where

1198970= ] + 119906

1+ 1205824minus ]1199012

1198971= 3] + 120582

1+ 1205822+ 1205823+ 1199062minus ]1199011

1198972= (] + 120582

2) (] + 120582

3+ 1199062minus ]1199011)

+ (] + 1205821) (2] + 120582

2+ 1205823+ 1199062minus ]1199011) minus 12058211205881198780

1198973= (] + 120582

1) (] + 120582

2) (] + 120582

3+ 1199062minus ]1199011)

minus 12058211205881198780(] + 120582

3+ 1199062+ 12057911990131205822minus ]1199011)

(19)

We need to verify the following two conditions

(a) 1198970 1198971 1198972 1198973gt 0

(b) 11989711198972minus 1198973gt 0

It is easy to see that 1198970 1198971gt 0 and 119897

2 1198973 11989711198972minus 1198973gt 0 if

1198770lt 1 It follows from the Routh-Hurwitz criterion that the

eigenvalues have negative real parts if 1198770lt 1 Hence the

disease-free equilibriumofmodel (2) is locally asymptoticallystable if 119877

0lt 1 and unstable if 119877

0gt 1

To discuss the properties of the endemic equilibriumpoint

Theorem 5 The endemic equilibrium point is locally asymp-totically stable if 119877

0gt 1

Proof We use Routh-Hurwitz criterion to establish the localstability of the endemic equilibrium The Jacobian matrix ofsystem (2) at endemic equilibrium is

Jlowast =[[[

[

minus (] + 1205824minus ]1199012+ 1199061+ 120588119868lowast) minus (120582

4minus ]1199012) minus (120588119878

lowast+ 1205824minus ]1199012) minus (]119901

1+ 1205824minus ]1199012+ 120588120579119878

lowast)

120588119868lowast+ 120588120579119862

lowastminus (] + 120582

1) 120588119878

lowast120588120579119878lowast

0 1205821

minus (] + 1205822) 0

0 0 11990131205822

]1199011minus ] minus 120582

3minus 1199062

]]]

]

(20)

Then the characteristic equation at 119864lowastlowast is 1205824 + 11989111205823+ 11989121205822+

1198913120582 + 1198914= 0 where

1198911= 1198861+ 1198872+ 1198883+ 1198894

1198912= 11988611198863+ 11988611198894+ 11988611198872+ 11988831198894+ 11988721198883+ 11988721198894minus 11988731198882minus 11988711198862

1198913= 119886111988831198894+ 119886111988721198883+ 119886111988721198894+ 119887211988831198894+ 119887411988821198893+ 119887111988821198863

minus 119886111988731198882minus 119887311988821198894minus 119887111988621198883minus 119887111988621198894

1198914= 1198861119887211988831198894+ 1198861119887411988821198893+ 1198871119888211988631198894minus 1198861119887311988821198894

minus 1198871119886211988831198894minus 1198871119886411988821198893

(21)

where

1198861= minus (] + 120582

4minus ]1199012+ 1199061+ 120588119868lowast)

1198862= minus (120582

4minus ]1199012) 119886

3= minus (120588119878

lowast+ 1205824minus ]1199012)

1198864= minus (]119901

1+ 1205824minus ]1199012+ 120588120579119878

lowast) 119887

1= 120588119868lowast+ 120588120579119862

lowast

1198872= minus (] + 120582

1) 119887

3= 120588119878lowast

1198874= 120588120579119878

lowast 119888

1= 0

1198882= 1205821 119888

3= minus (] + 120582

2)

1198884= 0 119889

1= 0

1198892= 0 119889

3= 11990131205822

1198894= ]1199011minus ] minus 120582

3minus 1199062

(22)

We need to verify the following three conditions

(a) 1198911 1198912 1198913 1198914gt 0

(b) 11989111198912minus 1198913gt 0

(c) 1198913(11989111198912minus 1198913) minus 1198912

11198914gt 0

Computational and Mathematical Methods in Medicine 7

It is easy to see that conditions (a) and (b) are satisfiedAfter computations we can prove that 119891

3(11989111198912minus 1198913) minus

1198912

11198914gt 0 is also validTheRouth-Hurwitz criterion and those

inequalities in (a)ndash(c) imply that the characteristic equationat 119864lowastlowast has only roots with negative real part which certifiesthe local stability of 119864lowastlowast

4 Optimal Control with Two Objectives

One of the early reasons for studying hepatitis B virus (HBV)infection is to improve the control variables and finally to putdown the infection of the population Optimal control is auseful mathematical tool that can be used to make decisionsin this case In the previous sections we have analyzed themodel with two control variables one is treatment and theother is vaccination and we consider their constant controlsthroughout the analysis But in fact these control variablesshould be time dependent In this section we consider thevaccination and treatment as time-dependent controls in acompact interval of time duration

Our goals here are to put down infection from the popu-lation by increasing the recovered individuals and reducingsusceptible exposed infected and carrier individuals in apopulation and to minimize the costs required to control thehepatitis B virus (HBV) infection by using vaccination andtreatment First of all we construct the objective functionalto be optimized as follows

119869 = int

119879

0

[1198601119878 (119905) + 119860

2119864 (119905) + 119860

3119868 (119905) + 119860

4119862 (119905)

+

1

2

(11986111199062

1(119905) + 119861

21199062

2(119905))] 119889119905

(23)

subject to

119878 (119905) = ] minus ]1199011119862 minus 120588 (119868 + 120579119862) 119878 minus ]119878 minus 119906

1119878

+ (1205824minus ]1199012) (1 minus 119878 minus 119864 minus 119868 minus 119862)

(119905) = 120588 (119868 + 120579119862) 119878 minus (] + 1205821) 119864

119868 (119905) = 1205821119864 minus (] + 120582

2) 119868

(119905) = ]1199011119862 + 119901

31205822119868 minus (] + 120582

3) 119862 minus 119906

2119862

(24)

Our object is to find (119906lowast1 119906lowast

2) such that

119869 (119906lowast

1 119906lowast

2) = min 119869 (119906

1 1199062) (25)

where

1199061(119905) 119906

2(119905) isin Γ

= 119906 (119905) | 0 le 119906 (119905) le 119906max (119905) le 1 0 le 119905 le 119879

119906 (119905) is Labesgue measurable

(26)

Here 119860119894 for 119894 = 1 4 are positive constants that

are represented to keep a balance in the size of 119878(119905) 119864(119905)119868(119905) and 119862(119905) respectively 119861

1and 119861

2 respectively are the

weights corresponding to the controls 1199061and 119906

2 119906max is the

maximum attainable value for controls (1199061max and 119906

2max)1199061max and 119906

2max will depend on the amount of resourcesavailable to implement each of the control measures [26]The 119861

1and 119861

2will depend on the relative importance of

each of the control measures in mitigating the spread of thedisease as well as the cost (human effort material resourcesetc) of implementing each of the control measures per unittime [26] Thus the terms 119861

11199062

1and 119861

21199062

2describe the costs

associated with vaccination and treatment respectively Thesquare of the control variables is taken here to remove theseverity of the side effects and overdoses of vaccination andtreatment [27]

41 Optimal Control Solution For existence of the solutionwe consider the control system (24) with initial condition

119878 (0) = 1198780 119864 (0) = 119864

0

119868 (0) = 1198680 119862 (0) = 119862

0

(27)

then we rewrite (2) in the following form

119889119909 (119905)

119889119905

= 119860119909 + 119865 (119909) (28)

where 119909(119905) = (119878(119905) 119864(119905) 119868(119905) 119862(119905)) is the vector of the statevariables and A and F(x) are defined as follows

A =

[[[[[[[

[

minus] minus 1199061minus 1205824+ ]1199012minus1205824+ ]1199012minus1205824+ ]1199012

minus1205824+ ]1199012

0 minus] minus 1205821

0 0

0 1205821

minus] minus 1205822

0

0 0 11990131205822

]1199011minus ] minus 120582

3minus 1199062

]]]]]]]

]

F (x) =[[[[[[

[

] + 120588 (119868 + 120579119862) 119878 + 1205824minus ]1199012

120588 (119868 + 120579119862) 119878

0

0

]]]]]]

]

(29)

8 Computational and Mathematical Methods in Medicine

We set

119866 (119909) = 119860119909 + 119865 (119909) (30)

The second term on the right-hand side of (30) satisfies

1003816100381610038161003816119865 (1199091) minus 119865 (119909

2)1003816100381610038161003816

le 119872 (10038161003816100381610038161198781minus 1198782

1003816100381610038161003816+10038161003816100381610038161198641minus 1198642

1003816100381610038161003816+10038161003816100381610038161198681minus 1198682

1003816100381610038161003816+10038161003816100381610038161198621minus 1198622

1003816100381610038161003816)

(31)

where the positive constant 119872 is independent of the statevariables 119909 and119872 le 1 Also we get

1003816100381610038161003816119866 (1199091) minus 119866 (119909

2)1003816100381610038161003816le 119871

10038161003816100381610038161199091minus 1199092

1003816100381610038161003816 (32)

where 119871 = max119872 119860 lt infin Therefore it follows thatthe function 119866 is uniformly Lipschitz continuous From thedefinition of the controls 119906

1(119905) and 119906

2(119905) and the restrictions

on the nonnegativeness of the state variables we see that asolution of the system (28) exists [24 28 29]

42The Lagrangian and Hamiltonian for the Control ProblemIn order to find an optimal solution pair first we shouldfind the Lagrangian and Hamiltonian for the optimal controlproblem (24) In fact the Lagrangian of problem is given by

119871 (119878 119864 119868 119862 1199061 1199062) = 119860

1119878 (119905) + 119860

2119864 (119905) + 119860

3119868 (119905)

+ 1198604119862 (119905) +

1

2

(11986111199062

1(119905) + 119861

21199062

2(119905))

(33)

We are looking for the minimal value of the LagrangianTo accomplish this we define Hamiltonian119867 for the controlproblem as follows

119867(119878 119864 119868 119862 1199061 1199062 1205721 1205722 1205723 1205724)

= 119871 + 1205721(119905)

119889119878

119889119905

+ 1205722(119905)

119889119864

119889119905

+ 1205723(119905)

119889119868

119889119905

+ 1205724(119905)

119889119862

119889119905

(34)

where 120572119894(119905) for 119894 = 1 2 3 4 are the adjoint variables and can

be determined by solving the following system of differentialequations

1(119905) = minus

120597119867

120597119878

= minus [1198601+ 1205721((]1199012minus 1205824) minus 120588 (119868 + 120579119862) minus 119906

1minus ])

+1205722120588 (119868 + 120579119862)]

2(119905) = minus

120597119867

120597119864

= minus [1198602+ 1205721(]1199012minus 1205824) + 1205722(minus] minus 120582

1) + 12057231205821]

3(119905) = minus

120597119867

120597119868

= minus [1198603+ 1205721((]1199012minus 1205824) minus 120588119878) + 120572

2120588119878

+1205723(minus] minus 120582

2) + 120572411990131205822]

4(119905) = minus

120597119867

120597119862

= minus [1198604+ 1205721(]1199012minus 1205824minus ]1199011minus 120588120579119878)

+1205722120588120579119878 + 120572

4(]1199011minus ] + 120582

3+ 1199062)]

(35)

satisfying the transversality conditions and optimality condi-tions

1205721(119879) = 120572

2(119879) = 120572

3(119879) = 120572

4(119879) = 0

120597119867

1205971199061

= 11986111199061minus 1205721119878 = 0

120597119867

1205971199062

= 11986121199062minus 1205724119862 = 0

(36)

We now state and prove the following theorem

Theorem 6 There is an optimal control (119906lowast1 119906lowast

2) such that

119869 (119878 119864 119868 119878 119906lowast

1 119906lowast

2) = min 119869 (119878 119864 119868 119878 119906

1 1199062) (37)

subject to the system of differential equation (24)

Proof Here both the control and state variables are non-negative values In this minimized problem the necessaryconvexity of the objective functional in 119906

1and 1199062are satisfied

The set of all control variables 1199061(119905) 1199062(119905) is also convex and

closed by definition The optimal system is bounded whichdetermines the compactness needed for the existence of theoptimal control In addition the integrand in the functional

int

119879

0

[1198601119878 (119905) + 119860

2119864 (119905) + 119860

3119868 (119905) + 119860

4119862 (119905)

+

1

2

(11986111199062

1(119905) + 119861

21199062

2(119905))] 119889119905

(38)

is convex on the control set Hence the theorem (the proof iscomplete)

Computational and Mathematical Methods in Medicine 9

43 Necessary and Sufficient Conditions for Optimal ControlsApplying Pontryaginrsquos maximum principle [30] on the con-structed Hamiltonian 119867 and the theorem (24) we obtainthe optimal steady-state solution and corresponding controlas follows If taking 119883 = (119878 119864 119868 119862) and 119880 = (119906

1 1199062)

then (119883lowast 119880lowast) is an optimal solution of an optimal controlproblem then we now state the following theorem

Theorem 7 The optimal control pair (119906lowast1 119906lowast

2) which mini-

mizes J over the region Γ is given by

119906lowast

1= max[0119879]

0min (1(119905) 1)

119906lowast

2= max[0119879]

0min (2(119905) 1)

(39)

where 1= 11198781198611and 2= 41198621198612and let

1 2 3 4 be

the solution of system (35)

Proof With the help of the optimality conditions we have

120597119867

1205971199061

= 11986111199061minus 1205721119878 = 0 997904rArr 119906

1=

1119878

1198611

= (1)

120597119867

1205971199062

= 11986121199062minus 1205724119862 = 0 997904rArr 119906

2=

4119862

1198612

= (2)

(40)

Using the property of the control space Γ the two controlswhich are bounded with upper and lower bounds arerespectively 1 and 0 that is

119906lowast

1=

0 if 1le 0

1

if 0 lt 1lt 1

1 if 1ge 1

(41)

This can be rewritten in compact notation

119906lowast

1= max[0119879]

0min (1(119905) 1) (42)

and similarly

119906lowast

2=

0 if 2le 0

2

if 0 lt 2lt 1

1 if 2ge 1

(43)

This can be rewritten in compact notation

119906lowast

2= max[0119879]

0min (2(119905) 1) (44)

Hence for these pair of controls (119906lowast1 119906lowast

2) we get the optimum

value of the functional 119869 given by (24)

5 Numerical Examples

Numerical solutions to the optimality system (24) arediscussed in this section We make several interestingobservation by numerically simulating (2) in the rangeof parameter values We consider the parameter set

Δ = ] 120588 120579 1205821 1205822 1205823 1205824 1199011 1199012 1199013 1198601 1198602 1198603 1198604 1198611 1198612

some of the parameters are taken from the published articlesand some are assumed with feasible values Moreover thetime interval for which the optimal control is applied istaken as 70 years also consider Ψ = 119878

0 1198640 1198680 1198620 as initial

condition for simulation of the model The main parametervalues are listed in Table 1 We compare the results having nocontrols only vaccination control only treatment controland both vaccination and treatment controls

With the parameter values in Table 1 the system asymp-totically approaches towards the equilibrium 119864

lowastlowast

1(00858

03327 04003 05349) where the basic reproduction ratio1198770= 39810 (Figure 2) For the parameters set the system

(2) has two feasible equilibria one is disease free and theother is endemic and the endemic equilibrium is locallyasymptotically stable The effect of two control measures ondisease dynamics may be understood well if we considerFigure 2 It explains how control reproduction ratio 119877

0

evolves with different rates of 1199061and 119906

2 It is seen that both

vaccination and treatment reduce the value of 1198770effectively

But an integrated control works better than either of thecontrol measures

In this section we use an iterative method to obtainresults for an optimal control problem of the proposedmodel(24) We use Runge-Kuttarsquos fourth-order procedure [23] hereto solve the optimality system consisting of eight ordinarydifferential equations having four state equations and fouradjoint equations and boundary conditions Because stateequations have initial conditions and adjoint equations haveconditions at the final time an iterative program was createdto numerically simulate solutions Given an initial guessfor the controls to compute the optimal state values theprogram solves (24) with initial conditions (27) forward intime interval [0 70]using aRungeKuttamethod of the fourthorder Resulting state values are placed in adjoint equations(35) These adjoint equations with given final conditions arethen solved backwards in time Again a fourth order RungeKutta method is employed Both state and adjoint values areused to update the control using the characterization (39)and the entire process repeats itself This iterative processterminates when current state adjoint and control valuesare sufficiently close to successive values Then we use thebackward Runge-Kutta fourth-order procedure to solve theadjoint variables in the same time interval with the help ofthe solution of the state variables transversality conditions

We have plotted susceptible exposed acute infectionindividuals and carriers individuals with andwithout controlby considering values of parameters We simulate the systemat different values of rate of 119906

1and 119906

2

From Figures 3 4 and 5 we see that after 20 years thenumber of susceptible population decreases than when thereis no control In this case most of this population tends to theinfected class Again when only treatment control is appliedthen the number of susceptible population is not muchdifferent than the population in the case having no controlBut the susceptible population differs much from these twostrategies if we apply the strategies of only vaccination controland both vaccination and treatment controls At a high rateof vaccination the sensitive population density is reduced

10 Computational and Mathematical Methods in Medicine

Table 1 Parameter values used in numerical simulations

Parameter Description Values range Reference] Birth (and death) rate 00121 [17 18]120588 Transmission rate 08ndash2049 [17]120579 Infectiousness of carriers relative to acute infections 0-1 [17]1205821

Rate of moving from exposed to acute 6 per year [16]1205822

Rate at which individuals leave the acute infection class 4 per year [16]1205823

Rate of moving from carrier to recovery 0025 per year [16]1205824

Loss of recovery rate 003ndash006 [17]1198751

Probability of infected newborns 011 [19]1198752

Probability of immune newborns 01 [19]1198753

Proportion of acute infection individuals becoming carriers 005ndash09 [17]1198601

Weight factor for susceptible individuals 0091 [31]1198602

Weight factor for exposed individuals 001 [31]1198603

Weight factor for infected individuals 004 [31]1198604

Weight factor for carrier individuals 005 [31]1198611

Weight factor for the controls 1199061

15 [31]1198612

Weight factor for the controls 1199062

27 [31]1198780

Susceptible individuals 0493 [4]1198640

Exposed individuals 00035 [4]1198680

Acute infection individuals 00035 [4]1198620

Chronic HBV carriers 0007 [4]

0 10 20 30 40 50 60 70Time (year)

Susc

eptib

le p

opul

atio

n

NV-NTNV-T

V-NTV-T

10minus4

10minus3

10minus2

10minus1

100

Figure 3 The plot shows the changes in sensitive populations(without) vaccination and (without) treatment

to a very low level initially and then it takes longer time torestore the steady-state value

From Figures 6 7 and 8 we see that the number ofexposed population increase thanwhen there is no control Inthis case most of this population tends to the acute infectedclass Again when only treatment control is applied thenthe number of exposed population is not much different

0 10 20 30 40 50 60 70Time (year)

Susc

eptib

le p

opul

atio

n

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0365 u2 = 0

u1 = 045 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

Figure 4 The plot shows the sensitivity of sensitive populations fordifferent values of control 119906

1

than the population in the case having no control But theexposed population differs much from these two strategies ifwe apply the strategies of only vaccination control and bothvaccination and treatment controls

From Figures 9 10 and 11 we see that the number ofacute infected population increases than when there is nocontrol In this case most of this population tends to thecarrier class Again when only treatment control is appliedthen the number of acute infected population is not muchdifferent than the population in the case having no controlBut the acute infected population differsmuch from these two

Computational and Mathematical Methods in Medicine 11

0 10 20 30 40 50 60 70Time (year)

Susc

eptib

le p

opul

atio

n

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Figure 5 The plot shows the sensitivity of sensitive populations fordifferent values of control 119906

2

Expo

sed

popu

latio

n

NV-NTNV-T

V-NTV-T

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 6 The plot shows the changes in exposed populations(without) vaccination and (without) treatment

strategies if we apply the strategies of only vaccination controland both vaccination and treatment controls

Again from Figures 12 13 and 14 we see that thenumber of carrier population increases than when there isno control We see that the application of only vaccinationcontrol gives better result than the application of no controlAgain application of treatment control would give betterresult than the application of vaccination control since thetreatment control is better than the vaccination control whilethe application of both vaccination and treatment controlsgive the best result as in this case the number of carrierpopulation would be the least in number

Expo

sed

popu

latio

n

u1 = 0 u2 = 0

u1 = 03635 u2 = 0

u1 = 045 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 7 The plot shows the sensitivity of exposed populations fordifferent values of control 119906

1

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Expo

sed

popu

latio

n

Figure 8 The plot shows the sensitivity of exposed populations fordifferent values of control 119906

2

Finally the effect of two control measures on diseasedynamics may be understood well if we consider Figures 1ndash14 15 16 17 and 18 Since our main purpose is to reducethe number of sensitive exposed acute infection and carrierindividuals therefore numerical simulation results show thatoptimal combination of vaccination and treatment is themosteffective way to control hepatitis B virus infection

12 Computational and Mathematical Methods in MedicineAc

ute i

nfec

tion

popu

latio

n

NV-NTNV-T

V-NTV-T

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 9The plot shows the changes in acute infection populations(without) vaccination and (without) treatment

Acut

e inf

ectio

n po

pulat

ion

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0365 u2 = 0

u1 = 045 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

Figure 10 The plot shows the sensitivity of acute infection popula-tions for different values of control 119906

1

6 Conclusion and Suggestions

In this paper we propose an S-119864-119868-119862-119877model of hepatitis Bvirus infection with two controls vaccination and treatmentFirst we analyze the dynamic behavior of the system for con-stant controls In the constant controls case we calculate thebasic reproduction number and investigate the existence andstability of equilibriaThere are two nonnegative equilibria of

Acut

e inf

ectio

n po

pulat

ion

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Figure 11 The plot shows the sensitivity of acute infection popula-tions for different values of control 119906

2

Carr

ier p

opul

atio

n

NV-NTNV-T

V-NTV-T

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 12 The plot shows the changes in carrier populations(without) vaccination and (without) treatment

the system namely the disease-free and endemicWe see thatthe disease-free equilibriumwhich always exists and is locallyasymptotically stable if 119877

0lt 1 and endemic equilibrium

which exists and is locally asymptotically stable if 1198770gt 1

After investigating the dynamic behavior of the systemwith constant controls we formulate an optimal controlproblem if the controls become time dependent and solve itby using Pontryaginrsquos maximum principle Different possiblecombinations of controls are used and their effectiveness iscompared by simulation works Also from the numericalresults it is very clear that a combination of mixed controlmeasures respond better than any other independent control

Computational and Mathematical Methods in Medicine 13Ca

rrie

r pop

ulat

ion

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0365 u2 = 0

u1 = 046 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

Figure 13 The plot shows the sensitivity of carrier populations fordifferent values of control 119906

1

Carr

ier p

opul

atio

n

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Figure 14 The plot shows the sensitivity of carrier populations fordifferent values of control 119906

2

There is still a tremendous amount of work needed tobe done in this area Parameters are rarely constant becausethey depend on environmental conditions We do not knowhowever the detailed relationship between these parametersand environmental conditions There may be a time lag as asusceptible populationmay take some time to be infected andalso a susceptible population may take some time to immune

Popu

latio

n (N

V-N

T)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 15 The plot shows the changes in total populations withoutvaccination and without treatment

Popu

latio

n (V

-NT)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 16 The plot shows the changes in total populations withvaccination and without treatment

Popu

latio

n (N

V-T)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 17 The plot shows the changes in total populations withoutvaccination and with treatment

14 Computational and Mathematical Methods in MedicinePo

pulat

ion

(V-T

)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 18 The plot shows the changes in total populations withvaccination and treatment

after vaccination We leave all these possible extensions forthe future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] WHO Hepatitis B Fact Sheet No 204 The World HealthOrganisation Geneva Switzerland 2013 httpwwwwhointmediacentrefactsheetsfs204en

[2] CanadianCentre forOccupationalHealth and Safety ldquoHepatitisBrdquo httpwwwccohscaoshanswersdiseaseshepatitis bhtml

[3] ldquoHealthcare stumbling in RIrsquos Hepatitis fightrdquoThe Jakarta PostJanuary 2011

[4] G F Medley N A Lindop W J Edmunds and D JNokes ldquoHepatitis-B virus endemicity heterogeneity catas-trophic dynamics and controlrdquo Nature Medicine vol 7 no 5pp 619ndash624 2001

[5] J Mann and M Roberts ldquoModelling the epidemiology ofhepatitis B in New Zealandrdquo Journal of Theoretical Biology vol269 no 1 pp 266ndash272 2011

[6] ldquoHepatitis B (HBV)rdquo httpkidshealthorgteensexual healthstdsstd hepatitishtml

[7] CDC ldquoHepatitis B virus a comprehensive strategy for elim-inating transmission in the United States through universalchildhood vaccination Recommendations of the Immuniza-tion Practices Advisory Committee (ACIP)rdquo MMWR Recom-mendations and Reports vol 40(RR-13) pp 1ndash25 1991

[8] M K Libbus and L M Phillips ldquoPublic health management ofperinatal hepatitis B virusrdquo Public Health Nursing vol 26 no 4pp 353ndash361 2009

[9] F B Hollinger and D T Lau ldquoHepatitis B the pathway torecovery through treatmentrdquo Gastroenterology Clinics of NorthAmerica vol 35 no 4 pp 895ndash931 2006

[10] C-L Lai andM-F Yuen ldquoThe natural history and treatment ofchronic hepatitis B a critical evaluation of standard treatment

criteria and end pointsrdquoAnnals of InternalMedicine vol 147 no1 pp 58ndash61 2007

[11] R M Anderson and R M May Infectious Disease of HumansDynamics and Control Oxford University Press Oxford UK1991

[12] S Thornley C Bullen and M Roberts ldquoHepatitis B in ahigh prevalence New Zealand population a mathematicalmodel applied to infection control policyrdquo Journal ofTheoreticalBiology vol 254 no 3 pp 599ndash603 2008

[13] S-J Zhao Z-Y Xu and Y Lu ldquoA mathematical model ofhepatitis B virus transmission and its application for vaccinationstrategy inChinardquo International Journal of Epidemiology vol 29no 4 pp 744ndash752 2000

[14] K Wang W Wang and S Song ldquoDynamics of an HBV modelwith diffusion and delayrdquo Journal ofTheoretical Biology vol 253no 1 pp 36ndash44 2008

[15] R Xu andZMa ldquoAnHBVmodelwith diffusion and time delayrdquoJournal of Theoretical Biology vol 257 no 3 pp 499ndash509 2009

[16] L Zou W Zhang and S Ruan ldquoModeling the transmissiondynamics and control of hepatitis B virus in Chinardquo Journal ofTheoretical Biology vol 262 no 2 pp 330ndash338 2010

[17] J Pang J-A Cui and X Zhou ldquoDynamical behavior of ahepatitis B virus transmission model with vaccinationrdquo Journalof Theoretical Biology vol 265 no 4 pp 572ndash578 2010

[18] S Zhang and Y Zhou ldquoThe analysis and application of an HBVmodelrdquoAppliedMathematicalModelling vol 36 no 3 pp 1302ndash1312 2012

[19] S Bhattacharyya and S Ghosh ldquoOptimal control of verticallytransmitted diseaserdquoComputational andMathematicalMethodsin Medicine vol 11 no 4 pp 369ndash387 2010

[20] T K Kar andA Batabyal ldquoStability analysis and optimal controlof an SIR epidemic model with vaccinationrdquo Biosystems vol104 no 2-3 pp 127ndash135 2011

[21] T-K Kar and S Jana ldquoA theoretical study on mathematicalmodelling of an infectious disease with application of optimalcontrolrdquo Biosystems vol 111 no 1 pp 37ndash50 2013

[22] M J Keeling and P Rohani Modeling Infectious Diseases inHumans and Animals Princeton University Press PrincetonNJ USA 2008

[23] E Jung S Lenhart and Z Feng ldquoOptimal control of treatmentsin a two-strain tuberculosis modelrdquo Discrete and ContinuousDynamical Systems B vol 2 no 4 pp 473ndash482 2002

[24] G Birkhoff and G-C Rota Ordinary Differential EquationsJohn Wiley amp Sons New York NY USA 4th edition 1989

[25] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[26] T T Yusuf and F Benyah ldquoOptimal control of vaccination andtreatment for an SIR epidemiological modelrdquo World Journal ofModelling and Simulation vol 8 no 3 pp 194ndash204 2008

[27] H-R Joshi S Lenhart M Y Li and L Wang ldquoOptimalcontrol methods applied to disease modelsrdquo in MathematicalStudies on Human Disease Dynamics vol 410 of ContemporaryMathematics pp 187ndash207 American Mathematical SocietyProvidence RI USA 2006

[28] G Zaman Y H Kang and I H Jung ldquoStability analysis andoptimal vaccination of an SIR epidemic modelrdquo Biosystems vol93 no 3 pp 240ndash249 2008

[29] D Iacoviello and N Stasiob ldquoOptimal control for SIRCepidemic outbreakrdquo Computer Methods and Programs inBiomedicine vol 110 no 3 pp 333ndash342 2013

Computational and Mathematical Methods in Medicine 15

[30] L S Pontryagin V G Boltyanskii R V Gamkrelidze and EF Mishchenko The Mathematical Theory of Optimal ProcessesJohn Wiley amp Sons New York NY USA 1962

[31] WHO Hepatitis B Fact Sheet No 204 The World HealthOrganisation Geneva Switzerland 2000 httpwwwwhointmediacentrefactsheetsfs204en

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

2 Computational and Mathematical Methods in Medicine

In adults

lt5 of otherwise healthy adults who areinfected will develop chronic infection15ndash25 of adults who become chronicallyinfected during childhood die from hepatitis Brelated liver cancer or cirrhosis [1 5]

Patients with chronic carrier often have no history ofacute illness but may develop cirrhosis (liver scarring) thatcan lead to liver failure and may also develop liver cancer[5] A small portion (1ndash6) of chronic carries clear thevirus naturally [5] Someone who is infected with hepatitisB may have symptoms similar to those caused by other viralinfections However many people infected with hepatitis Bdo not have any symptoms until when more serious sideeffects such as liver damage can occur Someone who hasbeen exposed to hepatitis B may have signs 2 to 5 monthslater Some people with hepatitis B do not notice warningsigns until they become very intense Some have few orno symptoms but even someone who does not notice anysymptoms can still transmit the disease to others and canstill develop complications later in life Some people carry thevirus in their bodies and are contagious for the rest of theirlives [6]

The risk of transmission from mother to newborn canbe reduced from 20ndash90 to 5ndash10 by administering to thenewborn hepatitis B vaccine (HBV) and hepatitis B immuneglobulin (HBIG) within 12 hours of birth followed by asecond dose of hepatitis B vaccine (HBV) at 1-2 months and athird dose at and not earlier than 6 months (24 weeks) [7 8]The hepatitis B infection does not usually require treatmentbecause most adults clear the infection spontaneously [9]Early antiviral treatment may be required in less than 1of people whose infection takes a very aggressive course(fulminant hepatitis) or who are immunocompromised Onthe other hand treatment of chronic infection may benecessary to reduce the risk of cirrhosis and liver cancerTreatment lasts from six months to a year depending onmedication and genotype [10]

One of the primary reasons for studying hepatitis Bvirus (HBV) infection is to improve control and finally toput down the infection from the population Mathematicalmodels can be a useful tool in this approach which helps usto optimize the use of finite sources or simply to goal (theincidence of infection) control measures more impressivelyAnderson and May [11] used a simple mathematical modelto illustrate the effects of carriers on the transmission ofHBV A hepatitis B mathematical model (Medley et al[4]) was used to develop a strategy for eliminating HBVin New Zealand [5 12] Zhao et al [13] proposed an agestructuremodel to predict the dynamics ofHBV transmissionand evaluate the long-term effectiveness of the vaccinationprogramme in ChinaWang et al [14] proposed and analyzedthe hepatitis B virus infection in a diffusion model confinedto a finite domain Xu and Ma [15] proposed a hepatitisB virus (HBV) model with spatial diffusion and saturationresponse of the infection rate was investigated Zou et al[16] also proposed a mathematical model to understand the

transmission dynamics and prevalence of HBV in mainlandChina Pang et al [17] developed a model to explore theimpact of vaccination and other controlling measures ofHBV infection Zhang and Zhou [18] proposed analysis andapplication of an HBV model Bhattacharyya and Ghosh[19] Kar and Batabyal [20] and Kar and Jana [21] proposedoptimal control of infectious diseases

In this paper we study the dynamics of hepatitis B virus(HBV) infection under administration of vaccination andtreatment where HBV infection is transmitted in two waysthrough vertical transmission and horizontal transmissionWhile the horizontal transmission is reduced through theadministration of vaccination to those susceptible individ-uals the vertical transmission gets reduced through theadministration of treatment to infected individuals thereforethe vaccine and the treatment play different roles in control-ling the HBV [19] In this study we analyze and apply optimalcontrol to determine the possible impacts of vaccination tosusceptible individuals and treatment to infected individualsSome numerical simulations of the model are also givento illustrate the results and to find optimal strategies incontrolling HBV infection

The paper is organized as follows In Section 2 weproposed an HBV infection model with vaccination andtreatment In Section 3 we analyzed the qualitative propertyof themodel In Section 4 we considered the optimal analysisof the model and in Section 5 we considered some numericalexperiments under special choice of parameter values Thepaper will be finished with a brief discussion and conclusion

2 Model and Preliminaries

To analyze and control hepatitis B virus (HBV) infection inthe present paper we consider a model with two controlsvaccination and treatment The model is constructed basedon the characteristics of HBV transmission and the modelof Medley et al [4] According to the natural history ofHBV this study has a unique characteristic that distinguishesit from Medleyrsquos model as follows Firstly in this studytwo controlling variables are considered (vaccination andtreatment) in order to prevent the spread of the HBVand finally to put down the infection from the populationwhereas in Medleyrsquos model only one controlling variablethe vaccination was employed Secondly in our model wehave considered two different ways of the transmission ofhepatitis B virus infection namely vertical transmission(hepatitis B virus infection transmits directly from the par-ents to the offspring) and horizontal transmission (hepatitisB virus infection transmits through contact with infectiveindividuals) [19]Thirdly we have considered a proportion ofthe vaccination in susceptible individuals to all age groupswhereas Medley et al [4] have utilized only the newbornsvaccination In fact the adults especially the teenagers areencouraged to hepatitis B vaccine [17] Fourthly ldquoimmunityrdquohas been considered as ldquopermanentrdquo in Medleyrsquos modelwhereas the effect of the vaccination may be decreased aftersometime and the person may be affected and be susceptible[22] Consequently we have considered a parameter 120582

4to

Computational and Mathematical Methods in Medicine 3

SS E I C R

minus p1C minus p2R u1S

120588(I + 120579C)S

E

1205821E

I

p31205822I

p1C C

(1 minus p3)1205822I

u2C

1205823C

R

p2R

1205824R

Figure 1 FlowdiagramofHBVdynamics under application of vaccine and treatment 119878119864 119868119862 and119877denote five compartment of susceptibleexposed acute infection chronic carriers and immune or recovered class respectively The 120588(119868 + 120579119862)119878 indicates the horizontal transmissionfrom compartment 119878 to 119864 whereas ]119901

1119862 denotes the vertical transmission from 119862 to 119862 by birth of offspring from a carrier individual

Similarly ]1199012119877 represents the proportion of immune newborn from recovered class 120582

3119862 shows individualsrsquo spontaneous recovery and move

from compartment 119862 to 119877 and 1199062119862 denotes proportions of carriers move to recovered class by treatment 120582

4119877 denotes a portion that moves

from compartment 119877 to 119878 due to loss of immunity 11990131205822119868 shows proportion of acute infection individuals who become carries and (1minus119901

3)1205822119868

shows proportion of acute infection individuals clear HBV and move from 119868 to 119877 1199061119878 denotes proportions of susceptible move to recovered

class by vaccination and each compartment has its own death rate

account for the turning back recovered people to susceptibleones [16 17]

We consider an 119878-119864-119868-119862-119877 epidemic model by dividingtotal population into five time-dependent classes namely thesusceptible individuals 119878(119905) infected but not yet infectiousindividuals (exposed) 119864(119905) acute infection individuals 119868(119905)chronic HBV carriers 119862(119905) and recovered 119877(119905) for hepatitisB virus (HBV) infection that propagates through contactbetween the infected and the susceptible individuals and alsothrough the infected parents to the offspring A flow chartof this compartmental model is shown in Figure 1 Theseassumptions however lead to the following dynamic model

119878 (119905) = ] minus ]1199011119862 minus ]119901

2119877 minus 120588 (119868 + 120579119862) 119878 minus ]119878 minus 119906

1119878 + 1205824119877

(119905) = 120588 (119868 + 120579119862) 119878 minus (] + 1205821) 119864

119868 (119905) = 1205821119864 minus (] + 120582

2) 119868

(119905) = ]1199011119862 + 119901

31205822119868 minus (] + 120582

3) 119862 minus 119906

2119862

(119905) = ]p2119877 + (1 minus 119901

3) 1205822119868 + 1205823119862 minus ]119877 minus 120582

4119877 + 1199061119878 + 1199062119862

(1)

In these equations all the parameters are nonnegativeWe assume stable population with equal per capita birth anddeath rate ] (as disease-induced death rate is not consideredin system) and 120582

1is the rate of exposed individuals becoming

infections 1205822is the rate at which individuals leave the

acute infection class 1205823is the rate of carrier individuals

who recover from the disease by natural way (spontaneousrecovery) and move from carrier to recovered [16 21] and 120579is the infectiousness of carriers relative to acute infections Aproportion 119901

3of acute infection individuals become carriers

and another clear HBV moves directly to immunity class[17] The horizontal transmission of disease propagation isdenoted by the mass action term 120588(119868 + 120579119862)119878 where 120588represents the contact rate For vertical transmission weassume that a fraction 119901

1of newborns from infected class

are infected and it is denoted by the term ]1199011119862 (1199011lt 1)

Similarly a fraction 1199012of newborns from recovered class

are immune and it is denoted by ]1199012119877 (1199012lt 1) [23]

Consequently the birth flux into the susceptible class is givenby ] minus ]119901

1119862 minus ]119901

2119877

For simplicity we normalize the population size to 1 thatis now 119878 119864 119868 119862 and 119877 are respectively the fractions of thesusceptible the exposed the acute infective the carriers andthe recovered individuals in the population and 119878+119864+119868+119862+119877 = 1 holds [17] Hence the fifth equation may be omittedand (1) becomes

119878 (119905) = ] minus ]1199011119862 minus 120588 (119868 + 120579119862) 119878 minus ]119878 minus 119906

1119878

+ (1205824minus ]1199012) (1 minus 119878 minus 119864 minus 119868 minus 119862)

(119905) = 120588 (119868 + 120579119862) 119878 minus (] + 1205821) 119864

119868 (119905) = 1205821119864 minus (] + 120582

2) 119868

(119905) = ]1199011119862 + 119901

31205822119868 minus (] + 120582

3) 119862 minus 119906

2119862

(2)

Among them 1199061is the proportion of the susceptible

that is vaccinated per unit time further we assume thatthe vaccination is not perfect that is although vaccinationoffers a very powerful method of disease control there aremany associated difficulties Generally vaccines are not 100effective and therefore only a proportion of vaccinated indi-viduals are protected then some proportion of the vaccinatedindividuals may be susceptible again to that disease [22]Similarly 119906

2is the proportion of the chronic HBV carriers

that is treated per unit time The transfer rate from therecovered class to the susceptible class is taken as 120582

4(1205824ge 0)

We assume that the population of newborn carriers born tocarriers is less than the sum of the death carriers and thepopulation moving from carrier to recovery state [16] In thiscase we have ]119901

1lt ] + 119906

2+ 1205823 otherwise carriers would

keep increasing rapidly as long as there is infection that is119889119862119889119905 gt 0 for 119862 = 0 or 119868 = 0 and 119905 ge 0

Let119883(119905) = 119878(119905) + 119862(119905) + 119868(119905) + 119862(119905) then we have

d119883119889119905

=

119889119878

119889119905

+

119889119864

119889119905

+

119889119868

119889119905

+

119889119862

119889119905

(3)

4 Computational and Mathematical Methods in Medicine

that is

119889119883

119889119905

+ (] + 1205824minus ]1199012)119883 le ] + 120582

4 (4)

Now integrating both sides of the above inequality and usingthe theory of differential inequality [21 24] we get

119883(119905) le 119890minus(]+120582

4minus]1199012)119905[

] + 1205824

] + 1205824minus ]1199012

119890(]+1205824minus]1199012)119905+ 119896] (5)

where 119896 is a constant and letting 119905 rarr infin we have

119878 + 119864 + 119868 + 119862 le

] + 1205824

] + 1205824minus ]1199012

119878 (119905) = ] minus ]1199011119862 minus 120588 (119868 + 120579119862) 119878 minus ]119878 minus 119906

1119878

+ (1205824minus ]1199012) (1 minus 119878 minus 119864 minus 119868 minus 119862)

(6)

then

119878 le

] + 1205824

] + 1205824+ 1199061minus ]1199012

(7)

then

Π = (119878 119864 119868 119862) isin R4

+| 119878 le

] + 1205824

] + 1199061+ 1205824minus ]1199012

119878 + 119864 + 119868 + 119862 le

] + 1205824

] + 1205824minus ]1199012

(8)

is positively invariant Hence the system is mathematicallywell-posed There for initial starting point 119909 isin R4

+ the

trajectory lies inΠTherefore wewill focus our attention onlyon the region Π

3 Analysis of the System forConstant Controls

In this section we assume that the control parameters 1199061(119905)

and 1199062(119905) are constant functions

31 Equilibrium Points and Basic Reproduction Number Wewill discuss the existence of all the possible equilibriumpointsof the system (2) We see that system (2) has two possiblenonnegative equilibria

(i) Disease-free equilibrium point1198640= (1198780 0 0 0) where

1198780=

] minus ]1199012+ 1205824

1199061+ ] + 120582

4minus ]1199012

(9)

it is always feasible In the absence of vaccination this isreduced to the equilibrium (1 0 0 0)

Definition 1 The basic reproduction number denoted by 1198770

is the expected number of secondary cases produced ina completely susceptible population by a typical infectiveindividual [25]

Using the notation in van den Driessche and Watmough[25] we have

F = [

[

0 12058811987801205881205791198780

0 0 0

0 0 0

]

]

V = [

[

] + 1205821

0 0

minus1205821

] + 1205822

0

0 minus11990131205822] + 1205823+ 1199062minus ]1199011

]

]

(10)

The reproduction number is given by 120588(119865119881minus1) and

1198770

=

1205881205821(] + 120582

3+ 1199062minus 1199011] + 120579119901

31205822) (] minus ]119901

2+ 1205824)

(] + 1205821) (] + 120582

2) (] + 120582

3+ 1199062minus 1199011]) (1199061+ ] + 120582

4minus ]1199012)

=

1205881205821(] + 120582

3+ 1199062minus 1199011] + 120579119901

31205822)

(] + 1205821) (] + 120582

2) (] + 120582

3+ 1199062minus 1199011])1198780

=

1205881205821

(] + 1205821) (] + 120582

2)

[1 +

12057911990131205822

(] + 1205823+ 1199062minus ]1199011)

] 1198780

=

1205881205821

(] + 1205821) (] + 120582

2)

times [1 +

12057911990131205822

(] + 1205823+ 1199062minus ]1199011)

] [

] minus ]1199012+ 1205824

1199061+ ] + 120582

4minus ]1199012

]

(11)

Define

1198771=

12058812058211198780

(] + 1205821) (] + 120582

2)

1198772=

1205881205791199013120582112058221198780

(] + 1205821) (] + 120582

2) (] + 120582

3+ 1199062minus ]1199011)

(12)

then we can see that 1198770= 1198771+ 1198772

Remark 2 We should note from (11) that the use both ofvaccine and treatment control to both reduce the value of 119877

0

and at the same time effects of both intervention strategieson 1198770are not simply the addition of two independent effects

rather they multiply together in order to improve the overalleffects of population level independently (Figure 2) Considerthe following

(ii) endemic equilibria 119864lowastlowast = (119878lowast 119864lowast 119868lowast 119862lowast) which hasfour different cases

Computational and Mathematical Methods in Medicine 5

0

1

2

3

4

5

6

Repr

oduc

tion

num

ber

0 01 02 03 04 05 06 07 08 09 1

Control (u1 u2)

u1 = 0 u2 = 0

u1 = 0 u2 = 1

u1 = 1 u2 = 0

u1 = 1 u2 = 1

Figure 2 Variation of different reproduction numbers by usingof two control variables (vaccination and treatment) Figure 2illustrates the impact of application of two controls (vaccination andtreatment) on different basic reproduction numbers 119877

0= 39810

when there is no vaccination and treatment (1199061= 0 119906

2= 0) The

figure shows that application of vaccination reduces1198770more rapidly

than treatment though mixed intervention strategies always workbetter to reduce the disease burden

(a) No vaccine no treatment control (1199061= 0 119906

2= 0)

119864lowastlowast

1= (119878lowast

1 119864lowast

1 119868lowast

1 119862lowast

1) where

119878lowast

1=

(] + 1205821) (] + 120582

2) (] + 120582

3minus 1199011])

1205881205821(] + 120582

3+ 12057912058221199013minus 1199011])

119864lowast

1=

120579120588 (] + 1205822) 119862lowast

1119878lowast

1

(] + 1205821) (] + 120582

2) minus 120588120582

1119878lowast

1

119868lowast

1=

1205791205881205821119862lowast

1119878lowast

1

(] + 1205821) (] + 120582

2) minus 120588120582

1119878lowast

1

119862lowast

1= (120582112058221199013(] + 120582

4minus ]1199012) 119878lowast

1(1198770minus 1))

times ( (] + 1205823minus 1199011])

times [(] + 1205824minus ]1199012) (] + 120582

2+ 1205821) + 12058211205822]

+120582112058221199013(]1199011minus ]1199012+ 1205824) )minus1

(13)

(b) No vaccine with treatment control (1199061= 0 119906

2= 0)

119864lowastlowast

2= (119878lowast

2 119864lowast

2 119868lowast

2 119862lowast

2) where

119878lowast

2=

(] + 1205821) (] + 120582

2) (] + 120582

3minus 1199011] + 1199062)

1205881205821(] + 120582

3+ 12057912058221199013minus 1199011] + 1199062)

119864lowast

2=

120579120588 (] + 1205822) 119862lowast

2119878lowast

2

(] + 1205821) (] + 120582

2) minus 120588120582

1119878lowast

2

119868lowast

2=

1205791205881205821119862lowast

2119878lowast

2

(] + 1205821) (] + 120582

2) minus 120588120582

1119878lowast

2

119862lowast

2= (120582112058221199013(] + 120582

4minus ]1199012) 119878lowast

2(1198770minus 1))

times ( (] + 1205823minus 1199011] + 1199062)

times [(] + 1205824minus ]1199012) (] + 120582

2+ 1205821) + 12058211205822]

+120582112058221199013(]1199011minus ]1199012+ 1205824) )minus1

(14)

(c) With vaccine no treatment control (1199061= 0 1199062= 0)

119864lowastlowast

3= (119878lowast

3 119864lowast

3 119868lowast

3 119862lowast

3) where

119878lowast

3=

(] + 1205821) (] + 120582

2) (] + 120582

3minus 1199011])

1205881205821(] + 120582

3+ 12057912058221199013minus 1199011])

119864lowast

3=

120579120588 (] + 1205822) 119862lowast

3119878lowast

3

(] + 1205821) (] + 120582

2) minus 120588120582

1119878lowast

3

119868lowast

3=

1205791205881205821119862lowast

3119878lowast

3

(] + 1205821) (] + 120582

2) minus 120588120582

1119878lowast

3

119862lowast

3= (120582112058221199013(] + 120582

4minus ]1199012+ 1199061) 119878lowast

3(1198770minus 1))

times ( (] + 1205823minus 1199011])

times [(] + 1205824minus ]1199012) (] + 120582

2+ 1205821) + 12058211205822]

+120582112058221199013(]1199011minus ]1199012+ 1205824) )minus1

(15)

(d) With vaccine with treatment control (1199061= 0 1199062= 0)

119864lowastlowast

4= (119878lowast

4 119864lowast

4 119868lowast

4 119862lowast

4) where

119878lowast

4=

(] + 1205821) (] + 120582

2) (] + 120582

3minus 1199011] + 1199062)

1205881205821(] + 120582

3+ 12057912058221199013minus 1199011] + 1199062)

119864lowast

4=

120579120573 (] + 1205822) 119862lowast

4119878lowast

4

(] + 1205821) (] + 120582

2) minus 120588120582

1119878lowast

4

119868lowast

4=

1205791205881205821119862lowast

4119878lowast

4

(] + 1205821) (] + 120582

2) minus 120588120582

1119878lowast

4

119862lowast

4= (120582112058221199013(] + 120582

4minus ]1199012+ 1199061) 119878lowast

4(1198770minus 1))

times ( (] + 1205823minus 1199011] + 1199062)

times [(] + 1205824minus ]1199012) (] + 120582

2+ 1205821) + 12058211205822]

+120582112058221199013(]1199011minus ]1199012+ 1205824) )minus1

(16)

Clearly 119864lowastlowast is feasible if 119862lowast gt 0 that is if 1198770gt 1 Also for

1198770= 1 the endemic equilibrium reduces to the disease-free

equilibrium and for 1198770lt 1 it becomes infeasible Hence we

may state the following theorem

Theorem 3 (i) If 1198770lt 1 then the system (2) has only one

equilibrium which is disease-free(ii) If 119877

0gt 1 then the system (2) has two equilibria one is

disease-free and the other is endemic equilibrium(iii) If 119877

0= 1 then the endemic equilibrium reduces to the

disease-free equilibrium

6 Computational and Mathematical Methods in Medicine

32 Stability Analysis In this section we will discuss thestability of different equilibria Firstly we analyse the localstability of the disease-free equilibrium

Theorem 4 If 1198770lt 1 then the disease-free equilibrium is

locally asymptotically stable

Proof The Jacobian matrix of system (2) at the disease-freeequilibrium is

J0 =[[[[

[

minus (] + 1205824minus ]1199012+ 1199061) minus (120582

4minus ]1199012) minus (120588119878

0+ 1205824minus ]1199012) minus (]119901

1+ 1205824minus ]1199012+ 120579120588119878

0)

0 minus (] + 1205821) 120588119878

01205791205881198780

0 1205821

minus (] + 1205822) 0

0 0 11990131205822

]1199011minus ] minus 120582

3minus 1199062

]]]]

]

(17)

The characteristic polynomial of 1198690given by

119875 (120582) = (120582 + 1198970) (1205823+ 11989711205822+ 1198972120582 + 1198973) (18)

where

1198970= ] + 119906

1+ 1205824minus ]1199012

1198971= 3] + 120582

1+ 1205822+ 1205823+ 1199062minus ]1199011

1198972= (] + 120582

2) (] + 120582

3+ 1199062minus ]1199011)

+ (] + 1205821) (2] + 120582

2+ 1205823+ 1199062minus ]1199011) minus 12058211205881198780

1198973= (] + 120582

1) (] + 120582

2) (] + 120582

3+ 1199062minus ]1199011)

minus 12058211205881198780(] + 120582

3+ 1199062+ 12057911990131205822minus ]1199011)

(19)

We need to verify the following two conditions

(a) 1198970 1198971 1198972 1198973gt 0

(b) 11989711198972minus 1198973gt 0

It is easy to see that 1198970 1198971gt 0 and 119897

2 1198973 11989711198972minus 1198973gt 0 if

1198770lt 1 It follows from the Routh-Hurwitz criterion that the

eigenvalues have negative real parts if 1198770lt 1 Hence the

disease-free equilibriumofmodel (2) is locally asymptoticallystable if 119877

0lt 1 and unstable if 119877

0gt 1

To discuss the properties of the endemic equilibriumpoint

Theorem 5 The endemic equilibrium point is locally asymp-totically stable if 119877

0gt 1

Proof We use Routh-Hurwitz criterion to establish the localstability of the endemic equilibrium The Jacobian matrix ofsystem (2) at endemic equilibrium is

Jlowast =[[[

[

minus (] + 1205824minus ]1199012+ 1199061+ 120588119868lowast) minus (120582

4minus ]1199012) minus (120588119878

lowast+ 1205824minus ]1199012) minus (]119901

1+ 1205824minus ]1199012+ 120588120579119878

lowast)

120588119868lowast+ 120588120579119862

lowastminus (] + 120582

1) 120588119878

lowast120588120579119878lowast

0 1205821

minus (] + 1205822) 0

0 0 11990131205822

]1199011minus ] minus 120582

3minus 1199062

]]]

]

(20)

Then the characteristic equation at 119864lowastlowast is 1205824 + 11989111205823+ 11989121205822+

1198913120582 + 1198914= 0 where

1198911= 1198861+ 1198872+ 1198883+ 1198894

1198912= 11988611198863+ 11988611198894+ 11988611198872+ 11988831198894+ 11988721198883+ 11988721198894minus 11988731198882minus 11988711198862

1198913= 119886111988831198894+ 119886111988721198883+ 119886111988721198894+ 119887211988831198894+ 119887411988821198893+ 119887111988821198863

minus 119886111988731198882minus 119887311988821198894minus 119887111988621198883minus 119887111988621198894

1198914= 1198861119887211988831198894+ 1198861119887411988821198893+ 1198871119888211988631198894minus 1198861119887311988821198894

minus 1198871119886211988831198894minus 1198871119886411988821198893

(21)

where

1198861= minus (] + 120582

4minus ]1199012+ 1199061+ 120588119868lowast)

1198862= minus (120582

4minus ]1199012) 119886

3= minus (120588119878

lowast+ 1205824minus ]1199012)

1198864= minus (]119901

1+ 1205824minus ]1199012+ 120588120579119878

lowast) 119887

1= 120588119868lowast+ 120588120579119862

lowast

1198872= minus (] + 120582

1) 119887

3= 120588119878lowast

1198874= 120588120579119878

lowast 119888

1= 0

1198882= 1205821 119888

3= minus (] + 120582

2)

1198884= 0 119889

1= 0

1198892= 0 119889

3= 11990131205822

1198894= ]1199011minus ] minus 120582

3minus 1199062

(22)

We need to verify the following three conditions

(a) 1198911 1198912 1198913 1198914gt 0

(b) 11989111198912minus 1198913gt 0

(c) 1198913(11989111198912minus 1198913) minus 1198912

11198914gt 0

Computational and Mathematical Methods in Medicine 7

It is easy to see that conditions (a) and (b) are satisfiedAfter computations we can prove that 119891

3(11989111198912minus 1198913) minus

1198912

11198914gt 0 is also validTheRouth-Hurwitz criterion and those

inequalities in (a)ndash(c) imply that the characteristic equationat 119864lowastlowast has only roots with negative real part which certifiesthe local stability of 119864lowastlowast

4 Optimal Control with Two Objectives

One of the early reasons for studying hepatitis B virus (HBV)infection is to improve the control variables and finally to putdown the infection of the population Optimal control is auseful mathematical tool that can be used to make decisionsin this case In the previous sections we have analyzed themodel with two control variables one is treatment and theother is vaccination and we consider their constant controlsthroughout the analysis But in fact these control variablesshould be time dependent In this section we consider thevaccination and treatment as time-dependent controls in acompact interval of time duration

Our goals here are to put down infection from the popu-lation by increasing the recovered individuals and reducingsusceptible exposed infected and carrier individuals in apopulation and to minimize the costs required to control thehepatitis B virus (HBV) infection by using vaccination andtreatment First of all we construct the objective functionalto be optimized as follows

119869 = int

119879

0

[1198601119878 (119905) + 119860

2119864 (119905) + 119860

3119868 (119905) + 119860

4119862 (119905)

+

1

2

(11986111199062

1(119905) + 119861

21199062

2(119905))] 119889119905

(23)

subject to

119878 (119905) = ] minus ]1199011119862 minus 120588 (119868 + 120579119862) 119878 minus ]119878 minus 119906

1119878

+ (1205824minus ]1199012) (1 minus 119878 minus 119864 minus 119868 minus 119862)

(119905) = 120588 (119868 + 120579119862) 119878 minus (] + 1205821) 119864

119868 (119905) = 1205821119864 minus (] + 120582

2) 119868

(119905) = ]1199011119862 + 119901

31205822119868 minus (] + 120582

3) 119862 minus 119906

2119862

(24)

Our object is to find (119906lowast1 119906lowast

2) such that

119869 (119906lowast

1 119906lowast

2) = min 119869 (119906

1 1199062) (25)

where

1199061(119905) 119906

2(119905) isin Γ

= 119906 (119905) | 0 le 119906 (119905) le 119906max (119905) le 1 0 le 119905 le 119879

119906 (119905) is Labesgue measurable

(26)

Here 119860119894 for 119894 = 1 4 are positive constants that

are represented to keep a balance in the size of 119878(119905) 119864(119905)119868(119905) and 119862(119905) respectively 119861

1and 119861

2 respectively are the

weights corresponding to the controls 1199061and 119906

2 119906max is the

maximum attainable value for controls (1199061max and 119906

2max)1199061max and 119906

2max will depend on the amount of resourcesavailable to implement each of the control measures [26]The 119861

1and 119861

2will depend on the relative importance of

each of the control measures in mitigating the spread of thedisease as well as the cost (human effort material resourcesetc) of implementing each of the control measures per unittime [26] Thus the terms 119861

11199062

1and 119861

21199062

2describe the costs

associated with vaccination and treatment respectively Thesquare of the control variables is taken here to remove theseverity of the side effects and overdoses of vaccination andtreatment [27]

41 Optimal Control Solution For existence of the solutionwe consider the control system (24) with initial condition

119878 (0) = 1198780 119864 (0) = 119864

0

119868 (0) = 1198680 119862 (0) = 119862

0

(27)

then we rewrite (2) in the following form

119889119909 (119905)

119889119905

= 119860119909 + 119865 (119909) (28)

where 119909(119905) = (119878(119905) 119864(119905) 119868(119905) 119862(119905)) is the vector of the statevariables and A and F(x) are defined as follows

A =

[[[[[[[

[

minus] minus 1199061minus 1205824+ ]1199012minus1205824+ ]1199012minus1205824+ ]1199012

minus1205824+ ]1199012

0 minus] minus 1205821

0 0

0 1205821

minus] minus 1205822

0

0 0 11990131205822

]1199011minus ] minus 120582

3minus 1199062

]]]]]]]

]

F (x) =[[[[[[

[

] + 120588 (119868 + 120579119862) 119878 + 1205824minus ]1199012

120588 (119868 + 120579119862) 119878

0

0

]]]]]]

]

(29)

8 Computational and Mathematical Methods in Medicine

We set

119866 (119909) = 119860119909 + 119865 (119909) (30)

The second term on the right-hand side of (30) satisfies

1003816100381610038161003816119865 (1199091) minus 119865 (119909

2)1003816100381610038161003816

le 119872 (10038161003816100381610038161198781minus 1198782

1003816100381610038161003816+10038161003816100381610038161198641minus 1198642

1003816100381610038161003816+10038161003816100381610038161198681minus 1198682

1003816100381610038161003816+10038161003816100381610038161198621minus 1198622

1003816100381610038161003816)

(31)

where the positive constant 119872 is independent of the statevariables 119909 and119872 le 1 Also we get

1003816100381610038161003816119866 (1199091) minus 119866 (119909

2)1003816100381610038161003816le 119871

10038161003816100381610038161199091minus 1199092

1003816100381610038161003816 (32)

where 119871 = max119872 119860 lt infin Therefore it follows thatthe function 119866 is uniformly Lipschitz continuous From thedefinition of the controls 119906

1(119905) and 119906

2(119905) and the restrictions

on the nonnegativeness of the state variables we see that asolution of the system (28) exists [24 28 29]

42The Lagrangian and Hamiltonian for the Control ProblemIn order to find an optimal solution pair first we shouldfind the Lagrangian and Hamiltonian for the optimal controlproblem (24) In fact the Lagrangian of problem is given by

119871 (119878 119864 119868 119862 1199061 1199062) = 119860

1119878 (119905) + 119860

2119864 (119905) + 119860

3119868 (119905)

+ 1198604119862 (119905) +

1

2

(11986111199062

1(119905) + 119861

21199062

2(119905))

(33)

We are looking for the minimal value of the LagrangianTo accomplish this we define Hamiltonian119867 for the controlproblem as follows

119867(119878 119864 119868 119862 1199061 1199062 1205721 1205722 1205723 1205724)

= 119871 + 1205721(119905)

119889119878

119889119905

+ 1205722(119905)

119889119864

119889119905

+ 1205723(119905)

119889119868

119889119905

+ 1205724(119905)

119889119862

119889119905

(34)

where 120572119894(119905) for 119894 = 1 2 3 4 are the adjoint variables and can

be determined by solving the following system of differentialequations

1(119905) = minus

120597119867

120597119878

= minus [1198601+ 1205721((]1199012minus 1205824) minus 120588 (119868 + 120579119862) minus 119906

1minus ])

+1205722120588 (119868 + 120579119862)]

2(119905) = minus

120597119867

120597119864

= minus [1198602+ 1205721(]1199012minus 1205824) + 1205722(minus] minus 120582

1) + 12057231205821]

3(119905) = minus

120597119867

120597119868

= minus [1198603+ 1205721((]1199012minus 1205824) minus 120588119878) + 120572

2120588119878

+1205723(minus] minus 120582

2) + 120572411990131205822]

4(119905) = minus

120597119867

120597119862

= minus [1198604+ 1205721(]1199012minus 1205824minus ]1199011minus 120588120579119878)

+1205722120588120579119878 + 120572

4(]1199011minus ] + 120582

3+ 1199062)]

(35)

satisfying the transversality conditions and optimality condi-tions

1205721(119879) = 120572

2(119879) = 120572

3(119879) = 120572

4(119879) = 0

120597119867

1205971199061

= 11986111199061minus 1205721119878 = 0

120597119867

1205971199062

= 11986121199062minus 1205724119862 = 0

(36)

We now state and prove the following theorem

Theorem 6 There is an optimal control (119906lowast1 119906lowast

2) such that

119869 (119878 119864 119868 119878 119906lowast

1 119906lowast

2) = min 119869 (119878 119864 119868 119878 119906

1 1199062) (37)

subject to the system of differential equation (24)

Proof Here both the control and state variables are non-negative values In this minimized problem the necessaryconvexity of the objective functional in 119906

1and 1199062are satisfied

The set of all control variables 1199061(119905) 1199062(119905) is also convex and

closed by definition The optimal system is bounded whichdetermines the compactness needed for the existence of theoptimal control In addition the integrand in the functional

int

119879

0

[1198601119878 (119905) + 119860

2119864 (119905) + 119860

3119868 (119905) + 119860

4119862 (119905)

+

1

2

(11986111199062

1(119905) + 119861

21199062

2(119905))] 119889119905

(38)

is convex on the control set Hence the theorem (the proof iscomplete)

Computational and Mathematical Methods in Medicine 9

43 Necessary and Sufficient Conditions for Optimal ControlsApplying Pontryaginrsquos maximum principle [30] on the con-structed Hamiltonian 119867 and the theorem (24) we obtainthe optimal steady-state solution and corresponding controlas follows If taking 119883 = (119878 119864 119868 119862) and 119880 = (119906

1 1199062)

then (119883lowast 119880lowast) is an optimal solution of an optimal controlproblem then we now state the following theorem

Theorem 7 The optimal control pair (119906lowast1 119906lowast

2) which mini-

mizes J over the region Γ is given by

119906lowast

1= max[0119879]

0min (1(119905) 1)

119906lowast

2= max[0119879]

0min (2(119905) 1)

(39)

where 1= 11198781198611and 2= 41198621198612and let

1 2 3 4 be

the solution of system (35)

Proof With the help of the optimality conditions we have

120597119867

1205971199061

= 11986111199061minus 1205721119878 = 0 997904rArr 119906

1=

1119878

1198611

= (1)

120597119867

1205971199062

= 11986121199062minus 1205724119862 = 0 997904rArr 119906

2=

4119862

1198612

= (2)

(40)

Using the property of the control space Γ the two controlswhich are bounded with upper and lower bounds arerespectively 1 and 0 that is

119906lowast

1=

0 if 1le 0

1

if 0 lt 1lt 1

1 if 1ge 1

(41)

This can be rewritten in compact notation

119906lowast

1= max[0119879]

0min (1(119905) 1) (42)

and similarly

119906lowast

2=

0 if 2le 0

2

if 0 lt 2lt 1

1 if 2ge 1

(43)

This can be rewritten in compact notation

119906lowast

2= max[0119879]

0min (2(119905) 1) (44)

Hence for these pair of controls (119906lowast1 119906lowast

2) we get the optimum

value of the functional 119869 given by (24)

5 Numerical Examples

Numerical solutions to the optimality system (24) arediscussed in this section We make several interestingobservation by numerically simulating (2) in the rangeof parameter values We consider the parameter set

Δ = ] 120588 120579 1205821 1205822 1205823 1205824 1199011 1199012 1199013 1198601 1198602 1198603 1198604 1198611 1198612

some of the parameters are taken from the published articlesand some are assumed with feasible values Moreover thetime interval for which the optimal control is applied istaken as 70 years also consider Ψ = 119878

0 1198640 1198680 1198620 as initial

condition for simulation of the model The main parametervalues are listed in Table 1 We compare the results having nocontrols only vaccination control only treatment controland both vaccination and treatment controls

With the parameter values in Table 1 the system asymp-totically approaches towards the equilibrium 119864

lowastlowast

1(00858

03327 04003 05349) where the basic reproduction ratio1198770= 39810 (Figure 2) For the parameters set the system

(2) has two feasible equilibria one is disease free and theother is endemic and the endemic equilibrium is locallyasymptotically stable The effect of two control measures ondisease dynamics may be understood well if we considerFigure 2 It explains how control reproduction ratio 119877

0

evolves with different rates of 1199061and 119906

2 It is seen that both

vaccination and treatment reduce the value of 1198770effectively

But an integrated control works better than either of thecontrol measures

In this section we use an iterative method to obtainresults for an optimal control problem of the proposedmodel(24) We use Runge-Kuttarsquos fourth-order procedure [23] hereto solve the optimality system consisting of eight ordinarydifferential equations having four state equations and fouradjoint equations and boundary conditions Because stateequations have initial conditions and adjoint equations haveconditions at the final time an iterative program was createdto numerically simulate solutions Given an initial guessfor the controls to compute the optimal state values theprogram solves (24) with initial conditions (27) forward intime interval [0 70]using aRungeKuttamethod of the fourthorder Resulting state values are placed in adjoint equations(35) These adjoint equations with given final conditions arethen solved backwards in time Again a fourth order RungeKutta method is employed Both state and adjoint values areused to update the control using the characterization (39)and the entire process repeats itself This iterative processterminates when current state adjoint and control valuesare sufficiently close to successive values Then we use thebackward Runge-Kutta fourth-order procedure to solve theadjoint variables in the same time interval with the help ofthe solution of the state variables transversality conditions

We have plotted susceptible exposed acute infectionindividuals and carriers individuals with andwithout controlby considering values of parameters We simulate the systemat different values of rate of 119906

1and 119906

2

From Figures 3 4 and 5 we see that after 20 years thenumber of susceptible population decreases than when thereis no control In this case most of this population tends to theinfected class Again when only treatment control is appliedthen the number of susceptible population is not muchdifferent than the population in the case having no controlBut the susceptible population differs much from these twostrategies if we apply the strategies of only vaccination controland both vaccination and treatment controls At a high rateof vaccination the sensitive population density is reduced

10 Computational and Mathematical Methods in Medicine

Table 1 Parameter values used in numerical simulations

Parameter Description Values range Reference] Birth (and death) rate 00121 [17 18]120588 Transmission rate 08ndash2049 [17]120579 Infectiousness of carriers relative to acute infections 0-1 [17]1205821

Rate of moving from exposed to acute 6 per year [16]1205822

Rate at which individuals leave the acute infection class 4 per year [16]1205823

Rate of moving from carrier to recovery 0025 per year [16]1205824

Loss of recovery rate 003ndash006 [17]1198751

Probability of infected newborns 011 [19]1198752

Probability of immune newborns 01 [19]1198753

Proportion of acute infection individuals becoming carriers 005ndash09 [17]1198601

Weight factor for susceptible individuals 0091 [31]1198602

Weight factor for exposed individuals 001 [31]1198603

Weight factor for infected individuals 004 [31]1198604

Weight factor for carrier individuals 005 [31]1198611

Weight factor for the controls 1199061

15 [31]1198612

Weight factor for the controls 1199062

27 [31]1198780

Susceptible individuals 0493 [4]1198640

Exposed individuals 00035 [4]1198680

Acute infection individuals 00035 [4]1198620

Chronic HBV carriers 0007 [4]

0 10 20 30 40 50 60 70Time (year)

Susc

eptib

le p

opul

atio

n

NV-NTNV-T

V-NTV-T

10minus4

10minus3

10minus2

10minus1

100

Figure 3 The plot shows the changes in sensitive populations(without) vaccination and (without) treatment

to a very low level initially and then it takes longer time torestore the steady-state value

From Figures 6 7 and 8 we see that the number ofexposed population increase thanwhen there is no control Inthis case most of this population tends to the acute infectedclass Again when only treatment control is applied thenthe number of exposed population is not much different

0 10 20 30 40 50 60 70Time (year)

Susc

eptib

le p

opul

atio

n

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0365 u2 = 0

u1 = 045 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

Figure 4 The plot shows the sensitivity of sensitive populations fordifferent values of control 119906

1

than the population in the case having no control But theexposed population differs much from these two strategies ifwe apply the strategies of only vaccination control and bothvaccination and treatment controls

From Figures 9 10 and 11 we see that the number ofacute infected population increases than when there is nocontrol In this case most of this population tends to thecarrier class Again when only treatment control is appliedthen the number of acute infected population is not muchdifferent than the population in the case having no controlBut the acute infected population differsmuch from these two

Computational and Mathematical Methods in Medicine 11

0 10 20 30 40 50 60 70Time (year)

Susc

eptib

le p

opul

atio

n

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Figure 5 The plot shows the sensitivity of sensitive populations fordifferent values of control 119906

2

Expo

sed

popu

latio

n

NV-NTNV-T

V-NTV-T

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 6 The plot shows the changes in exposed populations(without) vaccination and (without) treatment

strategies if we apply the strategies of only vaccination controland both vaccination and treatment controls

Again from Figures 12 13 and 14 we see that thenumber of carrier population increases than when there isno control We see that the application of only vaccinationcontrol gives better result than the application of no controlAgain application of treatment control would give betterresult than the application of vaccination control since thetreatment control is better than the vaccination control whilethe application of both vaccination and treatment controlsgive the best result as in this case the number of carrierpopulation would be the least in number

Expo

sed

popu

latio

n

u1 = 0 u2 = 0

u1 = 03635 u2 = 0

u1 = 045 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 7 The plot shows the sensitivity of exposed populations fordifferent values of control 119906

1

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Expo

sed

popu

latio

n

Figure 8 The plot shows the sensitivity of exposed populations fordifferent values of control 119906

2

Finally the effect of two control measures on diseasedynamics may be understood well if we consider Figures 1ndash14 15 16 17 and 18 Since our main purpose is to reducethe number of sensitive exposed acute infection and carrierindividuals therefore numerical simulation results show thatoptimal combination of vaccination and treatment is themosteffective way to control hepatitis B virus infection

12 Computational and Mathematical Methods in MedicineAc

ute i

nfec

tion

popu

latio

n

NV-NTNV-T

V-NTV-T

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 9The plot shows the changes in acute infection populations(without) vaccination and (without) treatment

Acut

e inf

ectio

n po

pulat

ion

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0365 u2 = 0

u1 = 045 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

Figure 10 The plot shows the sensitivity of acute infection popula-tions for different values of control 119906

1

6 Conclusion and Suggestions

In this paper we propose an S-119864-119868-119862-119877model of hepatitis Bvirus infection with two controls vaccination and treatmentFirst we analyze the dynamic behavior of the system for con-stant controls In the constant controls case we calculate thebasic reproduction number and investigate the existence andstability of equilibriaThere are two nonnegative equilibria of

Acut

e inf

ectio

n po

pulat

ion

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Figure 11 The plot shows the sensitivity of acute infection popula-tions for different values of control 119906

2

Carr

ier p

opul

atio

n

NV-NTNV-T

V-NTV-T

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 12 The plot shows the changes in carrier populations(without) vaccination and (without) treatment

the system namely the disease-free and endemicWe see thatthe disease-free equilibriumwhich always exists and is locallyasymptotically stable if 119877

0lt 1 and endemic equilibrium

which exists and is locally asymptotically stable if 1198770gt 1

After investigating the dynamic behavior of the systemwith constant controls we formulate an optimal controlproblem if the controls become time dependent and solve itby using Pontryaginrsquos maximum principle Different possiblecombinations of controls are used and their effectiveness iscompared by simulation works Also from the numericalresults it is very clear that a combination of mixed controlmeasures respond better than any other independent control

Computational and Mathematical Methods in Medicine 13Ca

rrie

r pop

ulat

ion

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0365 u2 = 0

u1 = 046 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

Figure 13 The plot shows the sensitivity of carrier populations fordifferent values of control 119906

1

Carr

ier p

opul

atio

n

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Figure 14 The plot shows the sensitivity of carrier populations fordifferent values of control 119906

2

There is still a tremendous amount of work needed tobe done in this area Parameters are rarely constant becausethey depend on environmental conditions We do not knowhowever the detailed relationship between these parametersand environmental conditions There may be a time lag as asusceptible populationmay take some time to be infected andalso a susceptible population may take some time to immune

Popu

latio

n (N

V-N

T)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 15 The plot shows the changes in total populations withoutvaccination and without treatment

Popu

latio

n (V

-NT)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 16 The plot shows the changes in total populations withvaccination and without treatment

Popu

latio

n (N

V-T)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 17 The plot shows the changes in total populations withoutvaccination and with treatment

14 Computational and Mathematical Methods in MedicinePo

pulat

ion

(V-T

)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 18 The plot shows the changes in total populations withvaccination and treatment

after vaccination We leave all these possible extensions forthe future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] WHO Hepatitis B Fact Sheet No 204 The World HealthOrganisation Geneva Switzerland 2013 httpwwwwhointmediacentrefactsheetsfs204en

[2] CanadianCentre forOccupationalHealth and Safety ldquoHepatitisBrdquo httpwwwccohscaoshanswersdiseaseshepatitis bhtml

[3] ldquoHealthcare stumbling in RIrsquos Hepatitis fightrdquoThe Jakarta PostJanuary 2011

[4] G F Medley N A Lindop W J Edmunds and D JNokes ldquoHepatitis-B virus endemicity heterogeneity catas-trophic dynamics and controlrdquo Nature Medicine vol 7 no 5pp 619ndash624 2001

[5] J Mann and M Roberts ldquoModelling the epidemiology ofhepatitis B in New Zealandrdquo Journal of Theoretical Biology vol269 no 1 pp 266ndash272 2011

[6] ldquoHepatitis B (HBV)rdquo httpkidshealthorgteensexual healthstdsstd hepatitishtml

[7] CDC ldquoHepatitis B virus a comprehensive strategy for elim-inating transmission in the United States through universalchildhood vaccination Recommendations of the Immuniza-tion Practices Advisory Committee (ACIP)rdquo MMWR Recom-mendations and Reports vol 40(RR-13) pp 1ndash25 1991

[8] M K Libbus and L M Phillips ldquoPublic health management ofperinatal hepatitis B virusrdquo Public Health Nursing vol 26 no 4pp 353ndash361 2009

[9] F B Hollinger and D T Lau ldquoHepatitis B the pathway torecovery through treatmentrdquo Gastroenterology Clinics of NorthAmerica vol 35 no 4 pp 895ndash931 2006

[10] C-L Lai andM-F Yuen ldquoThe natural history and treatment ofchronic hepatitis B a critical evaluation of standard treatment

criteria and end pointsrdquoAnnals of InternalMedicine vol 147 no1 pp 58ndash61 2007

[11] R M Anderson and R M May Infectious Disease of HumansDynamics and Control Oxford University Press Oxford UK1991

[12] S Thornley C Bullen and M Roberts ldquoHepatitis B in ahigh prevalence New Zealand population a mathematicalmodel applied to infection control policyrdquo Journal ofTheoreticalBiology vol 254 no 3 pp 599ndash603 2008

[13] S-J Zhao Z-Y Xu and Y Lu ldquoA mathematical model ofhepatitis B virus transmission and its application for vaccinationstrategy inChinardquo International Journal of Epidemiology vol 29no 4 pp 744ndash752 2000

[14] K Wang W Wang and S Song ldquoDynamics of an HBV modelwith diffusion and delayrdquo Journal ofTheoretical Biology vol 253no 1 pp 36ndash44 2008

[15] R Xu andZMa ldquoAnHBVmodelwith diffusion and time delayrdquoJournal of Theoretical Biology vol 257 no 3 pp 499ndash509 2009

[16] L Zou W Zhang and S Ruan ldquoModeling the transmissiondynamics and control of hepatitis B virus in Chinardquo Journal ofTheoretical Biology vol 262 no 2 pp 330ndash338 2010

[17] J Pang J-A Cui and X Zhou ldquoDynamical behavior of ahepatitis B virus transmission model with vaccinationrdquo Journalof Theoretical Biology vol 265 no 4 pp 572ndash578 2010

[18] S Zhang and Y Zhou ldquoThe analysis and application of an HBVmodelrdquoAppliedMathematicalModelling vol 36 no 3 pp 1302ndash1312 2012

[19] S Bhattacharyya and S Ghosh ldquoOptimal control of verticallytransmitted diseaserdquoComputational andMathematicalMethodsin Medicine vol 11 no 4 pp 369ndash387 2010

[20] T K Kar andA Batabyal ldquoStability analysis and optimal controlof an SIR epidemic model with vaccinationrdquo Biosystems vol104 no 2-3 pp 127ndash135 2011

[21] T-K Kar and S Jana ldquoA theoretical study on mathematicalmodelling of an infectious disease with application of optimalcontrolrdquo Biosystems vol 111 no 1 pp 37ndash50 2013

[22] M J Keeling and P Rohani Modeling Infectious Diseases inHumans and Animals Princeton University Press PrincetonNJ USA 2008

[23] E Jung S Lenhart and Z Feng ldquoOptimal control of treatmentsin a two-strain tuberculosis modelrdquo Discrete and ContinuousDynamical Systems B vol 2 no 4 pp 473ndash482 2002

[24] G Birkhoff and G-C Rota Ordinary Differential EquationsJohn Wiley amp Sons New York NY USA 4th edition 1989

[25] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[26] T T Yusuf and F Benyah ldquoOptimal control of vaccination andtreatment for an SIR epidemiological modelrdquo World Journal ofModelling and Simulation vol 8 no 3 pp 194ndash204 2008

[27] H-R Joshi S Lenhart M Y Li and L Wang ldquoOptimalcontrol methods applied to disease modelsrdquo in MathematicalStudies on Human Disease Dynamics vol 410 of ContemporaryMathematics pp 187ndash207 American Mathematical SocietyProvidence RI USA 2006

[28] G Zaman Y H Kang and I H Jung ldquoStability analysis andoptimal vaccination of an SIR epidemic modelrdquo Biosystems vol93 no 3 pp 240ndash249 2008

[29] D Iacoviello and N Stasiob ldquoOptimal control for SIRCepidemic outbreakrdquo Computer Methods and Programs inBiomedicine vol 110 no 3 pp 333ndash342 2013

Computational and Mathematical Methods in Medicine 15

[30] L S Pontryagin V G Boltyanskii R V Gamkrelidze and EF Mishchenko The Mathematical Theory of Optimal ProcessesJohn Wiley amp Sons New York NY USA 1962

[31] WHO Hepatitis B Fact Sheet No 204 The World HealthOrganisation Geneva Switzerland 2000 httpwwwwhointmediacentrefactsheetsfs204en

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

Computational and Mathematical Methods in Medicine 3

SS E I C R

minus p1C minus p2R u1S

120588(I + 120579C)S

E

1205821E

I

p31205822I

p1C C

(1 minus p3)1205822I

u2C

1205823C

R

p2R

1205824R

Figure 1 FlowdiagramofHBVdynamics under application of vaccine and treatment 119878119864 119868119862 and119877denote five compartment of susceptibleexposed acute infection chronic carriers and immune or recovered class respectively The 120588(119868 + 120579119862)119878 indicates the horizontal transmissionfrom compartment 119878 to 119864 whereas ]119901

1119862 denotes the vertical transmission from 119862 to 119862 by birth of offspring from a carrier individual

Similarly ]1199012119877 represents the proportion of immune newborn from recovered class 120582

3119862 shows individualsrsquo spontaneous recovery and move

from compartment 119862 to 119877 and 1199062119862 denotes proportions of carriers move to recovered class by treatment 120582

4119877 denotes a portion that moves

from compartment 119877 to 119878 due to loss of immunity 11990131205822119868 shows proportion of acute infection individuals who become carries and (1minus119901

3)1205822119868

shows proportion of acute infection individuals clear HBV and move from 119868 to 119877 1199061119878 denotes proportions of susceptible move to recovered

class by vaccination and each compartment has its own death rate

account for the turning back recovered people to susceptibleones [16 17]

We consider an 119878-119864-119868-119862-119877 epidemic model by dividingtotal population into five time-dependent classes namely thesusceptible individuals 119878(119905) infected but not yet infectiousindividuals (exposed) 119864(119905) acute infection individuals 119868(119905)chronic HBV carriers 119862(119905) and recovered 119877(119905) for hepatitisB virus (HBV) infection that propagates through contactbetween the infected and the susceptible individuals and alsothrough the infected parents to the offspring A flow chartof this compartmental model is shown in Figure 1 Theseassumptions however lead to the following dynamic model

119878 (119905) = ] minus ]1199011119862 minus ]119901

2119877 minus 120588 (119868 + 120579119862) 119878 minus ]119878 minus 119906

1119878 + 1205824119877

(119905) = 120588 (119868 + 120579119862) 119878 minus (] + 1205821) 119864

119868 (119905) = 1205821119864 minus (] + 120582

2) 119868

(119905) = ]1199011119862 + 119901

31205822119868 minus (] + 120582

3) 119862 minus 119906

2119862

(119905) = ]p2119877 + (1 minus 119901

3) 1205822119868 + 1205823119862 minus ]119877 minus 120582

4119877 + 1199061119878 + 1199062119862

(1)

In these equations all the parameters are nonnegativeWe assume stable population with equal per capita birth anddeath rate ] (as disease-induced death rate is not consideredin system) and 120582

1is the rate of exposed individuals becoming

infections 1205822is the rate at which individuals leave the

acute infection class 1205823is the rate of carrier individuals

who recover from the disease by natural way (spontaneousrecovery) and move from carrier to recovered [16 21] and 120579is the infectiousness of carriers relative to acute infections Aproportion 119901

3of acute infection individuals become carriers

and another clear HBV moves directly to immunity class[17] The horizontal transmission of disease propagation isdenoted by the mass action term 120588(119868 + 120579119862)119878 where 120588represents the contact rate For vertical transmission weassume that a fraction 119901

1of newborns from infected class

are infected and it is denoted by the term ]1199011119862 (1199011lt 1)

Similarly a fraction 1199012of newborns from recovered class

are immune and it is denoted by ]1199012119877 (1199012lt 1) [23]

Consequently the birth flux into the susceptible class is givenby ] minus ]119901

1119862 minus ]119901

2119877

For simplicity we normalize the population size to 1 thatis now 119878 119864 119868 119862 and 119877 are respectively the fractions of thesusceptible the exposed the acute infective the carriers andthe recovered individuals in the population and 119878+119864+119868+119862+119877 = 1 holds [17] Hence the fifth equation may be omittedand (1) becomes

119878 (119905) = ] minus ]1199011119862 minus 120588 (119868 + 120579119862) 119878 minus ]119878 minus 119906

1119878

+ (1205824minus ]1199012) (1 minus 119878 minus 119864 minus 119868 minus 119862)

(119905) = 120588 (119868 + 120579119862) 119878 minus (] + 1205821) 119864

119868 (119905) = 1205821119864 minus (] + 120582

2) 119868

(119905) = ]1199011119862 + 119901

31205822119868 minus (] + 120582

3) 119862 minus 119906

2119862

(2)

Among them 1199061is the proportion of the susceptible

that is vaccinated per unit time further we assume thatthe vaccination is not perfect that is although vaccinationoffers a very powerful method of disease control there aremany associated difficulties Generally vaccines are not 100effective and therefore only a proportion of vaccinated indi-viduals are protected then some proportion of the vaccinatedindividuals may be susceptible again to that disease [22]Similarly 119906

2is the proportion of the chronic HBV carriers

that is treated per unit time The transfer rate from therecovered class to the susceptible class is taken as 120582

4(1205824ge 0)

We assume that the population of newborn carriers born tocarriers is less than the sum of the death carriers and thepopulation moving from carrier to recovery state [16] In thiscase we have ]119901

1lt ] + 119906

2+ 1205823 otherwise carriers would

keep increasing rapidly as long as there is infection that is119889119862119889119905 gt 0 for 119862 = 0 or 119868 = 0 and 119905 ge 0

Let119883(119905) = 119878(119905) + 119862(119905) + 119868(119905) + 119862(119905) then we have

d119883119889119905

=

119889119878

119889119905

+

119889119864

119889119905

+

119889119868

119889119905

+

119889119862

119889119905

(3)

4 Computational and Mathematical Methods in Medicine

that is

119889119883

119889119905

+ (] + 1205824minus ]1199012)119883 le ] + 120582

4 (4)

Now integrating both sides of the above inequality and usingthe theory of differential inequality [21 24] we get

119883(119905) le 119890minus(]+120582

4minus]1199012)119905[

] + 1205824

] + 1205824minus ]1199012

119890(]+1205824minus]1199012)119905+ 119896] (5)

where 119896 is a constant and letting 119905 rarr infin we have

119878 + 119864 + 119868 + 119862 le

] + 1205824

] + 1205824minus ]1199012

119878 (119905) = ] minus ]1199011119862 minus 120588 (119868 + 120579119862) 119878 minus ]119878 minus 119906

1119878

+ (1205824minus ]1199012) (1 minus 119878 minus 119864 minus 119868 minus 119862)

(6)

then

119878 le

] + 1205824

] + 1205824+ 1199061minus ]1199012

(7)

then

Π = (119878 119864 119868 119862) isin R4

+| 119878 le

] + 1205824

] + 1199061+ 1205824minus ]1199012

119878 + 119864 + 119868 + 119862 le

] + 1205824

] + 1205824minus ]1199012

(8)

is positively invariant Hence the system is mathematicallywell-posed There for initial starting point 119909 isin R4

+ the

trajectory lies inΠTherefore wewill focus our attention onlyon the region Π

3 Analysis of the System forConstant Controls

In this section we assume that the control parameters 1199061(119905)

and 1199062(119905) are constant functions

31 Equilibrium Points and Basic Reproduction Number Wewill discuss the existence of all the possible equilibriumpointsof the system (2) We see that system (2) has two possiblenonnegative equilibria

(i) Disease-free equilibrium point1198640= (1198780 0 0 0) where

1198780=

] minus ]1199012+ 1205824

1199061+ ] + 120582

4minus ]1199012

(9)

it is always feasible In the absence of vaccination this isreduced to the equilibrium (1 0 0 0)

Definition 1 The basic reproduction number denoted by 1198770

is the expected number of secondary cases produced ina completely susceptible population by a typical infectiveindividual [25]

Using the notation in van den Driessche and Watmough[25] we have

F = [

[

0 12058811987801205881205791198780

0 0 0

0 0 0

]

]

V = [

[

] + 1205821

0 0

minus1205821

] + 1205822

0

0 minus11990131205822] + 1205823+ 1199062minus ]1199011

]

]

(10)

The reproduction number is given by 120588(119865119881minus1) and

1198770

=

1205881205821(] + 120582

3+ 1199062minus 1199011] + 120579119901

31205822) (] minus ]119901

2+ 1205824)

(] + 1205821) (] + 120582

2) (] + 120582

3+ 1199062minus 1199011]) (1199061+ ] + 120582

4minus ]1199012)

=

1205881205821(] + 120582

3+ 1199062minus 1199011] + 120579119901

31205822)

(] + 1205821) (] + 120582

2) (] + 120582

3+ 1199062minus 1199011])1198780

=

1205881205821

(] + 1205821) (] + 120582

2)

[1 +

12057911990131205822

(] + 1205823+ 1199062minus ]1199011)

] 1198780

=

1205881205821

(] + 1205821) (] + 120582

2)

times [1 +

12057911990131205822

(] + 1205823+ 1199062minus ]1199011)

] [

] minus ]1199012+ 1205824

1199061+ ] + 120582

4minus ]1199012

]

(11)

Define

1198771=

12058812058211198780

(] + 1205821) (] + 120582

2)

1198772=

1205881205791199013120582112058221198780

(] + 1205821) (] + 120582

2) (] + 120582

3+ 1199062minus ]1199011)

(12)

then we can see that 1198770= 1198771+ 1198772

Remark 2 We should note from (11) that the use both ofvaccine and treatment control to both reduce the value of 119877

0

and at the same time effects of both intervention strategieson 1198770are not simply the addition of two independent effects

rather they multiply together in order to improve the overalleffects of population level independently (Figure 2) Considerthe following

(ii) endemic equilibria 119864lowastlowast = (119878lowast 119864lowast 119868lowast 119862lowast) which hasfour different cases

Computational and Mathematical Methods in Medicine 5

0

1

2

3

4

5

6

Repr

oduc

tion

num

ber

0 01 02 03 04 05 06 07 08 09 1

Control (u1 u2)

u1 = 0 u2 = 0

u1 = 0 u2 = 1

u1 = 1 u2 = 0

u1 = 1 u2 = 1

Figure 2 Variation of different reproduction numbers by usingof two control variables (vaccination and treatment) Figure 2illustrates the impact of application of two controls (vaccination andtreatment) on different basic reproduction numbers 119877

0= 39810

when there is no vaccination and treatment (1199061= 0 119906

2= 0) The

figure shows that application of vaccination reduces1198770more rapidly

than treatment though mixed intervention strategies always workbetter to reduce the disease burden

(a) No vaccine no treatment control (1199061= 0 119906

2= 0)

119864lowastlowast

1= (119878lowast

1 119864lowast

1 119868lowast

1 119862lowast

1) where

119878lowast

1=

(] + 1205821) (] + 120582

2) (] + 120582

3minus 1199011])

1205881205821(] + 120582

3+ 12057912058221199013minus 1199011])

119864lowast

1=

120579120588 (] + 1205822) 119862lowast

1119878lowast

1

(] + 1205821) (] + 120582

2) minus 120588120582

1119878lowast

1

119868lowast

1=

1205791205881205821119862lowast

1119878lowast

1

(] + 1205821) (] + 120582

2) minus 120588120582

1119878lowast

1

119862lowast

1= (120582112058221199013(] + 120582

4minus ]1199012) 119878lowast

1(1198770minus 1))

times ( (] + 1205823minus 1199011])

times [(] + 1205824minus ]1199012) (] + 120582

2+ 1205821) + 12058211205822]

+120582112058221199013(]1199011minus ]1199012+ 1205824) )minus1

(13)

(b) No vaccine with treatment control (1199061= 0 119906

2= 0)

119864lowastlowast

2= (119878lowast

2 119864lowast

2 119868lowast

2 119862lowast

2) where

119878lowast

2=

(] + 1205821) (] + 120582

2) (] + 120582

3minus 1199011] + 1199062)

1205881205821(] + 120582

3+ 12057912058221199013minus 1199011] + 1199062)

119864lowast

2=

120579120588 (] + 1205822) 119862lowast

2119878lowast

2

(] + 1205821) (] + 120582

2) minus 120588120582

1119878lowast

2

119868lowast

2=

1205791205881205821119862lowast

2119878lowast

2

(] + 1205821) (] + 120582

2) minus 120588120582

1119878lowast

2

119862lowast

2= (120582112058221199013(] + 120582

4minus ]1199012) 119878lowast

2(1198770minus 1))

times ( (] + 1205823minus 1199011] + 1199062)

times [(] + 1205824minus ]1199012) (] + 120582

2+ 1205821) + 12058211205822]

+120582112058221199013(]1199011minus ]1199012+ 1205824) )minus1

(14)

(c) With vaccine no treatment control (1199061= 0 1199062= 0)

119864lowastlowast

3= (119878lowast

3 119864lowast

3 119868lowast

3 119862lowast

3) where

119878lowast

3=

(] + 1205821) (] + 120582

2) (] + 120582

3minus 1199011])

1205881205821(] + 120582

3+ 12057912058221199013minus 1199011])

119864lowast

3=

120579120588 (] + 1205822) 119862lowast

3119878lowast

3

(] + 1205821) (] + 120582

2) minus 120588120582

1119878lowast

3

119868lowast

3=

1205791205881205821119862lowast

3119878lowast

3

(] + 1205821) (] + 120582

2) minus 120588120582

1119878lowast

3

119862lowast

3= (120582112058221199013(] + 120582

4minus ]1199012+ 1199061) 119878lowast

3(1198770minus 1))

times ( (] + 1205823minus 1199011])

times [(] + 1205824minus ]1199012) (] + 120582

2+ 1205821) + 12058211205822]

+120582112058221199013(]1199011minus ]1199012+ 1205824) )minus1

(15)

(d) With vaccine with treatment control (1199061= 0 1199062= 0)

119864lowastlowast

4= (119878lowast

4 119864lowast

4 119868lowast

4 119862lowast

4) where

119878lowast

4=

(] + 1205821) (] + 120582

2) (] + 120582

3minus 1199011] + 1199062)

1205881205821(] + 120582

3+ 12057912058221199013minus 1199011] + 1199062)

119864lowast

4=

120579120573 (] + 1205822) 119862lowast

4119878lowast

4

(] + 1205821) (] + 120582

2) minus 120588120582

1119878lowast

4

119868lowast

4=

1205791205881205821119862lowast

4119878lowast

4

(] + 1205821) (] + 120582

2) minus 120588120582

1119878lowast

4

119862lowast

4= (120582112058221199013(] + 120582

4minus ]1199012+ 1199061) 119878lowast

4(1198770minus 1))

times ( (] + 1205823minus 1199011] + 1199062)

times [(] + 1205824minus ]1199012) (] + 120582

2+ 1205821) + 12058211205822]

+120582112058221199013(]1199011minus ]1199012+ 1205824) )minus1

(16)

Clearly 119864lowastlowast is feasible if 119862lowast gt 0 that is if 1198770gt 1 Also for

1198770= 1 the endemic equilibrium reduces to the disease-free

equilibrium and for 1198770lt 1 it becomes infeasible Hence we

may state the following theorem

Theorem 3 (i) If 1198770lt 1 then the system (2) has only one

equilibrium which is disease-free(ii) If 119877

0gt 1 then the system (2) has two equilibria one is

disease-free and the other is endemic equilibrium(iii) If 119877

0= 1 then the endemic equilibrium reduces to the

disease-free equilibrium

6 Computational and Mathematical Methods in Medicine

32 Stability Analysis In this section we will discuss thestability of different equilibria Firstly we analyse the localstability of the disease-free equilibrium

Theorem 4 If 1198770lt 1 then the disease-free equilibrium is

locally asymptotically stable

Proof The Jacobian matrix of system (2) at the disease-freeequilibrium is

J0 =[[[[

[

minus (] + 1205824minus ]1199012+ 1199061) minus (120582

4minus ]1199012) minus (120588119878

0+ 1205824minus ]1199012) minus (]119901

1+ 1205824minus ]1199012+ 120579120588119878

0)

0 minus (] + 1205821) 120588119878

01205791205881198780

0 1205821

minus (] + 1205822) 0

0 0 11990131205822

]1199011minus ] minus 120582

3minus 1199062

]]]]

]

(17)

The characteristic polynomial of 1198690given by

119875 (120582) = (120582 + 1198970) (1205823+ 11989711205822+ 1198972120582 + 1198973) (18)

where

1198970= ] + 119906

1+ 1205824minus ]1199012

1198971= 3] + 120582

1+ 1205822+ 1205823+ 1199062minus ]1199011

1198972= (] + 120582

2) (] + 120582

3+ 1199062minus ]1199011)

+ (] + 1205821) (2] + 120582

2+ 1205823+ 1199062minus ]1199011) minus 12058211205881198780

1198973= (] + 120582

1) (] + 120582

2) (] + 120582

3+ 1199062minus ]1199011)

minus 12058211205881198780(] + 120582

3+ 1199062+ 12057911990131205822minus ]1199011)

(19)

We need to verify the following two conditions

(a) 1198970 1198971 1198972 1198973gt 0

(b) 11989711198972minus 1198973gt 0

It is easy to see that 1198970 1198971gt 0 and 119897

2 1198973 11989711198972minus 1198973gt 0 if

1198770lt 1 It follows from the Routh-Hurwitz criterion that the

eigenvalues have negative real parts if 1198770lt 1 Hence the

disease-free equilibriumofmodel (2) is locally asymptoticallystable if 119877

0lt 1 and unstable if 119877

0gt 1

To discuss the properties of the endemic equilibriumpoint

Theorem 5 The endemic equilibrium point is locally asymp-totically stable if 119877

0gt 1

Proof We use Routh-Hurwitz criterion to establish the localstability of the endemic equilibrium The Jacobian matrix ofsystem (2) at endemic equilibrium is

Jlowast =[[[

[

minus (] + 1205824minus ]1199012+ 1199061+ 120588119868lowast) minus (120582

4minus ]1199012) minus (120588119878

lowast+ 1205824minus ]1199012) minus (]119901

1+ 1205824minus ]1199012+ 120588120579119878

lowast)

120588119868lowast+ 120588120579119862

lowastminus (] + 120582

1) 120588119878

lowast120588120579119878lowast

0 1205821

minus (] + 1205822) 0

0 0 11990131205822

]1199011minus ] minus 120582

3minus 1199062

]]]

]

(20)

Then the characteristic equation at 119864lowastlowast is 1205824 + 11989111205823+ 11989121205822+

1198913120582 + 1198914= 0 where

1198911= 1198861+ 1198872+ 1198883+ 1198894

1198912= 11988611198863+ 11988611198894+ 11988611198872+ 11988831198894+ 11988721198883+ 11988721198894minus 11988731198882minus 11988711198862

1198913= 119886111988831198894+ 119886111988721198883+ 119886111988721198894+ 119887211988831198894+ 119887411988821198893+ 119887111988821198863

minus 119886111988731198882minus 119887311988821198894minus 119887111988621198883minus 119887111988621198894

1198914= 1198861119887211988831198894+ 1198861119887411988821198893+ 1198871119888211988631198894minus 1198861119887311988821198894

minus 1198871119886211988831198894minus 1198871119886411988821198893

(21)

where

1198861= minus (] + 120582

4minus ]1199012+ 1199061+ 120588119868lowast)

1198862= minus (120582

4minus ]1199012) 119886

3= minus (120588119878

lowast+ 1205824minus ]1199012)

1198864= minus (]119901

1+ 1205824minus ]1199012+ 120588120579119878

lowast) 119887

1= 120588119868lowast+ 120588120579119862

lowast

1198872= minus (] + 120582

1) 119887

3= 120588119878lowast

1198874= 120588120579119878

lowast 119888

1= 0

1198882= 1205821 119888

3= minus (] + 120582

2)

1198884= 0 119889

1= 0

1198892= 0 119889

3= 11990131205822

1198894= ]1199011minus ] minus 120582

3minus 1199062

(22)

We need to verify the following three conditions

(a) 1198911 1198912 1198913 1198914gt 0

(b) 11989111198912minus 1198913gt 0

(c) 1198913(11989111198912minus 1198913) minus 1198912

11198914gt 0

Computational and Mathematical Methods in Medicine 7

It is easy to see that conditions (a) and (b) are satisfiedAfter computations we can prove that 119891

3(11989111198912minus 1198913) minus

1198912

11198914gt 0 is also validTheRouth-Hurwitz criterion and those

inequalities in (a)ndash(c) imply that the characteristic equationat 119864lowastlowast has only roots with negative real part which certifiesthe local stability of 119864lowastlowast

4 Optimal Control with Two Objectives

One of the early reasons for studying hepatitis B virus (HBV)infection is to improve the control variables and finally to putdown the infection of the population Optimal control is auseful mathematical tool that can be used to make decisionsin this case In the previous sections we have analyzed themodel with two control variables one is treatment and theother is vaccination and we consider their constant controlsthroughout the analysis But in fact these control variablesshould be time dependent In this section we consider thevaccination and treatment as time-dependent controls in acompact interval of time duration

Our goals here are to put down infection from the popu-lation by increasing the recovered individuals and reducingsusceptible exposed infected and carrier individuals in apopulation and to minimize the costs required to control thehepatitis B virus (HBV) infection by using vaccination andtreatment First of all we construct the objective functionalto be optimized as follows

119869 = int

119879

0

[1198601119878 (119905) + 119860

2119864 (119905) + 119860

3119868 (119905) + 119860

4119862 (119905)

+

1

2

(11986111199062

1(119905) + 119861

21199062

2(119905))] 119889119905

(23)

subject to

119878 (119905) = ] minus ]1199011119862 minus 120588 (119868 + 120579119862) 119878 minus ]119878 minus 119906

1119878

+ (1205824minus ]1199012) (1 minus 119878 minus 119864 minus 119868 minus 119862)

(119905) = 120588 (119868 + 120579119862) 119878 minus (] + 1205821) 119864

119868 (119905) = 1205821119864 minus (] + 120582

2) 119868

(119905) = ]1199011119862 + 119901

31205822119868 minus (] + 120582

3) 119862 minus 119906

2119862

(24)

Our object is to find (119906lowast1 119906lowast

2) such that

119869 (119906lowast

1 119906lowast

2) = min 119869 (119906

1 1199062) (25)

where

1199061(119905) 119906

2(119905) isin Γ

= 119906 (119905) | 0 le 119906 (119905) le 119906max (119905) le 1 0 le 119905 le 119879

119906 (119905) is Labesgue measurable

(26)

Here 119860119894 for 119894 = 1 4 are positive constants that

are represented to keep a balance in the size of 119878(119905) 119864(119905)119868(119905) and 119862(119905) respectively 119861

1and 119861

2 respectively are the

weights corresponding to the controls 1199061and 119906

2 119906max is the

maximum attainable value for controls (1199061max and 119906

2max)1199061max and 119906

2max will depend on the amount of resourcesavailable to implement each of the control measures [26]The 119861

1and 119861

2will depend on the relative importance of

each of the control measures in mitigating the spread of thedisease as well as the cost (human effort material resourcesetc) of implementing each of the control measures per unittime [26] Thus the terms 119861

11199062

1and 119861

21199062

2describe the costs

associated with vaccination and treatment respectively Thesquare of the control variables is taken here to remove theseverity of the side effects and overdoses of vaccination andtreatment [27]

41 Optimal Control Solution For existence of the solutionwe consider the control system (24) with initial condition

119878 (0) = 1198780 119864 (0) = 119864

0

119868 (0) = 1198680 119862 (0) = 119862

0

(27)

then we rewrite (2) in the following form

119889119909 (119905)

119889119905

= 119860119909 + 119865 (119909) (28)

where 119909(119905) = (119878(119905) 119864(119905) 119868(119905) 119862(119905)) is the vector of the statevariables and A and F(x) are defined as follows

A =

[[[[[[[

[

minus] minus 1199061minus 1205824+ ]1199012minus1205824+ ]1199012minus1205824+ ]1199012

minus1205824+ ]1199012

0 minus] minus 1205821

0 0

0 1205821

minus] minus 1205822

0

0 0 11990131205822

]1199011minus ] minus 120582

3minus 1199062

]]]]]]]

]

F (x) =[[[[[[

[

] + 120588 (119868 + 120579119862) 119878 + 1205824minus ]1199012

120588 (119868 + 120579119862) 119878

0

0

]]]]]]

]

(29)

8 Computational and Mathematical Methods in Medicine

We set

119866 (119909) = 119860119909 + 119865 (119909) (30)

The second term on the right-hand side of (30) satisfies

1003816100381610038161003816119865 (1199091) minus 119865 (119909

2)1003816100381610038161003816

le 119872 (10038161003816100381610038161198781minus 1198782

1003816100381610038161003816+10038161003816100381610038161198641minus 1198642

1003816100381610038161003816+10038161003816100381610038161198681minus 1198682

1003816100381610038161003816+10038161003816100381610038161198621minus 1198622

1003816100381610038161003816)

(31)

where the positive constant 119872 is independent of the statevariables 119909 and119872 le 1 Also we get

1003816100381610038161003816119866 (1199091) minus 119866 (119909

2)1003816100381610038161003816le 119871

10038161003816100381610038161199091minus 1199092

1003816100381610038161003816 (32)

where 119871 = max119872 119860 lt infin Therefore it follows thatthe function 119866 is uniformly Lipschitz continuous From thedefinition of the controls 119906

1(119905) and 119906

2(119905) and the restrictions

on the nonnegativeness of the state variables we see that asolution of the system (28) exists [24 28 29]

42The Lagrangian and Hamiltonian for the Control ProblemIn order to find an optimal solution pair first we shouldfind the Lagrangian and Hamiltonian for the optimal controlproblem (24) In fact the Lagrangian of problem is given by

119871 (119878 119864 119868 119862 1199061 1199062) = 119860

1119878 (119905) + 119860

2119864 (119905) + 119860

3119868 (119905)

+ 1198604119862 (119905) +

1

2

(11986111199062

1(119905) + 119861

21199062

2(119905))

(33)

We are looking for the minimal value of the LagrangianTo accomplish this we define Hamiltonian119867 for the controlproblem as follows

119867(119878 119864 119868 119862 1199061 1199062 1205721 1205722 1205723 1205724)

= 119871 + 1205721(119905)

119889119878

119889119905

+ 1205722(119905)

119889119864

119889119905

+ 1205723(119905)

119889119868

119889119905

+ 1205724(119905)

119889119862

119889119905

(34)

where 120572119894(119905) for 119894 = 1 2 3 4 are the adjoint variables and can

be determined by solving the following system of differentialequations

1(119905) = minus

120597119867

120597119878

= minus [1198601+ 1205721((]1199012minus 1205824) minus 120588 (119868 + 120579119862) minus 119906

1minus ])

+1205722120588 (119868 + 120579119862)]

2(119905) = minus

120597119867

120597119864

= minus [1198602+ 1205721(]1199012minus 1205824) + 1205722(minus] minus 120582

1) + 12057231205821]

3(119905) = minus

120597119867

120597119868

= minus [1198603+ 1205721((]1199012minus 1205824) minus 120588119878) + 120572

2120588119878

+1205723(minus] minus 120582

2) + 120572411990131205822]

4(119905) = minus

120597119867

120597119862

= minus [1198604+ 1205721(]1199012minus 1205824minus ]1199011minus 120588120579119878)

+1205722120588120579119878 + 120572

4(]1199011minus ] + 120582

3+ 1199062)]

(35)

satisfying the transversality conditions and optimality condi-tions

1205721(119879) = 120572

2(119879) = 120572

3(119879) = 120572

4(119879) = 0

120597119867

1205971199061

= 11986111199061minus 1205721119878 = 0

120597119867

1205971199062

= 11986121199062minus 1205724119862 = 0

(36)

We now state and prove the following theorem

Theorem 6 There is an optimal control (119906lowast1 119906lowast

2) such that

119869 (119878 119864 119868 119878 119906lowast

1 119906lowast

2) = min 119869 (119878 119864 119868 119878 119906

1 1199062) (37)

subject to the system of differential equation (24)

Proof Here both the control and state variables are non-negative values In this minimized problem the necessaryconvexity of the objective functional in 119906

1and 1199062are satisfied

The set of all control variables 1199061(119905) 1199062(119905) is also convex and

closed by definition The optimal system is bounded whichdetermines the compactness needed for the existence of theoptimal control In addition the integrand in the functional

int

119879

0

[1198601119878 (119905) + 119860

2119864 (119905) + 119860

3119868 (119905) + 119860

4119862 (119905)

+

1

2

(11986111199062

1(119905) + 119861

21199062

2(119905))] 119889119905

(38)

is convex on the control set Hence the theorem (the proof iscomplete)

Computational and Mathematical Methods in Medicine 9

43 Necessary and Sufficient Conditions for Optimal ControlsApplying Pontryaginrsquos maximum principle [30] on the con-structed Hamiltonian 119867 and the theorem (24) we obtainthe optimal steady-state solution and corresponding controlas follows If taking 119883 = (119878 119864 119868 119862) and 119880 = (119906

1 1199062)

then (119883lowast 119880lowast) is an optimal solution of an optimal controlproblem then we now state the following theorem

Theorem 7 The optimal control pair (119906lowast1 119906lowast

2) which mini-

mizes J over the region Γ is given by

119906lowast

1= max[0119879]

0min (1(119905) 1)

119906lowast

2= max[0119879]

0min (2(119905) 1)

(39)

where 1= 11198781198611and 2= 41198621198612and let

1 2 3 4 be

the solution of system (35)

Proof With the help of the optimality conditions we have

120597119867

1205971199061

= 11986111199061minus 1205721119878 = 0 997904rArr 119906

1=

1119878

1198611

= (1)

120597119867

1205971199062

= 11986121199062minus 1205724119862 = 0 997904rArr 119906

2=

4119862

1198612

= (2)

(40)

Using the property of the control space Γ the two controlswhich are bounded with upper and lower bounds arerespectively 1 and 0 that is

119906lowast

1=

0 if 1le 0

1

if 0 lt 1lt 1

1 if 1ge 1

(41)

This can be rewritten in compact notation

119906lowast

1= max[0119879]

0min (1(119905) 1) (42)

and similarly

119906lowast

2=

0 if 2le 0

2

if 0 lt 2lt 1

1 if 2ge 1

(43)

This can be rewritten in compact notation

119906lowast

2= max[0119879]

0min (2(119905) 1) (44)

Hence for these pair of controls (119906lowast1 119906lowast

2) we get the optimum

value of the functional 119869 given by (24)

5 Numerical Examples

Numerical solutions to the optimality system (24) arediscussed in this section We make several interestingobservation by numerically simulating (2) in the rangeof parameter values We consider the parameter set

Δ = ] 120588 120579 1205821 1205822 1205823 1205824 1199011 1199012 1199013 1198601 1198602 1198603 1198604 1198611 1198612

some of the parameters are taken from the published articlesand some are assumed with feasible values Moreover thetime interval for which the optimal control is applied istaken as 70 years also consider Ψ = 119878

0 1198640 1198680 1198620 as initial

condition for simulation of the model The main parametervalues are listed in Table 1 We compare the results having nocontrols only vaccination control only treatment controland both vaccination and treatment controls

With the parameter values in Table 1 the system asymp-totically approaches towards the equilibrium 119864

lowastlowast

1(00858

03327 04003 05349) where the basic reproduction ratio1198770= 39810 (Figure 2) For the parameters set the system

(2) has two feasible equilibria one is disease free and theother is endemic and the endemic equilibrium is locallyasymptotically stable The effect of two control measures ondisease dynamics may be understood well if we considerFigure 2 It explains how control reproduction ratio 119877

0

evolves with different rates of 1199061and 119906

2 It is seen that both

vaccination and treatment reduce the value of 1198770effectively

But an integrated control works better than either of thecontrol measures

In this section we use an iterative method to obtainresults for an optimal control problem of the proposedmodel(24) We use Runge-Kuttarsquos fourth-order procedure [23] hereto solve the optimality system consisting of eight ordinarydifferential equations having four state equations and fouradjoint equations and boundary conditions Because stateequations have initial conditions and adjoint equations haveconditions at the final time an iterative program was createdto numerically simulate solutions Given an initial guessfor the controls to compute the optimal state values theprogram solves (24) with initial conditions (27) forward intime interval [0 70]using aRungeKuttamethod of the fourthorder Resulting state values are placed in adjoint equations(35) These adjoint equations with given final conditions arethen solved backwards in time Again a fourth order RungeKutta method is employed Both state and adjoint values areused to update the control using the characterization (39)and the entire process repeats itself This iterative processterminates when current state adjoint and control valuesare sufficiently close to successive values Then we use thebackward Runge-Kutta fourth-order procedure to solve theadjoint variables in the same time interval with the help ofthe solution of the state variables transversality conditions

We have plotted susceptible exposed acute infectionindividuals and carriers individuals with andwithout controlby considering values of parameters We simulate the systemat different values of rate of 119906

1and 119906

2

From Figures 3 4 and 5 we see that after 20 years thenumber of susceptible population decreases than when thereis no control In this case most of this population tends to theinfected class Again when only treatment control is appliedthen the number of susceptible population is not muchdifferent than the population in the case having no controlBut the susceptible population differs much from these twostrategies if we apply the strategies of only vaccination controland both vaccination and treatment controls At a high rateof vaccination the sensitive population density is reduced

10 Computational and Mathematical Methods in Medicine

Table 1 Parameter values used in numerical simulations

Parameter Description Values range Reference] Birth (and death) rate 00121 [17 18]120588 Transmission rate 08ndash2049 [17]120579 Infectiousness of carriers relative to acute infections 0-1 [17]1205821

Rate of moving from exposed to acute 6 per year [16]1205822

Rate at which individuals leave the acute infection class 4 per year [16]1205823

Rate of moving from carrier to recovery 0025 per year [16]1205824

Loss of recovery rate 003ndash006 [17]1198751

Probability of infected newborns 011 [19]1198752

Probability of immune newborns 01 [19]1198753

Proportion of acute infection individuals becoming carriers 005ndash09 [17]1198601

Weight factor for susceptible individuals 0091 [31]1198602

Weight factor for exposed individuals 001 [31]1198603

Weight factor for infected individuals 004 [31]1198604

Weight factor for carrier individuals 005 [31]1198611

Weight factor for the controls 1199061

15 [31]1198612

Weight factor for the controls 1199062

27 [31]1198780

Susceptible individuals 0493 [4]1198640

Exposed individuals 00035 [4]1198680

Acute infection individuals 00035 [4]1198620

Chronic HBV carriers 0007 [4]

0 10 20 30 40 50 60 70Time (year)

Susc

eptib

le p

opul

atio

n

NV-NTNV-T

V-NTV-T

10minus4

10minus3

10minus2

10minus1

100

Figure 3 The plot shows the changes in sensitive populations(without) vaccination and (without) treatment

to a very low level initially and then it takes longer time torestore the steady-state value

From Figures 6 7 and 8 we see that the number ofexposed population increase thanwhen there is no control Inthis case most of this population tends to the acute infectedclass Again when only treatment control is applied thenthe number of exposed population is not much different

0 10 20 30 40 50 60 70Time (year)

Susc

eptib

le p

opul

atio

n

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0365 u2 = 0

u1 = 045 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

Figure 4 The plot shows the sensitivity of sensitive populations fordifferent values of control 119906

1

than the population in the case having no control But theexposed population differs much from these two strategies ifwe apply the strategies of only vaccination control and bothvaccination and treatment controls

From Figures 9 10 and 11 we see that the number ofacute infected population increases than when there is nocontrol In this case most of this population tends to thecarrier class Again when only treatment control is appliedthen the number of acute infected population is not muchdifferent than the population in the case having no controlBut the acute infected population differsmuch from these two

Computational and Mathematical Methods in Medicine 11

0 10 20 30 40 50 60 70Time (year)

Susc

eptib

le p

opul

atio

n

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Figure 5 The plot shows the sensitivity of sensitive populations fordifferent values of control 119906

2

Expo

sed

popu

latio

n

NV-NTNV-T

V-NTV-T

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 6 The plot shows the changes in exposed populations(without) vaccination and (without) treatment

strategies if we apply the strategies of only vaccination controland both vaccination and treatment controls

Again from Figures 12 13 and 14 we see that thenumber of carrier population increases than when there isno control We see that the application of only vaccinationcontrol gives better result than the application of no controlAgain application of treatment control would give betterresult than the application of vaccination control since thetreatment control is better than the vaccination control whilethe application of both vaccination and treatment controlsgive the best result as in this case the number of carrierpopulation would be the least in number

Expo

sed

popu

latio

n

u1 = 0 u2 = 0

u1 = 03635 u2 = 0

u1 = 045 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 7 The plot shows the sensitivity of exposed populations fordifferent values of control 119906

1

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Expo

sed

popu

latio

n

Figure 8 The plot shows the sensitivity of exposed populations fordifferent values of control 119906

2

Finally the effect of two control measures on diseasedynamics may be understood well if we consider Figures 1ndash14 15 16 17 and 18 Since our main purpose is to reducethe number of sensitive exposed acute infection and carrierindividuals therefore numerical simulation results show thatoptimal combination of vaccination and treatment is themosteffective way to control hepatitis B virus infection

12 Computational and Mathematical Methods in MedicineAc

ute i

nfec

tion

popu

latio

n

NV-NTNV-T

V-NTV-T

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 9The plot shows the changes in acute infection populations(without) vaccination and (without) treatment

Acut

e inf

ectio

n po

pulat

ion

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0365 u2 = 0

u1 = 045 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

Figure 10 The plot shows the sensitivity of acute infection popula-tions for different values of control 119906

1

6 Conclusion and Suggestions

In this paper we propose an S-119864-119868-119862-119877model of hepatitis Bvirus infection with two controls vaccination and treatmentFirst we analyze the dynamic behavior of the system for con-stant controls In the constant controls case we calculate thebasic reproduction number and investigate the existence andstability of equilibriaThere are two nonnegative equilibria of

Acut

e inf

ectio

n po

pulat

ion

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Figure 11 The plot shows the sensitivity of acute infection popula-tions for different values of control 119906

2

Carr

ier p

opul

atio

n

NV-NTNV-T

V-NTV-T

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 12 The plot shows the changes in carrier populations(without) vaccination and (without) treatment

the system namely the disease-free and endemicWe see thatthe disease-free equilibriumwhich always exists and is locallyasymptotically stable if 119877

0lt 1 and endemic equilibrium

which exists and is locally asymptotically stable if 1198770gt 1

After investigating the dynamic behavior of the systemwith constant controls we formulate an optimal controlproblem if the controls become time dependent and solve itby using Pontryaginrsquos maximum principle Different possiblecombinations of controls are used and their effectiveness iscompared by simulation works Also from the numericalresults it is very clear that a combination of mixed controlmeasures respond better than any other independent control

Computational and Mathematical Methods in Medicine 13Ca

rrie

r pop

ulat

ion

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0365 u2 = 0

u1 = 046 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

Figure 13 The plot shows the sensitivity of carrier populations fordifferent values of control 119906

1

Carr

ier p

opul

atio

n

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Figure 14 The plot shows the sensitivity of carrier populations fordifferent values of control 119906

2

There is still a tremendous amount of work needed tobe done in this area Parameters are rarely constant becausethey depend on environmental conditions We do not knowhowever the detailed relationship between these parametersand environmental conditions There may be a time lag as asusceptible populationmay take some time to be infected andalso a susceptible population may take some time to immune

Popu

latio

n (N

V-N

T)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 15 The plot shows the changes in total populations withoutvaccination and without treatment

Popu

latio

n (V

-NT)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 16 The plot shows the changes in total populations withvaccination and without treatment

Popu

latio

n (N

V-T)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 17 The plot shows the changes in total populations withoutvaccination and with treatment

14 Computational and Mathematical Methods in MedicinePo

pulat

ion

(V-T

)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 18 The plot shows the changes in total populations withvaccination and treatment

after vaccination We leave all these possible extensions forthe future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] WHO Hepatitis B Fact Sheet No 204 The World HealthOrganisation Geneva Switzerland 2013 httpwwwwhointmediacentrefactsheetsfs204en

[2] CanadianCentre forOccupationalHealth and Safety ldquoHepatitisBrdquo httpwwwccohscaoshanswersdiseaseshepatitis bhtml

[3] ldquoHealthcare stumbling in RIrsquos Hepatitis fightrdquoThe Jakarta PostJanuary 2011

[4] G F Medley N A Lindop W J Edmunds and D JNokes ldquoHepatitis-B virus endemicity heterogeneity catas-trophic dynamics and controlrdquo Nature Medicine vol 7 no 5pp 619ndash624 2001

[5] J Mann and M Roberts ldquoModelling the epidemiology ofhepatitis B in New Zealandrdquo Journal of Theoretical Biology vol269 no 1 pp 266ndash272 2011

[6] ldquoHepatitis B (HBV)rdquo httpkidshealthorgteensexual healthstdsstd hepatitishtml

[7] CDC ldquoHepatitis B virus a comprehensive strategy for elim-inating transmission in the United States through universalchildhood vaccination Recommendations of the Immuniza-tion Practices Advisory Committee (ACIP)rdquo MMWR Recom-mendations and Reports vol 40(RR-13) pp 1ndash25 1991

[8] M K Libbus and L M Phillips ldquoPublic health management ofperinatal hepatitis B virusrdquo Public Health Nursing vol 26 no 4pp 353ndash361 2009

[9] F B Hollinger and D T Lau ldquoHepatitis B the pathway torecovery through treatmentrdquo Gastroenterology Clinics of NorthAmerica vol 35 no 4 pp 895ndash931 2006

[10] C-L Lai andM-F Yuen ldquoThe natural history and treatment ofchronic hepatitis B a critical evaluation of standard treatment

criteria and end pointsrdquoAnnals of InternalMedicine vol 147 no1 pp 58ndash61 2007

[11] R M Anderson and R M May Infectious Disease of HumansDynamics and Control Oxford University Press Oxford UK1991

[12] S Thornley C Bullen and M Roberts ldquoHepatitis B in ahigh prevalence New Zealand population a mathematicalmodel applied to infection control policyrdquo Journal ofTheoreticalBiology vol 254 no 3 pp 599ndash603 2008

[13] S-J Zhao Z-Y Xu and Y Lu ldquoA mathematical model ofhepatitis B virus transmission and its application for vaccinationstrategy inChinardquo International Journal of Epidemiology vol 29no 4 pp 744ndash752 2000

[14] K Wang W Wang and S Song ldquoDynamics of an HBV modelwith diffusion and delayrdquo Journal ofTheoretical Biology vol 253no 1 pp 36ndash44 2008

[15] R Xu andZMa ldquoAnHBVmodelwith diffusion and time delayrdquoJournal of Theoretical Biology vol 257 no 3 pp 499ndash509 2009

[16] L Zou W Zhang and S Ruan ldquoModeling the transmissiondynamics and control of hepatitis B virus in Chinardquo Journal ofTheoretical Biology vol 262 no 2 pp 330ndash338 2010

[17] J Pang J-A Cui and X Zhou ldquoDynamical behavior of ahepatitis B virus transmission model with vaccinationrdquo Journalof Theoretical Biology vol 265 no 4 pp 572ndash578 2010

[18] S Zhang and Y Zhou ldquoThe analysis and application of an HBVmodelrdquoAppliedMathematicalModelling vol 36 no 3 pp 1302ndash1312 2012

[19] S Bhattacharyya and S Ghosh ldquoOptimal control of verticallytransmitted diseaserdquoComputational andMathematicalMethodsin Medicine vol 11 no 4 pp 369ndash387 2010

[20] T K Kar andA Batabyal ldquoStability analysis and optimal controlof an SIR epidemic model with vaccinationrdquo Biosystems vol104 no 2-3 pp 127ndash135 2011

[21] T-K Kar and S Jana ldquoA theoretical study on mathematicalmodelling of an infectious disease with application of optimalcontrolrdquo Biosystems vol 111 no 1 pp 37ndash50 2013

[22] M J Keeling and P Rohani Modeling Infectious Diseases inHumans and Animals Princeton University Press PrincetonNJ USA 2008

[23] E Jung S Lenhart and Z Feng ldquoOptimal control of treatmentsin a two-strain tuberculosis modelrdquo Discrete and ContinuousDynamical Systems B vol 2 no 4 pp 473ndash482 2002

[24] G Birkhoff and G-C Rota Ordinary Differential EquationsJohn Wiley amp Sons New York NY USA 4th edition 1989

[25] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[26] T T Yusuf and F Benyah ldquoOptimal control of vaccination andtreatment for an SIR epidemiological modelrdquo World Journal ofModelling and Simulation vol 8 no 3 pp 194ndash204 2008

[27] H-R Joshi S Lenhart M Y Li and L Wang ldquoOptimalcontrol methods applied to disease modelsrdquo in MathematicalStudies on Human Disease Dynamics vol 410 of ContemporaryMathematics pp 187ndash207 American Mathematical SocietyProvidence RI USA 2006

[28] G Zaman Y H Kang and I H Jung ldquoStability analysis andoptimal vaccination of an SIR epidemic modelrdquo Biosystems vol93 no 3 pp 240ndash249 2008

[29] D Iacoviello and N Stasiob ldquoOptimal control for SIRCepidemic outbreakrdquo Computer Methods and Programs inBiomedicine vol 110 no 3 pp 333ndash342 2013

Computational and Mathematical Methods in Medicine 15

[30] L S Pontryagin V G Boltyanskii R V Gamkrelidze and EF Mishchenko The Mathematical Theory of Optimal ProcessesJohn Wiley amp Sons New York NY USA 1962

[31] WHO Hepatitis B Fact Sheet No 204 The World HealthOrganisation Geneva Switzerland 2000 httpwwwwhointmediacentrefactsheetsfs204en

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

4 Computational and Mathematical Methods in Medicine

that is

119889119883

119889119905

+ (] + 1205824minus ]1199012)119883 le ] + 120582

4 (4)

Now integrating both sides of the above inequality and usingthe theory of differential inequality [21 24] we get

119883(119905) le 119890minus(]+120582

4minus]1199012)119905[

] + 1205824

] + 1205824minus ]1199012

119890(]+1205824minus]1199012)119905+ 119896] (5)

where 119896 is a constant and letting 119905 rarr infin we have

119878 + 119864 + 119868 + 119862 le

] + 1205824

] + 1205824minus ]1199012

119878 (119905) = ] minus ]1199011119862 minus 120588 (119868 + 120579119862) 119878 minus ]119878 minus 119906

1119878

+ (1205824minus ]1199012) (1 minus 119878 minus 119864 minus 119868 minus 119862)

(6)

then

119878 le

] + 1205824

] + 1205824+ 1199061minus ]1199012

(7)

then

Π = (119878 119864 119868 119862) isin R4

+| 119878 le

] + 1205824

] + 1199061+ 1205824minus ]1199012

119878 + 119864 + 119868 + 119862 le

] + 1205824

] + 1205824minus ]1199012

(8)

is positively invariant Hence the system is mathematicallywell-posed There for initial starting point 119909 isin R4

+ the

trajectory lies inΠTherefore wewill focus our attention onlyon the region Π

3 Analysis of the System forConstant Controls

In this section we assume that the control parameters 1199061(119905)

and 1199062(119905) are constant functions

31 Equilibrium Points and Basic Reproduction Number Wewill discuss the existence of all the possible equilibriumpointsof the system (2) We see that system (2) has two possiblenonnegative equilibria

(i) Disease-free equilibrium point1198640= (1198780 0 0 0) where

1198780=

] minus ]1199012+ 1205824

1199061+ ] + 120582

4minus ]1199012

(9)

it is always feasible In the absence of vaccination this isreduced to the equilibrium (1 0 0 0)

Definition 1 The basic reproduction number denoted by 1198770

is the expected number of secondary cases produced ina completely susceptible population by a typical infectiveindividual [25]

Using the notation in van den Driessche and Watmough[25] we have

F = [

[

0 12058811987801205881205791198780

0 0 0

0 0 0

]

]

V = [

[

] + 1205821

0 0

minus1205821

] + 1205822

0

0 minus11990131205822] + 1205823+ 1199062minus ]1199011

]

]

(10)

The reproduction number is given by 120588(119865119881minus1) and

1198770

=

1205881205821(] + 120582

3+ 1199062minus 1199011] + 120579119901

31205822) (] minus ]119901

2+ 1205824)

(] + 1205821) (] + 120582

2) (] + 120582

3+ 1199062minus 1199011]) (1199061+ ] + 120582

4minus ]1199012)

=

1205881205821(] + 120582

3+ 1199062minus 1199011] + 120579119901

31205822)

(] + 1205821) (] + 120582

2) (] + 120582

3+ 1199062minus 1199011])1198780

=

1205881205821

(] + 1205821) (] + 120582

2)

[1 +

12057911990131205822

(] + 1205823+ 1199062minus ]1199011)

] 1198780

=

1205881205821

(] + 1205821) (] + 120582

2)

times [1 +

12057911990131205822

(] + 1205823+ 1199062minus ]1199011)

] [

] minus ]1199012+ 1205824

1199061+ ] + 120582

4minus ]1199012

]

(11)

Define

1198771=

12058812058211198780

(] + 1205821) (] + 120582

2)

1198772=

1205881205791199013120582112058221198780

(] + 1205821) (] + 120582

2) (] + 120582

3+ 1199062minus ]1199011)

(12)

then we can see that 1198770= 1198771+ 1198772

Remark 2 We should note from (11) that the use both ofvaccine and treatment control to both reduce the value of 119877

0

and at the same time effects of both intervention strategieson 1198770are not simply the addition of two independent effects

rather they multiply together in order to improve the overalleffects of population level independently (Figure 2) Considerthe following

(ii) endemic equilibria 119864lowastlowast = (119878lowast 119864lowast 119868lowast 119862lowast) which hasfour different cases

Computational and Mathematical Methods in Medicine 5

0

1

2

3

4

5

6

Repr

oduc

tion

num

ber

0 01 02 03 04 05 06 07 08 09 1

Control (u1 u2)

u1 = 0 u2 = 0

u1 = 0 u2 = 1

u1 = 1 u2 = 0

u1 = 1 u2 = 1

Figure 2 Variation of different reproduction numbers by usingof two control variables (vaccination and treatment) Figure 2illustrates the impact of application of two controls (vaccination andtreatment) on different basic reproduction numbers 119877

0= 39810

when there is no vaccination and treatment (1199061= 0 119906

2= 0) The

figure shows that application of vaccination reduces1198770more rapidly

than treatment though mixed intervention strategies always workbetter to reduce the disease burden

(a) No vaccine no treatment control (1199061= 0 119906

2= 0)

119864lowastlowast

1= (119878lowast

1 119864lowast

1 119868lowast

1 119862lowast

1) where

119878lowast

1=

(] + 1205821) (] + 120582

2) (] + 120582

3minus 1199011])

1205881205821(] + 120582

3+ 12057912058221199013minus 1199011])

119864lowast

1=

120579120588 (] + 1205822) 119862lowast

1119878lowast

1

(] + 1205821) (] + 120582

2) minus 120588120582

1119878lowast

1

119868lowast

1=

1205791205881205821119862lowast

1119878lowast

1

(] + 1205821) (] + 120582

2) minus 120588120582

1119878lowast

1

119862lowast

1= (120582112058221199013(] + 120582

4minus ]1199012) 119878lowast

1(1198770minus 1))

times ( (] + 1205823minus 1199011])

times [(] + 1205824minus ]1199012) (] + 120582

2+ 1205821) + 12058211205822]

+120582112058221199013(]1199011minus ]1199012+ 1205824) )minus1

(13)

(b) No vaccine with treatment control (1199061= 0 119906

2= 0)

119864lowastlowast

2= (119878lowast

2 119864lowast

2 119868lowast

2 119862lowast

2) where

119878lowast

2=

(] + 1205821) (] + 120582

2) (] + 120582

3minus 1199011] + 1199062)

1205881205821(] + 120582

3+ 12057912058221199013minus 1199011] + 1199062)

119864lowast

2=

120579120588 (] + 1205822) 119862lowast

2119878lowast

2

(] + 1205821) (] + 120582

2) minus 120588120582

1119878lowast

2

119868lowast

2=

1205791205881205821119862lowast

2119878lowast

2

(] + 1205821) (] + 120582

2) minus 120588120582

1119878lowast

2

119862lowast

2= (120582112058221199013(] + 120582

4minus ]1199012) 119878lowast

2(1198770minus 1))

times ( (] + 1205823minus 1199011] + 1199062)

times [(] + 1205824minus ]1199012) (] + 120582

2+ 1205821) + 12058211205822]

+120582112058221199013(]1199011minus ]1199012+ 1205824) )minus1

(14)

(c) With vaccine no treatment control (1199061= 0 1199062= 0)

119864lowastlowast

3= (119878lowast

3 119864lowast

3 119868lowast

3 119862lowast

3) where

119878lowast

3=

(] + 1205821) (] + 120582

2) (] + 120582

3minus 1199011])

1205881205821(] + 120582

3+ 12057912058221199013minus 1199011])

119864lowast

3=

120579120588 (] + 1205822) 119862lowast

3119878lowast

3

(] + 1205821) (] + 120582

2) minus 120588120582

1119878lowast

3

119868lowast

3=

1205791205881205821119862lowast

3119878lowast

3

(] + 1205821) (] + 120582

2) minus 120588120582

1119878lowast

3

119862lowast

3= (120582112058221199013(] + 120582

4minus ]1199012+ 1199061) 119878lowast

3(1198770minus 1))

times ( (] + 1205823minus 1199011])

times [(] + 1205824minus ]1199012) (] + 120582

2+ 1205821) + 12058211205822]

+120582112058221199013(]1199011minus ]1199012+ 1205824) )minus1

(15)

(d) With vaccine with treatment control (1199061= 0 1199062= 0)

119864lowastlowast

4= (119878lowast

4 119864lowast

4 119868lowast

4 119862lowast

4) where

119878lowast

4=

(] + 1205821) (] + 120582

2) (] + 120582

3minus 1199011] + 1199062)

1205881205821(] + 120582

3+ 12057912058221199013minus 1199011] + 1199062)

119864lowast

4=

120579120573 (] + 1205822) 119862lowast

4119878lowast

4

(] + 1205821) (] + 120582

2) minus 120588120582

1119878lowast

4

119868lowast

4=

1205791205881205821119862lowast

4119878lowast

4

(] + 1205821) (] + 120582

2) minus 120588120582

1119878lowast

4

119862lowast

4= (120582112058221199013(] + 120582

4minus ]1199012+ 1199061) 119878lowast

4(1198770minus 1))

times ( (] + 1205823minus 1199011] + 1199062)

times [(] + 1205824minus ]1199012) (] + 120582

2+ 1205821) + 12058211205822]

+120582112058221199013(]1199011minus ]1199012+ 1205824) )minus1

(16)

Clearly 119864lowastlowast is feasible if 119862lowast gt 0 that is if 1198770gt 1 Also for

1198770= 1 the endemic equilibrium reduces to the disease-free

equilibrium and for 1198770lt 1 it becomes infeasible Hence we

may state the following theorem

Theorem 3 (i) If 1198770lt 1 then the system (2) has only one

equilibrium which is disease-free(ii) If 119877

0gt 1 then the system (2) has two equilibria one is

disease-free and the other is endemic equilibrium(iii) If 119877

0= 1 then the endemic equilibrium reduces to the

disease-free equilibrium

6 Computational and Mathematical Methods in Medicine

32 Stability Analysis In this section we will discuss thestability of different equilibria Firstly we analyse the localstability of the disease-free equilibrium

Theorem 4 If 1198770lt 1 then the disease-free equilibrium is

locally asymptotically stable

Proof The Jacobian matrix of system (2) at the disease-freeequilibrium is

J0 =[[[[

[

minus (] + 1205824minus ]1199012+ 1199061) minus (120582

4minus ]1199012) minus (120588119878

0+ 1205824minus ]1199012) minus (]119901

1+ 1205824minus ]1199012+ 120579120588119878

0)

0 minus (] + 1205821) 120588119878

01205791205881198780

0 1205821

minus (] + 1205822) 0

0 0 11990131205822

]1199011minus ] minus 120582

3minus 1199062

]]]]

]

(17)

The characteristic polynomial of 1198690given by

119875 (120582) = (120582 + 1198970) (1205823+ 11989711205822+ 1198972120582 + 1198973) (18)

where

1198970= ] + 119906

1+ 1205824minus ]1199012

1198971= 3] + 120582

1+ 1205822+ 1205823+ 1199062minus ]1199011

1198972= (] + 120582

2) (] + 120582

3+ 1199062minus ]1199011)

+ (] + 1205821) (2] + 120582

2+ 1205823+ 1199062minus ]1199011) minus 12058211205881198780

1198973= (] + 120582

1) (] + 120582

2) (] + 120582

3+ 1199062minus ]1199011)

minus 12058211205881198780(] + 120582

3+ 1199062+ 12057911990131205822minus ]1199011)

(19)

We need to verify the following two conditions

(a) 1198970 1198971 1198972 1198973gt 0

(b) 11989711198972minus 1198973gt 0

It is easy to see that 1198970 1198971gt 0 and 119897

2 1198973 11989711198972minus 1198973gt 0 if

1198770lt 1 It follows from the Routh-Hurwitz criterion that the

eigenvalues have negative real parts if 1198770lt 1 Hence the

disease-free equilibriumofmodel (2) is locally asymptoticallystable if 119877

0lt 1 and unstable if 119877

0gt 1

To discuss the properties of the endemic equilibriumpoint

Theorem 5 The endemic equilibrium point is locally asymp-totically stable if 119877

0gt 1

Proof We use Routh-Hurwitz criterion to establish the localstability of the endemic equilibrium The Jacobian matrix ofsystem (2) at endemic equilibrium is

Jlowast =[[[

[

minus (] + 1205824minus ]1199012+ 1199061+ 120588119868lowast) minus (120582

4minus ]1199012) minus (120588119878

lowast+ 1205824minus ]1199012) minus (]119901

1+ 1205824minus ]1199012+ 120588120579119878

lowast)

120588119868lowast+ 120588120579119862

lowastminus (] + 120582

1) 120588119878

lowast120588120579119878lowast

0 1205821

minus (] + 1205822) 0

0 0 11990131205822

]1199011minus ] minus 120582

3minus 1199062

]]]

]

(20)

Then the characteristic equation at 119864lowastlowast is 1205824 + 11989111205823+ 11989121205822+

1198913120582 + 1198914= 0 where

1198911= 1198861+ 1198872+ 1198883+ 1198894

1198912= 11988611198863+ 11988611198894+ 11988611198872+ 11988831198894+ 11988721198883+ 11988721198894minus 11988731198882minus 11988711198862

1198913= 119886111988831198894+ 119886111988721198883+ 119886111988721198894+ 119887211988831198894+ 119887411988821198893+ 119887111988821198863

minus 119886111988731198882minus 119887311988821198894minus 119887111988621198883minus 119887111988621198894

1198914= 1198861119887211988831198894+ 1198861119887411988821198893+ 1198871119888211988631198894minus 1198861119887311988821198894

minus 1198871119886211988831198894minus 1198871119886411988821198893

(21)

where

1198861= minus (] + 120582

4minus ]1199012+ 1199061+ 120588119868lowast)

1198862= minus (120582

4minus ]1199012) 119886

3= minus (120588119878

lowast+ 1205824minus ]1199012)

1198864= minus (]119901

1+ 1205824minus ]1199012+ 120588120579119878

lowast) 119887

1= 120588119868lowast+ 120588120579119862

lowast

1198872= minus (] + 120582

1) 119887

3= 120588119878lowast

1198874= 120588120579119878

lowast 119888

1= 0

1198882= 1205821 119888

3= minus (] + 120582

2)

1198884= 0 119889

1= 0

1198892= 0 119889

3= 11990131205822

1198894= ]1199011minus ] minus 120582

3minus 1199062

(22)

We need to verify the following three conditions

(a) 1198911 1198912 1198913 1198914gt 0

(b) 11989111198912minus 1198913gt 0

(c) 1198913(11989111198912minus 1198913) minus 1198912

11198914gt 0

Computational and Mathematical Methods in Medicine 7

It is easy to see that conditions (a) and (b) are satisfiedAfter computations we can prove that 119891

3(11989111198912minus 1198913) minus

1198912

11198914gt 0 is also validTheRouth-Hurwitz criterion and those

inequalities in (a)ndash(c) imply that the characteristic equationat 119864lowastlowast has only roots with negative real part which certifiesthe local stability of 119864lowastlowast

4 Optimal Control with Two Objectives

One of the early reasons for studying hepatitis B virus (HBV)infection is to improve the control variables and finally to putdown the infection of the population Optimal control is auseful mathematical tool that can be used to make decisionsin this case In the previous sections we have analyzed themodel with two control variables one is treatment and theother is vaccination and we consider their constant controlsthroughout the analysis But in fact these control variablesshould be time dependent In this section we consider thevaccination and treatment as time-dependent controls in acompact interval of time duration

Our goals here are to put down infection from the popu-lation by increasing the recovered individuals and reducingsusceptible exposed infected and carrier individuals in apopulation and to minimize the costs required to control thehepatitis B virus (HBV) infection by using vaccination andtreatment First of all we construct the objective functionalto be optimized as follows

119869 = int

119879

0

[1198601119878 (119905) + 119860

2119864 (119905) + 119860

3119868 (119905) + 119860

4119862 (119905)

+

1

2

(11986111199062

1(119905) + 119861

21199062

2(119905))] 119889119905

(23)

subject to

119878 (119905) = ] minus ]1199011119862 minus 120588 (119868 + 120579119862) 119878 minus ]119878 minus 119906

1119878

+ (1205824minus ]1199012) (1 minus 119878 minus 119864 minus 119868 minus 119862)

(119905) = 120588 (119868 + 120579119862) 119878 minus (] + 1205821) 119864

119868 (119905) = 1205821119864 minus (] + 120582

2) 119868

(119905) = ]1199011119862 + 119901

31205822119868 minus (] + 120582

3) 119862 minus 119906

2119862

(24)

Our object is to find (119906lowast1 119906lowast

2) such that

119869 (119906lowast

1 119906lowast

2) = min 119869 (119906

1 1199062) (25)

where

1199061(119905) 119906

2(119905) isin Γ

= 119906 (119905) | 0 le 119906 (119905) le 119906max (119905) le 1 0 le 119905 le 119879

119906 (119905) is Labesgue measurable

(26)

Here 119860119894 for 119894 = 1 4 are positive constants that

are represented to keep a balance in the size of 119878(119905) 119864(119905)119868(119905) and 119862(119905) respectively 119861

1and 119861

2 respectively are the

weights corresponding to the controls 1199061and 119906

2 119906max is the

maximum attainable value for controls (1199061max and 119906

2max)1199061max and 119906

2max will depend on the amount of resourcesavailable to implement each of the control measures [26]The 119861

1and 119861

2will depend on the relative importance of

each of the control measures in mitigating the spread of thedisease as well as the cost (human effort material resourcesetc) of implementing each of the control measures per unittime [26] Thus the terms 119861

11199062

1and 119861

21199062

2describe the costs

associated with vaccination and treatment respectively Thesquare of the control variables is taken here to remove theseverity of the side effects and overdoses of vaccination andtreatment [27]

41 Optimal Control Solution For existence of the solutionwe consider the control system (24) with initial condition

119878 (0) = 1198780 119864 (0) = 119864

0

119868 (0) = 1198680 119862 (0) = 119862

0

(27)

then we rewrite (2) in the following form

119889119909 (119905)

119889119905

= 119860119909 + 119865 (119909) (28)

where 119909(119905) = (119878(119905) 119864(119905) 119868(119905) 119862(119905)) is the vector of the statevariables and A and F(x) are defined as follows

A =

[[[[[[[

[

minus] minus 1199061minus 1205824+ ]1199012minus1205824+ ]1199012minus1205824+ ]1199012

minus1205824+ ]1199012

0 minus] minus 1205821

0 0

0 1205821

minus] minus 1205822

0

0 0 11990131205822

]1199011minus ] minus 120582

3minus 1199062

]]]]]]]

]

F (x) =[[[[[[

[

] + 120588 (119868 + 120579119862) 119878 + 1205824minus ]1199012

120588 (119868 + 120579119862) 119878

0

0

]]]]]]

]

(29)

8 Computational and Mathematical Methods in Medicine

We set

119866 (119909) = 119860119909 + 119865 (119909) (30)

The second term on the right-hand side of (30) satisfies

1003816100381610038161003816119865 (1199091) minus 119865 (119909

2)1003816100381610038161003816

le 119872 (10038161003816100381610038161198781minus 1198782

1003816100381610038161003816+10038161003816100381610038161198641minus 1198642

1003816100381610038161003816+10038161003816100381610038161198681minus 1198682

1003816100381610038161003816+10038161003816100381610038161198621minus 1198622

1003816100381610038161003816)

(31)

where the positive constant 119872 is independent of the statevariables 119909 and119872 le 1 Also we get

1003816100381610038161003816119866 (1199091) minus 119866 (119909

2)1003816100381610038161003816le 119871

10038161003816100381610038161199091minus 1199092

1003816100381610038161003816 (32)

where 119871 = max119872 119860 lt infin Therefore it follows thatthe function 119866 is uniformly Lipschitz continuous From thedefinition of the controls 119906

1(119905) and 119906

2(119905) and the restrictions

on the nonnegativeness of the state variables we see that asolution of the system (28) exists [24 28 29]

42The Lagrangian and Hamiltonian for the Control ProblemIn order to find an optimal solution pair first we shouldfind the Lagrangian and Hamiltonian for the optimal controlproblem (24) In fact the Lagrangian of problem is given by

119871 (119878 119864 119868 119862 1199061 1199062) = 119860

1119878 (119905) + 119860

2119864 (119905) + 119860

3119868 (119905)

+ 1198604119862 (119905) +

1

2

(11986111199062

1(119905) + 119861

21199062

2(119905))

(33)

We are looking for the minimal value of the LagrangianTo accomplish this we define Hamiltonian119867 for the controlproblem as follows

119867(119878 119864 119868 119862 1199061 1199062 1205721 1205722 1205723 1205724)

= 119871 + 1205721(119905)

119889119878

119889119905

+ 1205722(119905)

119889119864

119889119905

+ 1205723(119905)

119889119868

119889119905

+ 1205724(119905)

119889119862

119889119905

(34)

where 120572119894(119905) for 119894 = 1 2 3 4 are the adjoint variables and can

be determined by solving the following system of differentialequations

1(119905) = minus

120597119867

120597119878

= minus [1198601+ 1205721((]1199012minus 1205824) minus 120588 (119868 + 120579119862) minus 119906

1minus ])

+1205722120588 (119868 + 120579119862)]

2(119905) = minus

120597119867

120597119864

= minus [1198602+ 1205721(]1199012minus 1205824) + 1205722(minus] minus 120582

1) + 12057231205821]

3(119905) = minus

120597119867

120597119868

= minus [1198603+ 1205721((]1199012minus 1205824) minus 120588119878) + 120572

2120588119878

+1205723(minus] minus 120582

2) + 120572411990131205822]

4(119905) = minus

120597119867

120597119862

= minus [1198604+ 1205721(]1199012minus 1205824minus ]1199011minus 120588120579119878)

+1205722120588120579119878 + 120572

4(]1199011minus ] + 120582

3+ 1199062)]

(35)

satisfying the transversality conditions and optimality condi-tions

1205721(119879) = 120572

2(119879) = 120572

3(119879) = 120572

4(119879) = 0

120597119867

1205971199061

= 11986111199061minus 1205721119878 = 0

120597119867

1205971199062

= 11986121199062minus 1205724119862 = 0

(36)

We now state and prove the following theorem

Theorem 6 There is an optimal control (119906lowast1 119906lowast

2) such that

119869 (119878 119864 119868 119878 119906lowast

1 119906lowast

2) = min 119869 (119878 119864 119868 119878 119906

1 1199062) (37)

subject to the system of differential equation (24)

Proof Here both the control and state variables are non-negative values In this minimized problem the necessaryconvexity of the objective functional in 119906

1and 1199062are satisfied

The set of all control variables 1199061(119905) 1199062(119905) is also convex and

closed by definition The optimal system is bounded whichdetermines the compactness needed for the existence of theoptimal control In addition the integrand in the functional

int

119879

0

[1198601119878 (119905) + 119860

2119864 (119905) + 119860

3119868 (119905) + 119860

4119862 (119905)

+

1

2

(11986111199062

1(119905) + 119861

21199062

2(119905))] 119889119905

(38)

is convex on the control set Hence the theorem (the proof iscomplete)

Computational and Mathematical Methods in Medicine 9

43 Necessary and Sufficient Conditions for Optimal ControlsApplying Pontryaginrsquos maximum principle [30] on the con-structed Hamiltonian 119867 and the theorem (24) we obtainthe optimal steady-state solution and corresponding controlas follows If taking 119883 = (119878 119864 119868 119862) and 119880 = (119906

1 1199062)

then (119883lowast 119880lowast) is an optimal solution of an optimal controlproblem then we now state the following theorem

Theorem 7 The optimal control pair (119906lowast1 119906lowast

2) which mini-

mizes J over the region Γ is given by

119906lowast

1= max[0119879]

0min (1(119905) 1)

119906lowast

2= max[0119879]

0min (2(119905) 1)

(39)

where 1= 11198781198611and 2= 41198621198612and let

1 2 3 4 be

the solution of system (35)

Proof With the help of the optimality conditions we have

120597119867

1205971199061

= 11986111199061minus 1205721119878 = 0 997904rArr 119906

1=

1119878

1198611

= (1)

120597119867

1205971199062

= 11986121199062minus 1205724119862 = 0 997904rArr 119906

2=

4119862

1198612

= (2)

(40)

Using the property of the control space Γ the two controlswhich are bounded with upper and lower bounds arerespectively 1 and 0 that is

119906lowast

1=

0 if 1le 0

1

if 0 lt 1lt 1

1 if 1ge 1

(41)

This can be rewritten in compact notation

119906lowast

1= max[0119879]

0min (1(119905) 1) (42)

and similarly

119906lowast

2=

0 if 2le 0

2

if 0 lt 2lt 1

1 if 2ge 1

(43)

This can be rewritten in compact notation

119906lowast

2= max[0119879]

0min (2(119905) 1) (44)

Hence for these pair of controls (119906lowast1 119906lowast

2) we get the optimum

value of the functional 119869 given by (24)

5 Numerical Examples

Numerical solutions to the optimality system (24) arediscussed in this section We make several interestingobservation by numerically simulating (2) in the rangeof parameter values We consider the parameter set

Δ = ] 120588 120579 1205821 1205822 1205823 1205824 1199011 1199012 1199013 1198601 1198602 1198603 1198604 1198611 1198612

some of the parameters are taken from the published articlesand some are assumed with feasible values Moreover thetime interval for which the optimal control is applied istaken as 70 years also consider Ψ = 119878

0 1198640 1198680 1198620 as initial

condition for simulation of the model The main parametervalues are listed in Table 1 We compare the results having nocontrols only vaccination control only treatment controland both vaccination and treatment controls

With the parameter values in Table 1 the system asymp-totically approaches towards the equilibrium 119864

lowastlowast

1(00858

03327 04003 05349) where the basic reproduction ratio1198770= 39810 (Figure 2) For the parameters set the system

(2) has two feasible equilibria one is disease free and theother is endemic and the endemic equilibrium is locallyasymptotically stable The effect of two control measures ondisease dynamics may be understood well if we considerFigure 2 It explains how control reproduction ratio 119877

0

evolves with different rates of 1199061and 119906

2 It is seen that both

vaccination and treatment reduce the value of 1198770effectively

But an integrated control works better than either of thecontrol measures

In this section we use an iterative method to obtainresults for an optimal control problem of the proposedmodel(24) We use Runge-Kuttarsquos fourth-order procedure [23] hereto solve the optimality system consisting of eight ordinarydifferential equations having four state equations and fouradjoint equations and boundary conditions Because stateequations have initial conditions and adjoint equations haveconditions at the final time an iterative program was createdto numerically simulate solutions Given an initial guessfor the controls to compute the optimal state values theprogram solves (24) with initial conditions (27) forward intime interval [0 70]using aRungeKuttamethod of the fourthorder Resulting state values are placed in adjoint equations(35) These adjoint equations with given final conditions arethen solved backwards in time Again a fourth order RungeKutta method is employed Both state and adjoint values areused to update the control using the characterization (39)and the entire process repeats itself This iterative processterminates when current state adjoint and control valuesare sufficiently close to successive values Then we use thebackward Runge-Kutta fourth-order procedure to solve theadjoint variables in the same time interval with the help ofthe solution of the state variables transversality conditions

We have plotted susceptible exposed acute infectionindividuals and carriers individuals with andwithout controlby considering values of parameters We simulate the systemat different values of rate of 119906

1and 119906

2

From Figures 3 4 and 5 we see that after 20 years thenumber of susceptible population decreases than when thereis no control In this case most of this population tends to theinfected class Again when only treatment control is appliedthen the number of susceptible population is not muchdifferent than the population in the case having no controlBut the susceptible population differs much from these twostrategies if we apply the strategies of only vaccination controland both vaccination and treatment controls At a high rateof vaccination the sensitive population density is reduced

10 Computational and Mathematical Methods in Medicine

Table 1 Parameter values used in numerical simulations

Parameter Description Values range Reference] Birth (and death) rate 00121 [17 18]120588 Transmission rate 08ndash2049 [17]120579 Infectiousness of carriers relative to acute infections 0-1 [17]1205821

Rate of moving from exposed to acute 6 per year [16]1205822

Rate at which individuals leave the acute infection class 4 per year [16]1205823

Rate of moving from carrier to recovery 0025 per year [16]1205824

Loss of recovery rate 003ndash006 [17]1198751

Probability of infected newborns 011 [19]1198752

Probability of immune newborns 01 [19]1198753

Proportion of acute infection individuals becoming carriers 005ndash09 [17]1198601

Weight factor for susceptible individuals 0091 [31]1198602

Weight factor for exposed individuals 001 [31]1198603

Weight factor for infected individuals 004 [31]1198604

Weight factor for carrier individuals 005 [31]1198611

Weight factor for the controls 1199061

15 [31]1198612

Weight factor for the controls 1199062

27 [31]1198780

Susceptible individuals 0493 [4]1198640

Exposed individuals 00035 [4]1198680

Acute infection individuals 00035 [4]1198620

Chronic HBV carriers 0007 [4]

0 10 20 30 40 50 60 70Time (year)

Susc

eptib

le p

opul

atio

n

NV-NTNV-T

V-NTV-T

10minus4

10minus3

10minus2

10minus1

100

Figure 3 The plot shows the changes in sensitive populations(without) vaccination and (without) treatment

to a very low level initially and then it takes longer time torestore the steady-state value

From Figures 6 7 and 8 we see that the number ofexposed population increase thanwhen there is no control Inthis case most of this population tends to the acute infectedclass Again when only treatment control is applied thenthe number of exposed population is not much different

0 10 20 30 40 50 60 70Time (year)

Susc

eptib

le p

opul

atio

n

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0365 u2 = 0

u1 = 045 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

Figure 4 The plot shows the sensitivity of sensitive populations fordifferent values of control 119906

1

than the population in the case having no control But theexposed population differs much from these two strategies ifwe apply the strategies of only vaccination control and bothvaccination and treatment controls

From Figures 9 10 and 11 we see that the number ofacute infected population increases than when there is nocontrol In this case most of this population tends to thecarrier class Again when only treatment control is appliedthen the number of acute infected population is not muchdifferent than the population in the case having no controlBut the acute infected population differsmuch from these two

Computational and Mathematical Methods in Medicine 11

0 10 20 30 40 50 60 70Time (year)

Susc

eptib

le p

opul

atio

n

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Figure 5 The plot shows the sensitivity of sensitive populations fordifferent values of control 119906

2

Expo

sed

popu

latio

n

NV-NTNV-T

V-NTV-T

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 6 The plot shows the changes in exposed populations(without) vaccination and (without) treatment

strategies if we apply the strategies of only vaccination controland both vaccination and treatment controls

Again from Figures 12 13 and 14 we see that thenumber of carrier population increases than when there isno control We see that the application of only vaccinationcontrol gives better result than the application of no controlAgain application of treatment control would give betterresult than the application of vaccination control since thetreatment control is better than the vaccination control whilethe application of both vaccination and treatment controlsgive the best result as in this case the number of carrierpopulation would be the least in number

Expo

sed

popu

latio

n

u1 = 0 u2 = 0

u1 = 03635 u2 = 0

u1 = 045 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 7 The plot shows the sensitivity of exposed populations fordifferent values of control 119906

1

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Expo

sed

popu

latio

n

Figure 8 The plot shows the sensitivity of exposed populations fordifferent values of control 119906

2

Finally the effect of two control measures on diseasedynamics may be understood well if we consider Figures 1ndash14 15 16 17 and 18 Since our main purpose is to reducethe number of sensitive exposed acute infection and carrierindividuals therefore numerical simulation results show thatoptimal combination of vaccination and treatment is themosteffective way to control hepatitis B virus infection

12 Computational and Mathematical Methods in MedicineAc

ute i

nfec

tion

popu

latio

n

NV-NTNV-T

V-NTV-T

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 9The plot shows the changes in acute infection populations(without) vaccination and (without) treatment

Acut

e inf

ectio

n po

pulat

ion

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0365 u2 = 0

u1 = 045 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

Figure 10 The plot shows the sensitivity of acute infection popula-tions for different values of control 119906

1

6 Conclusion and Suggestions

In this paper we propose an S-119864-119868-119862-119877model of hepatitis Bvirus infection with two controls vaccination and treatmentFirst we analyze the dynamic behavior of the system for con-stant controls In the constant controls case we calculate thebasic reproduction number and investigate the existence andstability of equilibriaThere are two nonnegative equilibria of

Acut

e inf

ectio

n po

pulat

ion

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Figure 11 The plot shows the sensitivity of acute infection popula-tions for different values of control 119906

2

Carr

ier p

opul

atio

n

NV-NTNV-T

V-NTV-T

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 12 The plot shows the changes in carrier populations(without) vaccination and (without) treatment

the system namely the disease-free and endemicWe see thatthe disease-free equilibriumwhich always exists and is locallyasymptotically stable if 119877

0lt 1 and endemic equilibrium

which exists and is locally asymptotically stable if 1198770gt 1

After investigating the dynamic behavior of the systemwith constant controls we formulate an optimal controlproblem if the controls become time dependent and solve itby using Pontryaginrsquos maximum principle Different possiblecombinations of controls are used and their effectiveness iscompared by simulation works Also from the numericalresults it is very clear that a combination of mixed controlmeasures respond better than any other independent control

Computational and Mathematical Methods in Medicine 13Ca

rrie

r pop

ulat

ion

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0365 u2 = 0

u1 = 046 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

Figure 13 The plot shows the sensitivity of carrier populations fordifferent values of control 119906

1

Carr

ier p

opul

atio

n

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Figure 14 The plot shows the sensitivity of carrier populations fordifferent values of control 119906

2

There is still a tremendous amount of work needed tobe done in this area Parameters are rarely constant becausethey depend on environmental conditions We do not knowhowever the detailed relationship between these parametersand environmental conditions There may be a time lag as asusceptible populationmay take some time to be infected andalso a susceptible population may take some time to immune

Popu

latio

n (N

V-N

T)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 15 The plot shows the changes in total populations withoutvaccination and without treatment

Popu

latio

n (V

-NT)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 16 The plot shows the changes in total populations withvaccination and without treatment

Popu

latio

n (N

V-T)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 17 The plot shows the changes in total populations withoutvaccination and with treatment

14 Computational and Mathematical Methods in MedicinePo

pulat

ion

(V-T

)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 18 The plot shows the changes in total populations withvaccination and treatment

after vaccination We leave all these possible extensions forthe future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] WHO Hepatitis B Fact Sheet No 204 The World HealthOrganisation Geneva Switzerland 2013 httpwwwwhointmediacentrefactsheetsfs204en

[2] CanadianCentre forOccupationalHealth and Safety ldquoHepatitisBrdquo httpwwwccohscaoshanswersdiseaseshepatitis bhtml

[3] ldquoHealthcare stumbling in RIrsquos Hepatitis fightrdquoThe Jakarta PostJanuary 2011

[4] G F Medley N A Lindop W J Edmunds and D JNokes ldquoHepatitis-B virus endemicity heterogeneity catas-trophic dynamics and controlrdquo Nature Medicine vol 7 no 5pp 619ndash624 2001

[5] J Mann and M Roberts ldquoModelling the epidemiology ofhepatitis B in New Zealandrdquo Journal of Theoretical Biology vol269 no 1 pp 266ndash272 2011

[6] ldquoHepatitis B (HBV)rdquo httpkidshealthorgteensexual healthstdsstd hepatitishtml

[7] CDC ldquoHepatitis B virus a comprehensive strategy for elim-inating transmission in the United States through universalchildhood vaccination Recommendations of the Immuniza-tion Practices Advisory Committee (ACIP)rdquo MMWR Recom-mendations and Reports vol 40(RR-13) pp 1ndash25 1991

[8] M K Libbus and L M Phillips ldquoPublic health management ofperinatal hepatitis B virusrdquo Public Health Nursing vol 26 no 4pp 353ndash361 2009

[9] F B Hollinger and D T Lau ldquoHepatitis B the pathway torecovery through treatmentrdquo Gastroenterology Clinics of NorthAmerica vol 35 no 4 pp 895ndash931 2006

[10] C-L Lai andM-F Yuen ldquoThe natural history and treatment ofchronic hepatitis B a critical evaluation of standard treatment

criteria and end pointsrdquoAnnals of InternalMedicine vol 147 no1 pp 58ndash61 2007

[11] R M Anderson and R M May Infectious Disease of HumansDynamics and Control Oxford University Press Oxford UK1991

[12] S Thornley C Bullen and M Roberts ldquoHepatitis B in ahigh prevalence New Zealand population a mathematicalmodel applied to infection control policyrdquo Journal ofTheoreticalBiology vol 254 no 3 pp 599ndash603 2008

[13] S-J Zhao Z-Y Xu and Y Lu ldquoA mathematical model ofhepatitis B virus transmission and its application for vaccinationstrategy inChinardquo International Journal of Epidemiology vol 29no 4 pp 744ndash752 2000

[14] K Wang W Wang and S Song ldquoDynamics of an HBV modelwith diffusion and delayrdquo Journal ofTheoretical Biology vol 253no 1 pp 36ndash44 2008

[15] R Xu andZMa ldquoAnHBVmodelwith diffusion and time delayrdquoJournal of Theoretical Biology vol 257 no 3 pp 499ndash509 2009

[16] L Zou W Zhang and S Ruan ldquoModeling the transmissiondynamics and control of hepatitis B virus in Chinardquo Journal ofTheoretical Biology vol 262 no 2 pp 330ndash338 2010

[17] J Pang J-A Cui and X Zhou ldquoDynamical behavior of ahepatitis B virus transmission model with vaccinationrdquo Journalof Theoretical Biology vol 265 no 4 pp 572ndash578 2010

[18] S Zhang and Y Zhou ldquoThe analysis and application of an HBVmodelrdquoAppliedMathematicalModelling vol 36 no 3 pp 1302ndash1312 2012

[19] S Bhattacharyya and S Ghosh ldquoOptimal control of verticallytransmitted diseaserdquoComputational andMathematicalMethodsin Medicine vol 11 no 4 pp 369ndash387 2010

[20] T K Kar andA Batabyal ldquoStability analysis and optimal controlof an SIR epidemic model with vaccinationrdquo Biosystems vol104 no 2-3 pp 127ndash135 2011

[21] T-K Kar and S Jana ldquoA theoretical study on mathematicalmodelling of an infectious disease with application of optimalcontrolrdquo Biosystems vol 111 no 1 pp 37ndash50 2013

[22] M J Keeling and P Rohani Modeling Infectious Diseases inHumans and Animals Princeton University Press PrincetonNJ USA 2008

[23] E Jung S Lenhart and Z Feng ldquoOptimal control of treatmentsin a two-strain tuberculosis modelrdquo Discrete and ContinuousDynamical Systems B vol 2 no 4 pp 473ndash482 2002

[24] G Birkhoff and G-C Rota Ordinary Differential EquationsJohn Wiley amp Sons New York NY USA 4th edition 1989

[25] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[26] T T Yusuf and F Benyah ldquoOptimal control of vaccination andtreatment for an SIR epidemiological modelrdquo World Journal ofModelling and Simulation vol 8 no 3 pp 194ndash204 2008

[27] H-R Joshi S Lenhart M Y Li and L Wang ldquoOptimalcontrol methods applied to disease modelsrdquo in MathematicalStudies on Human Disease Dynamics vol 410 of ContemporaryMathematics pp 187ndash207 American Mathematical SocietyProvidence RI USA 2006

[28] G Zaman Y H Kang and I H Jung ldquoStability analysis andoptimal vaccination of an SIR epidemic modelrdquo Biosystems vol93 no 3 pp 240ndash249 2008

[29] D Iacoviello and N Stasiob ldquoOptimal control for SIRCepidemic outbreakrdquo Computer Methods and Programs inBiomedicine vol 110 no 3 pp 333ndash342 2013

Computational and Mathematical Methods in Medicine 15

[30] L S Pontryagin V G Boltyanskii R V Gamkrelidze and EF Mishchenko The Mathematical Theory of Optimal ProcessesJohn Wiley amp Sons New York NY USA 1962

[31] WHO Hepatitis B Fact Sheet No 204 The World HealthOrganisation Geneva Switzerland 2000 httpwwwwhointmediacentrefactsheetsfs204en

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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MEDIATORSINFLAMMATION

of

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Behavioural Neurology

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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ObesityJournal of

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Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

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Research and TreatmentAIDS

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Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

Computational and Mathematical Methods in Medicine 5

0

1

2

3

4

5

6

Repr

oduc

tion

num

ber

0 01 02 03 04 05 06 07 08 09 1

Control (u1 u2)

u1 = 0 u2 = 0

u1 = 0 u2 = 1

u1 = 1 u2 = 0

u1 = 1 u2 = 1

Figure 2 Variation of different reproduction numbers by usingof two control variables (vaccination and treatment) Figure 2illustrates the impact of application of two controls (vaccination andtreatment) on different basic reproduction numbers 119877

0= 39810

when there is no vaccination and treatment (1199061= 0 119906

2= 0) The

figure shows that application of vaccination reduces1198770more rapidly

than treatment though mixed intervention strategies always workbetter to reduce the disease burden

(a) No vaccine no treatment control (1199061= 0 119906

2= 0)

119864lowastlowast

1= (119878lowast

1 119864lowast

1 119868lowast

1 119862lowast

1) where

119878lowast

1=

(] + 1205821) (] + 120582

2) (] + 120582

3minus 1199011])

1205881205821(] + 120582

3+ 12057912058221199013minus 1199011])

119864lowast

1=

120579120588 (] + 1205822) 119862lowast

1119878lowast

1

(] + 1205821) (] + 120582

2) minus 120588120582

1119878lowast

1

119868lowast

1=

1205791205881205821119862lowast

1119878lowast

1

(] + 1205821) (] + 120582

2) minus 120588120582

1119878lowast

1

119862lowast

1= (120582112058221199013(] + 120582

4minus ]1199012) 119878lowast

1(1198770minus 1))

times ( (] + 1205823minus 1199011])

times [(] + 1205824minus ]1199012) (] + 120582

2+ 1205821) + 12058211205822]

+120582112058221199013(]1199011minus ]1199012+ 1205824) )minus1

(13)

(b) No vaccine with treatment control (1199061= 0 119906

2= 0)

119864lowastlowast

2= (119878lowast

2 119864lowast

2 119868lowast

2 119862lowast

2) where

119878lowast

2=

(] + 1205821) (] + 120582

2) (] + 120582

3minus 1199011] + 1199062)

1205881205821(] + 120582

3+ 12057912058221199013minus 1199011] + 1199062)

119864lowast

2=

120579120588 (] + 1205822) 119862lowast

2119878lowast

2

(] + 1205821) (] + 120582

2) minus 120588120582

1119878lowast

2

119868lowast

2=

1205791205881205821119862lowast

2119878lowast

2

(] + 1205821) (] + 120582

2) minus 120588120582

1119878lowast

2

119862lowast

2= (120582112058221199013(] + 120582

4minus ]1199012) 119878lowast

2(1198770minus 1))

times ( (] + 1205823minus 1199011] + 1199062)

times [(] + 1205824minus ]1199012) (] + 120582

2+ 1205821) + 12058211205822]

+120582112058221199013(]1199011minus ]1199012+ 1205824) )minus1

(14)

(c) With vaccine no treatment control (1199061= 0 1199062= 0)

119864lowastlowast

3= (119878lowast

3 119864lowast

3 119868lowast

3 119862lowast

3) where

119878lowast

3=

(] + 1205821) (] + 120582

2) (] + 120582

3minus 1199011])

1205881205821(] + 120582

3+ 12057912058221199013minus 1199011])

119864lowast

3=

120579120588 (] + 1205822) 119862lowast

3119878lowast

3

(] + 1205821) (] + 120582

2) minus 120588120582

1119878lowast

3

119868lowast

3=

1205791205881205821119862lowast

3119878lowast

3

(] + 1205821) (] + 120582

2) minus 120588120582

1119878lowast

3

119862lowast

3= (120582112058221199013(] + 120582

4minus ]1199012+ 1199061) 119878lowast

3(1198770minus 1))

times ( (] + 1205823minus 1199011])

times [(] + 1205824minus ]1199012) (] + 120582

2+ 1205821) + 12058211205822]

+120582112058221199013(]1199011minus ]1199012+ 1205824) )minus1

(15)

(d) With vaccine with treatment control (1199061= 0 1199062= 0)

119864lowastlowast

4= (119878lowast

4 119864lowast

4 119868lowast

4 119862lowast

4) where

119878lowast

4=

(] + 1205821) (] + 120582

2) (] + 120582

3minus 1199011] + 1199062)

1205881205821(] + 120582

3+ 12057912058221199013minus 1199011] + 1199062)

119864lowast

4=

120579120573 (] + 1205822) 119862lowast

4119878lowast

4

(] + 1205821) (] + 120582

2) minus 120588120582

1119878lowast

4

119868lowast

4=

1205791205881205821119862lowast

4119878lowast

4

(] + 1205821) (] + 120582

2) minus 120588120582

1119878lowast

4

119862lowast

4= (120582112058221199013(] + 120582

4minus ]1199012+ 1199061) 119878lowast

4(1198770minus 1))

times ( (] + 1205823minus 1199011] + 1199062)

times [(] + 1205824minus ]1199012) (] + 120582

2+ 1205821) + 12058211205822]

+120582112058221199013(]1199011minus ]1199012+ 1205824) )minus1

(16)

Clearly 119864lowastlowast is feasible if 119862lowast gt 0 that is if 1198770gt 1 Also for

1198770= 1 the endemic equilibrium reduces to the disease-free

equilibrium and for 1198770lt 1 it becomes infeasible Hence we

may state the following theorem

Theorem 3 (i) If 1198770lt 1 then the system (2) has only one

equilibrium which is disease-free(ii) If 119877

0gt 1 then the system (2) has two equilibria one is

disease-free and the other is endemic equilibrium(iii) If 119877

0= 1 then the endemic equilibrium reduces to the

disease-free equilibrium

6 Computational and Mathematical Methods in Medicine

32 Stability Analysis In this section we will discuss thestability of different equilibria Firstly we analyse the localstability of the disease-free equilibrium

Theorem 4 If 1198770lt 1 then the disease-free equilibrium is

locally asymptotically stable

Proof The Jacobian matrix of system (2) at the disease-freeequilibrium is

J0 =[[[[

[

minus (] + 1205824minus ]1199012+ 1199061) minus (120582

4minus ]1199012) minus (120588119878

0+ 1205824minus ]1199012) minus (]119901

1+ 1205824minus ]1199012+ 120579120588119878

0)

0 minus (] + 1205821) 120588119878

01205791205881198780

0 1205821

minus (] + 1205822) 0

0 0 11990131205822

]1199011minus ] minus 120582

3minus 1199062

]]]]

]

(17)

The characteristic polynomial of 1198690given by

119875 (120582) = (120582 + 1198970) (1205823+ 11989711205822+ 1198972120582 + 1198973) (18)

where

1198970= ] + 119906

1+ 1205824minus ]1199012

1198971= 3] + 120582

1+ 1205822+ 1205823+ 1199062minus ]1199011

1198972= (] + 120582

2) (] + 120582

3+ 1199062minus ]1199011)

+ (] + 1205821) (2] + 120582

2+ 1205823+ 1199062minus ]1199011) minus 12058211205881198780

1198973= (] + 120582

1) (] + 120582

2) (] + 120582

3+ 1199062minus ]1199011)

minus 12058211205881198780(] + 120582

3+ 1199062+ 12057911990131205822minus ]1199011)

(19)

We need to verify the following two conditions

(a) 1198970 1198971 1198972 1198973gt 0

(b) 11989711198972minus 1198973gt 0

It is easy to see that 1198970 1198971gt 0 and 119897

2 1198973 11989711198972minus 1198973gt 0 if

1198770lt 1 It follows from the Routh-Hurwitz criterion that the

eigenvalues have negative real parts if 1198770lt 1 Hence the

disease-free equilibriumofmodel (2) is locally asymptoticallystable if 119877

0lt 1 and unstable if 119877

0gt 1

To discuss the properties of the endemic equilibriumpoint

Theorem 5 The endemic equilibrium point is locally asymp-totically stable if 119877

0gt 1

Proof We use Routh-Hurwitz criterion to establish the localstability of the endemic equilibrium The Jacobian matrix ofsystem (2) at endemic equilibrium is

Jlowast =[[[

[

minus (] + 1205824minus ]1199012+ 1199061+ 120588119868lowast) minus (120582

4minus ]1199012) minus (120588119878

lowast+ 1205824minus ]1199012) minus (]119901

1+ 1205824minus ]1199012+ 120588120579119878

lowast)

120588119868lowast+ 120588120579119862

lowastminus (] + 120582

1) 120588119878

lowast120588120579119878lowast

0 1205821

minus (] + 1205822) 0

0 0 11990131205822

]1199011minus ] minus 120582

3minus 1199062

]]]

]

(20)

Then the characteristic equation at 119864lowastlowast is 1205824 + 11989111205823+ 11989121205822+

1198913120582 + 1198914= 0 where

1198911= 1198861+ 1198872+ 1198883+ 1198894

1198912= 11988611198863+ 11988611198894+ 11988611198872+ 11988831198894+ 11988721198883+ 11988721198894minus 11988731198882minus 11988711198862

1198913= 119886111988831198894+ 119886111988721198883+ 119886111988721198894+ 119887211988831198894+ 119887411988821198893+ 119887111988821198863

minus 119886111988731198882minus 119887311988821198894minus 119887111988621198883minus 119887111988621198894

1198914= 1198861119887211988831198894+ 1198861119887411988821198893+ 1198871119888211988631198894minus 1198861119887311988821198894

minus 1198871119886211988831198894minus 1198871119886411988821198893

(21)

where

1198861= minus (] + 120582

4minus ]1199012+ 1199061+ 120588119868lowast)

1198862= minus (120582

4minus ]1199012) 119886

3= minus (120588119878

lowast+ 1205824minus ]1199012)

1198864= minus (]119901

1+ 1205824minus ]1199012+ 120588120579119878

lowast) 119887

1= 120588119868lowast+ 120588120579119862

lowast

1198872= minus (] + 120582

1) 119887

3= 120588119878lowast

1198874= 120588120579119878

lowast 119888

1= 0

1198882= 1205821 119888

3= minus (] + 120582

2)

1198884= 0 119889

1= 0

1198892= 0 119889

3= 11990131205822

1198894= ]1199011minus ] minus 120582

3minus 1199062

(22)

We need to verify the following three conditions

(a) 1198911 1198912 1198913 1198914gt 0

(b) 11989111198912minus 1198913gt 0

(c) 1198913(11989111198912minus 1198913) minus 1198912

11198914gt 0

Computational and Mathematical Methods in Medicine 7

It is easy to see that conditions (a) and (b) are satisfiedAfter computations we can prove that 119891

3(11989111198912minus 1198913) minus

1198912

11198914gt 0 is also validTheRouth-Hurwitz criterion and those

inequalities in (a)ndash(c) imply that the characteristic equationat 119864lowastlowast has only roots with negative real part which certifiesthe local stability of 119864lowastlowast

4 Optimal Control with Two Objectives

One of the early reasons for studying hepatitis B virus (HBV)infection is to improve the control variables and finally to putdown the infection of the population Optimal control is auseful mathematical tool that can be used to make decisionsin this case In the previous sections we have analyzed themodel with two control variables one is treatment and theother is vaccination and we consider their constant controlsthroughout the analysis But in fact these control variablesshould be time dependent In this section we consider thevaccination and treatment as time-dependent controls in acompact interval of time duration

Our goals here are to put down infection from the popu-lation by increasing the recovered individuals and reducingsusceptible exposed infected and carrier individuals in apopulation and to minimize the costs required to control thehepatitis B virus (HBV) infection by using vaccination andtreatment First of all we construct the objective functionalto be optimized as follows

119869 = int

119879

0

[1198601119878 (119905) + 119860

2119864 (119905) + 119860

3119868 (119905) + 119860

4119862 (119905)

+

1

2

(11986111199062

1(119905) + 119861

21199062

2(119905))] 119889119905

(23)

subject to

119878 (119905) = ] minus ]1199011119862 minus 120588 (119868 + 120579119862) 119878 minus ]119878 minus 119906

1119878

+ (1205824minus ]1199012) (1 minus 119878 minus 119864 minus 119868 minus 119862)

(119905) = 120588 (119868 + 120579119862) 119878 minus (] + 1205821) 119864

119868 (119905) = 1205821119864 minus (] + 120582

2) 119868

(119905) = ]1199011119862 + 119901

31205822119868 minus (] + 120582

3) 119862 minus 119906

2119862

(24)

Our object is to find (119906lowast1 119906lowast

2) such that

119869 (119906lowast

1 119906lowast

2) = min 119869 (119906

1 1199062) (25)

where

1199061(119905) 119906

2(119905) isin Γ

= 119906 (119905) | 0 le 119906 (119905) le 119906max (119905) le 1 0 le 119905 le 119879

119906 (119905) is Labesgue measurable

(26)

Here 119860119894 for 119894 = 1 4 are positive constants that

are represented to keep a balance in the size of 119878(119905) 119864(119905)119868(119905) and 119862(119905) respectively 119861

1and 119861

2 respectively are the

weights corresponding to the controls 1199061and 119906

2 119906max is the

maximum attainable value for controls (1199061max and 119906

2max)1199061max and 119906

2max will depend on the amount of resourcesavailable to implement each of the control measures [26]The 119861

1and 119861

2will depend on the relative importance of

each of the control measures in mitigating the spread of thedisease as well as the cost (human effort material resourcesetc) of implementing each of the control measures per unittime [26] Thus the terms 119861

11199062

1and 119861

21199062

2describe the costs

associated with vaccination and treatment respectively Thesquare of the control variables is taken here to remove theseverity of the side effects and overdoses of vaccination andtreatment [27]

41 Optimal Control Solution For existence of the solutionwe consider the control system (24) with initial condition

119878 (0) = 1198780 119864 (0) = 119864

0

119868 (0) = 1198680 119862 (0) = 119862

0

(27)

then we rewrite (2) in the following form

119889119909 (119905)

119889119905

= 119860119909 + 119865 (119909) (28)

where 119909(119905) = (119878(119905) 119864(119905) 119868(119905) 119862(119905)) is the vector of the statevariables and A and F(x) are defined as follows

A =

[[[[[[[

[

minus] minus 1199061minus 1205824+ ]1199012minus1205824+ ]1199012minus1205824+ ]1199012

minus1205824+ ]1199012

0 minus] minus 1205821

0 0

0 1205821

minus] minus 1205822

0

0 0 11990131205822

]1199011minus ] minus 120582

3minus 1199062

]]]]]]]

]

F (x) =[[[[[[

[

] + 120588 (119868 + 120579119862) 119878 + 1205824minus ]1199012

120588 (119868 + 120579119862) 119878

0

0

]]]]]]

]

(29)

8 Computational and Mathematical Methods in Medicine

We set

119866 (119909) = 119860119909 + 119865 (119909) (30)

The second term on the right-hand side of (30) satisfies

1003816100381610038161003816119865 (1199091) minus 119865 (119909

2)1003816100381610038161003816

le 119872 (10038161003816100381610038161198781minus 1198782

1003816100381610038161003816+10038161003816100381610038161198641minus 1198642

1003816100381610038161003816+10038161003816100381610038161198681minus 1198682

1003816100381610038161003816+10038161003816100381610038161198621minus 1198622

1003816100381610038161003816)

(31)

where the positive constant 119872 is independent of the statevariables 119909 and119872 le 1 Also we get

1003816100381610038161003816119866 (1199091) minus 119866 (119909

2)1003816100381610038161003816le 119871

10038161003816100381610038161199091minus 1199092

1003816100381610038161003816 (32)

where 119871 = max119872 119860 lt infin Therefore it follows thatthe function 119866 is uniformly Lipschitz continuous From thedefinition of the controls 119906

1(119905) and 119906

2(119905) and the restrictions

on the nonnegativeness of the state variables we see that asolution of the system (28) exists [24 28 29]

42The Lagrangian and Hamiltonian for the Control ProblemIn order to find an optimal solution pair first we shouldfind the Lagrangian and Hamiltonian for the optimal controlproblem (24) In fact the Lagrangian of problem is given by

119871 (119878 119864 119868 119862 1199061 1199062) = 119860

1119878 (119905) + 119860

2119864 (119905) + 119860

3119868 (119905)

+ 1198604119862 (119905) +

1

2

(11986111199062

1(119905) + 119861

21199062

2(119905))

(33)

We are looking for the minimal value of the LagrangianTo accomplish this we define Hamiltonian119867 for the controlproblem as follows

119867(119878 119864 119868 119862 1199061 1199062 1205721 1205722 1205723 1205724)

= 119871 + 1205721(119905)

119889119878

119889119905

+ 1205722(119905)

119889119864

119889119905

+ 1205723(119905)

119889119868

119889119905

+ 1205724(119905)

119889119862

119889119905

(34)

where 120572119894(119905) for 119894 = 1 2 3 4 are the adjoint variables and can

be determined by solving the following system of differentialequations

1(119905) = minus

120597119867

120597119878

= minus [1198601+ 1205721((]1199012minus 1205824) minus 120588 (119868 + 120579119862) minus 119906

1minus ])

+1205722120588 (119868 + 120579119862)]

2(119905) = minus

120597119867

120597119864

= minus [1198602+ 1205721(]1199012minus 1205824) + 1205722(minus] minus 120582

1) + 12057231205821]

3(119905) = minus

120597119867

120597119868

= minus [1198603+ 1205721((]1199012minus 1205824) minus 120588119878) + 120572

2120588119878

+1205723(minus] minus 120582

2) + 120572411990131205822]

4(119905) = minus

120597119867

120597119862

= minus [1198604+ 1205721(]1199012minus 1205824minus ]1199011minus 120588120579119878)

+1205722120588120579119878 + 120572

4(]1199011minus ] + 120582

3+ 1199062)]

(35)

satisfying the transversality conditions and optimality condi-tions

1205721(119879) = 120572

2(119879) = 120572

3(119879) = 120572

4(119879) = 0

120597119867

1205971199061

= 11986111199061minus 1205721119878 = 0

120597119867

1205971199062

= 11986121199062minus 1205724119862 = 0

(36)

We now state and prove the following theorem

Theorem 6 There is an optimal control (119906lowast1 119906lowast

2) such that

119869 (119878 119864 119868 119878 119906lowast

1 119906lowast

2) = min 119869 (119878 119864 119868 119878 119906

1 1199062) (37)

subject to the system of differential equation (24)

Proof Here both the control and state variables are non-negative values In this minimized problem the necessaryconvexity of the objective functional in 119906

1and 1199062are satisfied

The set of all control variables 1199061(119905) 1199062(119905) is also convex and

closed by definition The optimal system is bounded whichdetermines the compactness needed for the existence of theoptimal control In addition the integrand in the functional

int

119879

0

[1198601119878 (119905) + 119860

2119864 (119905) + 119860

3119868 (119905) + 119860

4119862 (119905)

+

1

2

(11986111199062

1(119905) + 119861

21199062

2(119905))] 119889119905

(38)

is convex on the control set Hence the theorem (the proof iscomplete)

Computational and Mathematical Methods in Medicine 9

43 Necessary and Sufficient Conditions for Optimal ControlsApplying Pontryaginrsquos maximum principle [30] on the con-structed Hamiltonian 119867 and the theorem (24) we obtainthe optimal steady-state solution and corresponding controlas follows If taking 119883 = (119878 119864 119868 119862) and 119880 = (119906

1 1199062)

then (119883lowast 119880lowast) is an optimal solution of an optimal controlproblem then we now state the following theorem

Theorem 7 The optimal control pair (119906lowast1 119906lowast

2) which mini-

mizes J over the region Γ is given by

119906lowast

1= max[0119879]

0min (1(119905) 1)

119906lowast

2= max[0119879]

0min (2(119905) 1)

(39)

where 1= 11198781198611and 2= 41198621198612and let

1 2 3 4 be

the solution of system (35)

Proof With the help of the optimality conditions we have

120597119867

1205971199061

= 11986111199061minus 1205721119878 = 0 997904rArr 119906

1=

1119878

1198611

= (1)

120597119867

1205971199062

= 11986121199062minus 1205724119862 = 0 997904rArr 119906

2=

4119862

1198612

= (2)

(40)

Using the property of the control space Γ the two controlswhich are bounded with upper and lower bounds arerespectively 1 and 0 that is

119906lowast

1=

0 if 1le 0

1

if 0 lt 1lt 1

1 if 1ge 1

(41)

This can be rewritten in compact notation

119906lowast

1= max[0119879]

0min (1(119905) 1) (42)

and similarly

119906lowast

2=

0 if 2le 0

2

if 0 lt 2lt 1

1 if 2ge 1

(43)

This can be rewritten in compact notation

119906lowast

2= max[0119879]

0min (2(119905) 1) (44)

Hence for these pair of controls (119906lowast1 119906lowast

2) we get the optimum

value of the functional 119869 given by (24)

5 Numerical Examples

Numerical solutions to the optimality system (24) arediscussed in this section We make several interestingobservation by numerically simulating (2) in the rangeof parameter values We consider the parameter set

Δ = ] 120588 120579 1205821 1205822 1205823 1205824 1199011 1199012 1199013 1198601 1198602 1198603 1198604 1198611 1198612

some of the parameters are taken from the published articlesand some are assumed with feasible values Moreover thetime interval for which the optimal control is applied istaken as 70 years also consider Ψ = 119878

0 1198640 1198680 1198620 as initial

condition for simulation of the model The main parametervalues are listed in Table 1 We compare the results having nocontrols only vaccination control only treatment controland both vaccination and treatment controls

With the parameter values in Table 1 the system asymp-totically approaches towards the equilibrium 119864

lowastlowast

1(00858

03327 04003 05349) where the basic reproduction ratio1198770= 39810 (Figure 2) For the parameters set the system

(2) has two feasible equilibria one is disease free and theother is endemic and the endemic equilibrium is locallyasymptotically stable The effect of two control measures ondisease dynamics may be understood well if we considerFigure 2 It explains how control reproduction ratio 119877

0

evolves with different rates of 1199061and 119906

2 It is seen that both

vaccination and treatment reduce the value of 1198770effectively

But an integrated control works better than either of thecontrol measures

In this section we use an iterative method to obtainresults for an optimal control problem of the proposedmodel(24) We use Runge-Kuttarsquos fourth-order procedure [23] hereto solve the optimality system consisting of eight ordinarydifferential equations having four state equations and fouradjoint equations and boundary conditions Because stateequations have initial conditions and adjoint equations haveconditions at the final time an iterative program was createdto numerically simulate solutions Given an initial guessfor the controls to compute the optimal state values theprogram solves (24) with initial conditions (27) forward intime interval [0 70]using aRungeKuttamethod of the fourthorder Resulting state values are placed in adjoint equations(35) These adjoint equations with given final conditions arethen solved backwards in time Again a fourth order RungeKutta method is employed Both state and adjoint values areused to update the control using the characterization (39)and the entire process repeats itself This iterative processterminates when current state adjoint and control valuesare sufficiently close to successive values Then we use thebackward Runge-Kutta fourth-order procedure to solve theadjoint variables in the same time interval with the help ofthe solution of the state variables transversality conditions

We have plotted susceptible exposed acute infectionindividuals and carriers individuals with andwithout controlby considering values of parameters We simulate the systemat different values of rate of 119906

1and 119906

2

From Figures 3 4 and 5 we see that after 20 years thenumber of susceptible population decreases than when thereis no control In this case most of this population tends to theinfected class Again when only treatment control is appliedthen the number of susceptible population is not muchdifferent than the population in the case having no controlBut the susceptible population differs much from these twostrategies if we apply the strategies of only vaccination controland both vaccination and treatment controls At a high rateof vaccination the sensitive population density is reduced

10 Computational and Mathematical Methods in Medicine

Table 1 Parameter values used in numerical simulations

Parameter Description Values range Reference] Birth (and death) rate 00121 [17 18]120588 Transmission rate 08ndash2049 [17]120579 Infectiousness of carriers relative to acute infections 0-1 [17]1205821

Rate of moving from exposed to acute 6 per year [16]1205822

Rate at which individuals leave the acute infection class 4 per year [16]1205823

Rate of moving from carrier to recovery 0025 per year [16]1205824

Loss of recovery rate 003ndash006 [17]1198751

Probability of infected newborns 011 [19]1198752

Probability of immune newborns 01 [19]1198753

Proportion of acute infection individuals becoming carriers 005ndash09 [17]1198601

Weight factor for susceptible individuals 0091 [31]1198602

Weight factor for exposed individuals 001 [31]1198603

Weight factor for infected individuals 004 [31]1198604

Weight factor for carrier individuals 005 [31]1198611

Weight factor for the controls 1199061

15 [31]1198612

Weight factor for the controls 1199062

27 [31]1198780

Susceptible individuals 0493 [4]1198640

Exposed individuals 00035 [4]1198680

Acute infection individuals 00035 [4]1198620

Chronic HBV carriers 0007 [4]

0 10 20 30 40 50 60 70Time (year)

Susc

eptib

le p

opul

atio

n

NV-NTNV-T

V-NTV-T

10minus4

10minus3

10minus2

10minus1

100

Figure 3 The plot shows the changes in sensitive populations(without) vaccination and (without) treatment

to a very low level initially and then it takes longer time torestore the steady-state value

From Figures 6 7 and 8 we see that the number ofexposed population increase thanwhen there is no control Inthis case most of this population tends to the acute infectedclass Again when only treatment control is applied thenthe number of exposed population is not much different

0 10 20 30 40 50 60 70Time (year)

Susc

eptib

le p

opul

atio

n

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0365 u2 = 0

u1 = 045 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

Figure 4 The plot shows the sensitivity of sensitive populations fordifferent values of control 119906

1

than the population in the case having no control But theexposed population differs much from these two strategies ifwe apply the strategies of only vaccination control and bothvaccination and treatment controls

From Figures 9 10 and 11 we see that the number ofacute infected population increases than when there is nocontrol In this case most of this population tends to thecarrier class Again when only treatment control is appliedthen the number of acute infected population is not muchdifferent than the population in the case having no controlBut the acute infected population differsmuch from these two

Computational and Mathematical Methods in Medicine 11

0 10 20 30 40 50 60 70Time (year)

Susc

eptib

le p

opul

atio

n

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Figure 5 The plot shows the sensitivity of sensitive populations fordifferent values of control 119906

2

Expo

sed

popu

latio

n

NV-NTNV-T

V-NTV-T

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 6 The plot shows the changes in exposed populations(without) vaccination and (without) treatment

strategies if we apply the strategies of only vaccination controland both vaccination and treatment controls

Again from Figures 12 13 and 14 we see that thenumber of carrier population increases than when there isno control We see that the application of only vaccinationcontrol gives better result than the application of no controlAgain application of treatment control would give betterresult than the application of vaccination control since thetreatment control is better than the vaccination control whilethe application of both vaccination and treatment controlsgive the best result as in this case the number of carrierpopulation would be the least in number

Expo

sed

popu

latio

n

u1 = 0 u2 = 0

u1 = 03635 u2 = 0

u1 = 045 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 7 The plot shows the sensitivity of exposed populations fordifferent values of control 119906

1

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Expo

sed

popu

latio

n

Figure 8 The plot shows the sensitivity of exposed populations fordifferent values of control 119906

2

Finally the effect of two control measures on diseasedynamics may be understood well if we consider Figures 1ndash14 15 16 17 and 18 Since our main purpose is to reducethe number of sensitive exposed acute infection and carrierindividuals therefore numerical simulation results show thatoptimal combination of vaccination and treatment is themosteffective way to control hepatitis B virus infection

12 Computational and Mathematical Methods in MedicineAc

ute i

nfec

tion

popu

latio

n

NV-NTNV-T

V-NTV-T

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 9The plot shows the changes in acute infection populations(without) vaccination and (without) treatment

Acut

e inf

ectio

n po

pulat

ion

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0365 u2 = 0

u1 = 045 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

Figure 10 The plot shows the sensitivity of acute infection popula-tions for different values of control 119906

1

6 Conclusion and Suggestions

In this paper we propose an S-119864-119868-119862-119877model of hepatitis Bvirus infection with two controls vaccination and treatmentFirst we analyze the dynamic behavior of the system for con-stant controls In the constant controls case we calculate thebasic reproduction number and investigate the existence andstability of equilibriaThere are two nonnegative equilibria of

Acut

e inf

ectio

n po

pulat

ion

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Figure 11 The plot shows the sensitivity of acute infection popula-tions for different values of control 119906

2

Carr

ier p

opul

atio

n

NV-NTNV-T

V-NTV-T

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 12 The plot shows the changes in carrier populations(without) vaccination and (without) treatment

the system namely the disease-free and endemicWe see thatthe disease-free equilibriumwhich always exists and is locallyasymptotically stable if 119877

0lt 1 and endemic equilibrium

which exists and is locally asymptotically stable if 1198770gt 1

After investigating the dynamic behavior of the systemwith constant controls we formulate an optimal controlproblem if the controls become time dependent and solve itby using Pontryaginrsquos maximum principle Different possiblecombinations of controls are used and their effectiveness iscompared by simulation works Also from the numericalresults it is very clear that a combination of mixed controlmeasures respond better than any other independent control

Computational and Mathematical Methods in Medicine 13Ca

rrie

r pop

ulat

ion

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0365 u2 = 0

u1 = 046 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

Figure 13 The plot shows the sensitivity of carrier populations fordifferent values of control 119906

1

Carr

ier p

opul

atio

n

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Figure 14 The plot shows the sensitivity of carrier populations fordifferent values of control 119906

2

There is still a tremendous amount of work needed tobe done in this area Parameters are rarely constant becausethey depend on environmental conditions We do not knowhowever the detailed relationship between these parametersand environmental conditions There may be a time lag as asusceptible populationmay take some time to be infected andalso a susceptible population may take some time to immune

Popu

latio

n (N

V-N

T)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 15 The plot shows the changes in total populations withoutvaccination and without treatment

Popu

latio

n (V

-NT)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 16 The plot shows the changes in total populations withvaccination and without treatment

Popu

latio

n (N

V-T)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 17 The plot shows the changes in total populations withoutvaccination and with treatment

14 Computational and Mathematical Methods in MedicinePo

pulat

ion

(V-T

)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 18 The plot shows the changes in total populations withvaccination and treatment

after vaccination We leave all these possible extensions forthe future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] WHO Hepatitis B Fact Sheet No 204 The World HealthOrganisation Geneva Switzerland 2013 httpwwwwhointmediacentrefactsheetsfs204en

[2] CanadianCentre forOccupationalHealth and Safety ldquoHepatitisBrdquo httpwwwccohscaoshanswersdiseaseshepatitis bhtml

[3] ldquoHealthcare stumbling in RIrsquos Hepatitis fightrdquoThe Jakarta PostJanuary 2011

[4] G F Medley N A Lindop W J Edmunds and D JNokes ldquoHepatitis-B virus endemicity heterogeneity catas-trophic dynamics and controlrdquo Nature Medicine vol 7 no 5pp 619ndash624 2001

[5] J Mann and M Roberts ldquoModelling the epidemiology ofhepatitis B in New Zealandrdquo Journal of Theoretical Biology vol269 no 1 pp 266ndash272 2011

[6] ldquoHepatitis B (HBV)rdquo httpkidshealthorgteensexual healthstdsstd hepatitishtml

[7] CDC ldquoHepatitis B virus a comprehensive strategy for elim-inating transmission in the United States through universalchildhood vaccination Recommendations of the Immuniza-tion Practices Advisory Committee (ACIP)rdquo MMWR Recom-mendations and Reports vol 40(RR-13) pp 1ndash25 1991

[8] M K Libbus and L M Phillips ldquoPublic health management ofperinatal hepatitis B virusrdquo Public Health Nursing vol 26 no 4pp 353ndash361 2009

[9] F B Hollinger and D T Lau ldquoHepatitis B the pathway torecovery through treatmentrdquo Gastroenterology Clinics of NorthAmerica vol 35 no 4 pp 895ndash931 2006

[10] C-L Lai andM-F Yuen ldquoThe natural history and treatment ofchronic hepatitis B a critical evaluation of standard treatment

criteria and end pointsrdquoAnnals of InternalMedicine vol 147 no1 pp 58ndash61 2007

[11] R M Anderson and R M May Infectious Disease of HumansDynamics and Control Oxford University Press Oxford UK1991

[12] S Thornley C Bullen and M Roberts ldquoHepatitis B in ahigh prevalence New Zealand population a mathematicalmodel applied to infection control policyrdquo Journal ofTheoreticalBiology vol 254 no 3 pp 599ndash603 2008

[13] S-J Zhao Z-Y Xu and Y Lu ldquoA mathematical model ofhepatitis B virus transmission and its application for vaccinationstrategy inChinardquo International Journal of Epidemiology vol 29no 4 pp 744ndash752 2000

[14] K Wang W Wang and S Song ldquoDynamics of an HBV modelwith diffusion and delayrdquo Journal ofTheoretical Biology vol 253no 1 pp 36ndash44 2008

[15] R Xu andZMa ldquoAnHBVmodelwith diffusion and time delayrdquoJournal of Theoretical Biology vol 257 no 3 pp 499ndash509 2009

[16] L Zou W Zhang and S Ruan ldquoModeling the transmissiondynamics and control of hepatitis B virus in Chinardquo Journal ofTheoretical Biology vol 262 no 2 pp 330ndash338 2010

[17] J Pang J-A Cui and X Zhou ldquoDynamical behavior of ahepatitis B virus transmission model with vaccinationrdquo Journalof Theoretical Biology vol 265 no 4 pp 572ndash578 2010

[18] S Zhang and Y Zhou ldquoThe analysis and application of an HBVmodelrdquoAppliedMathematicalModelling vol 36 no 3 pp 1302ndash1312 2012

[19] S Bhattacharyya and S Ghosh ldquoOptimal control of verticallytransmitted diseaserdquoComputational andMathematicalMethodsin Medicine vol 11 no 4 pp 369ndash387 2010

[20] T K Kar andA Batabyal ldquoStability analysis and optimal controlof an SIR epidemic model with vaccinationrdquo Biosystems vol104 no 2-3 pp 127ndash135 2011

[21] T-K Kar and S Jana ldquoA theoretical study on mathematicalmodelling of an infectious disease with application of optimalcontrolrdquo Biosystems vol 111 no 1 pp 37ndash50 2013

[22] M J Keeling and P Rohani Modeling Infectious Diseases inHumans and Animals Princeton University Press PrincetonNJ USA 2008

[23] E Jung S Lenhart and Z Feng ldquoOptimal control of treatmentsin a two-strain tuberculosis modelrdquo Discrete and ContinuousDynamical Systems B vol 2 no 4 pp 473ndash482 2002

[24] G Birkhoff and G-C Rota Ordinary Differential EquationsJohn Wiley amp Sons New York NY USA 4th edition 1989

[25] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[26] T T Yusuf and F Benyah ldquoOptimal control of vaccination andtreatment for an SIR epidemiological modelrdquo World Journal ofModelling and Simulation vol 8 no 3 pp 194ndash204 2008

[27] H-R Joshi S Lenhart M Y Li and L Wang ldquoOptimalcontrol methods applied to disease modelsrdquo in MathematicalStudies on Human Disease Dynamics vol 410 of ContemporaryMathematics pp 187ndash207 American Mathematical SocietyProvidence RI USA 2006

[28] G Zaman Y H Kang and I H Jung ldquoStability analysis andoptimal vaccination of an SIR epidemic modelrdquo Biosystems vol93 no 3 pp 240ndash249 2008

[29] D Iacoviello and N Stasiob ldquoOptimal control for SIRCepidemic outbreakrdquo Computer Methods and Programs inBiomedicine vol 110 no 3 pp 333ndash342 2013

Computational and Mathematical Methods in Medicine 15

[30] L S Pontryagin V G Boltyanskii R V Gamkrelidze and EF Mishchenko The Mathematical Theory of Optimal ProcessesJohn Wiley amp Sons New York NY USA 1962

[31] WHO Hepatitis B Fact Sheet No 204 The World HealthOrganisation Geneva Switzerland 2000 httpwwwwhointmediacentrefactsheetsfs204en

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

6 Computational and Mathematical Methods in Medicine

32 Stability Analysis In this section we will discuss thestability of different equilibria Firstly we analyse the localstability of the disease-free equilibrium

Theorem 4 If 1198770lt 1 then the disease-free equilibrium is

locally asymptotically stable

Proof The Jacobian matrix of system (2) at the disease-freeequilibrium is

J0 =[[[[

[

minus (] + 1205824minus ]1199012+ 1199061) minus (120582

4minus ]1199012) minus (120588119878

0+ 1205824minus ]1199012) minus (]119901

1+ 1205824minus ]1199012+ 120579120588119878

0)

0 minus (] + 1205821) 120588119878

01205791205881198780

0 1205821

minus (] + 1205822) 0

0 0 11990131205822

]1199011minus ] minus 120582

3minus 1199062

]]]]

]

(17)

The characteristic polynomial of 1198690given by

119875 (120582) = (120582 + 1198970) (1205823+ 11989711205822+ 1198972120582 + 1198973) (18)

where

1198970= ] + 119906

1+ 1205824minus ]1199012

1198971= 3] + 120582

1+ 1205822+ 1205823+ 1199062minus ]1199011

1198972= (] + 120582

2) (] + 120582

3+ 1199062minus ]1199011)

+ (] + 1205821) (2] + 120582

2+ 1205823+ 1199062minus ]1199011) minus 12058211205881198780

1198973= (] + 120582

1) (] + 120582

2) (] + 120582

3+ 1199062minus ]1199011)

minus 12058211205881198780(] + 120582

3+ 1199062+ 12057911990131205822minus ]1199011)

(19)

We need to verify the following two conditions

(a) 1198970 1198971 1198972 1198973gt 0

(b) 11989711198972minus 1198973gt 0

It is easy to see that 1198970 1198971gt 0 and 119897

2 1198973 11989711198972minus 1198973gt 0 if

1198770lt 1 It follows from the Routh-Hurwitz criterion that the

eigenvalues have negative real parts if 1198770lt 1 Hence the

disease-free equilibriumofmodel (2) is locally asymptoticallystable if 119877

0lt 1 and unstable if 119877

0gt 1

To discuss the properties of the endemic equilibriumpoint

Theorem 5 The endemic equilibrium point is locally asymp-totically stable if 119877

0gt 1

Proof We use Routh-Hurwitz criterion to establish the localstability of the endemic equilibrium The Jacobian matrix ofsystem (2) at endemic equilibrium is

Jlowast =[[[

[

minus (] + 1205824minus ]1199012+ 1199061+ 120588119868lowast) minus (120582

4minus ]1199012) minus (120588119878

lowast+ 1205824minus ]1199012) minus (]119901

1+ 1205824minus ]1199012+ 120588120579119878

lowast)

120588119868lowast+ 120588120579119862

lowastminus (] + 120582

1) 120588119878

lowast120588120579119878lowast

0 1205821

minus (] + 1205822) 0

0 0 11990131205822

]1199011minus ] minus 120582

3minus 1199062

]]]

]

(20)

Then the characteristic equation at 119864lowastlowast is 1205824 + 11989111205823+ 11989121205822+

1198913120582 + 1198914= 0 where

1198911= 1198861+ 1198872+ 1198883+ 1198894

1198912= 11988611198863+ 11988611198894+ 11988611198872+ 11988831198894+ 11988721198883+ 11988721198894minus 11988731198882minus 11988711198862

1198913= 119886111988831198894+ 119886111988721198883+ 119886111988721198894+ 119887211988831198894+ 119887411988821198893+ 119887111988821198863

minus 119886111988731198882minus 119887311988821198894minus 119887111988621198883minus 119887111988621198894

1198914= 1198861119887211988831198894+ 1198861119887411988821198893+ 1198871119888211988631198894minus 1198861119887311988821198894

minus 1198871119886211988831198894minus 1198871119886411988821198893

(21)

where

1198861= minus (] + 120582

4minus ]1199012+ 1199061+ 120588119868lowast)

1198862= minus (120582

4minus ]1199012) 119886

3= minus (120588119878

lowast+ 1205824minus ]1199012)

1198864= minus (]119901

1+ 1205824minus ]1199012+ 120588120579119878

lowast) 119887

1= 120588119868lowast+ 120588120579119862

lowast

1198872= minus (] + 120582

1) 119887

3= 120588119878lowast

1198874= 120588120579119878

lowast 119888

1= 0

1198882= 1205821 119888

3= minus (] + 120582

2)

1198884= 0 119889

1= 0

1198892= 0 119889

3= 11990131205822

1198894= ]1199011minus ] minus 120582

3minus 1199062

(22)

We need to verify the following three conditions

(a) 1198911 1198912 1198913 1198914gt 0

(b) 11989111198912minus 1198913gt 0

(c) 1198913(11989111198912minus 1198913) minus 1198912

11198914gt 0

Computational and Mathematical Methods in Medicine 7

It is easy to see that conditions (a) and (b) are satisfiedAfter computations we can prove that 119891

3(11989111198912minus 1198913) minus

1198912

11198914gt 0 is also validTheRouth-Hurwitz criterion and those

inequalities in (a)ndash(c) imply that the characteristic equationat 119864lowastlowast has only roots with negative real part which certifiesthe local stability of 119864lowastlowast

4 Optimal Control with Two Objectives

One of the early reasons for studying hepatitis B virus (HBV)infection is to improve the control variables and finally to putdown the infection of the population Optimal control is auseful mathematical tool that can be used to make decisionsin this case In the previous sections we have analyzed themodel with two control variables one is treatment and theother is vaccination and we consider their constant controlsthroughout the analysis But in fact these control variablesshould be time dependent In this section we consider thevaccination and treatment as time-dependent controls in acompact interval of time duration

Our goals here are to put down infection from the popu-lation by increasing the recovered individuals and reducingsusceptible exposed infected and carrier individuals in apopulation and to minimize the costs required to control thehepatitis B virus (HBV) infection by using vaccination andtreatment First of all we construct the objective functionalto be optimized as follows

119869 = int

119879

0

[1198601119878 (119905) + 119860

2119864 (119905) + 119860

3119868 (119905) + 119860

4119862 (119905)

+

1

2

(11986111199062

1(119905) + 119861

21199062

2(119905))] 119889119905

(23)

subject to

119878 (119905) = ] minus ]1199011119862 minus 120588 (119868 + 120579119862) 119878 minus ]119878 minus 119906

1119878

+ (1205824minus ]1199012) (1 minus 119878 minus 119864 minus 119868 minus 119862)

(119905) = 120588 (119868 + 120579119862) 119878 minus (] + 1205821) 119864

119868 (119905) = 1205821119864 minus (] + 120582

2) 119868

(119905) = ]1199011119862 + 119901

31205822119868 minus (] + 120582

3) 119862 minus 119906

2119862

(24)

Our object is to find (119906lowast1 119906lowast

2) such that

119869 (119906lowast

1 119906lowast

2) = min 119869 (119906

1 1199062) (25)

where

1199061(119905) 119906

2(119905) isin Γ

= 119906 (119905) | 0 le 119906 (119905) le 119906max (119905) le 1 0 le 119905 le 119879

119906 (119905) is Labesgue measurable

(26)

Here 119860119894 for 119894 = 1 4 are positive constants that

are represented to keep a balance in the size of 119878(119905) 119864(119905)119868(119905) and 119862(119905) respectively 119861

1and 119861

2 respectively are the

weights corresponding to the controls 1199061and 119906

2 119906max is the

maximum attainable value for controls (1199061max and 119906

2max)1199061max and 119906

2max will depend on the amount of resourcesavailable to implement each of the control measures [26]The 119861

1and 119861

2will depend on the relative importance of

each of the control measures in mitigating the spread of thedisease as well as the cost (human effort material resourcesetc) of implementing each of the control measures per unittime [26] Thus the terms 119861

11199062

1and 119861

21199062

2describe the costs

associated with vaccination and treatment respectively Thesquare of the control variables is taken here to remove theseverity of the side effects and overdoses of vaccination andtreatment [27]

41 Optimal Control Solution For existence of the solutionwe consider the control system (24) with initial condition

119878 (0) = 1198780 119864 (0) = 119864

0

119868 (0) = 1198680 119862 (0) = 119862

0

(27)

then we rewrite (2) in the following form

119889119909 (119905)

119889119905

= 119860119909 + 119865 (119909) (28)

where 119909(119905) = (119878(119905) 119864(119905) 119868(119905) 119862(119905)) is the vector of the statevariables and A and F(x) are defined as follows

A =

[[[[[[[

[

minus] minus 1199061minus 1205824+ ]1199012minus1205824+ ]1199012minus1205824+ ]1199012

minus1205824+ ]1199012

0 minus] minus 1205821

0 0

0 1205821

minus] minus 1205822

0

0 0 11990131205822

]1199011minus ] minus 120582

3minus 1199062

]]]]]]]

]

F (x) =[[[[[[

[

] + 120588 (119868 + 120579119862) 119878 + 1205824minus ]1199012

120588 (119868 + 120579119862) 119878

0

0

]]]]]]

]

(29)

8 Computational and Mathematical Methods in Medicine

We set

119866 (119909) = 119860119909 + 119865 (119909) (30)

The second term on the right-hand side of (30) satisfies

1003816100381610038161003816119865 (1199091) minus 119865 (119909

2)1003816100381610038161003816

le 119872 (10038161003816100381610038161198781minus 1198782

1003816100381610038161003816+10038161003816100381610038161198641minus 1198642

1003816100381610038161003816+10038161003816100381610038161198681minus 1198682

1003816100381610038161003816+10038161003816100381610038161198621minus 1198622

1003816100381610038161003816)

(31)

where the positive constant 119872 is independent of the statevariables 119909 and119872 le 1 Also we get

1003816100381610038161003816119866 (1199091) minus 119866 (119909

2)1003816100381610038161003816le 119871

10038161003816100381610038161199091minus 1199092

1003816100381610038161003816 (32)

where 119871 = max119872 119860 lt infin Therefore it follows thatthe function 119866 is uniformly Lipschitz continuous From thedefinition of the controls 119906

1(119905) and 119906

2(119905) and the restrictions

on the nonnegativeness of the state variables we see that asolution of the system (28) exists [24 28 29]

42The Lagrangian and Hamiltonian for the Control ProblemIn order to find an optimal solution pair first we shouldfind the Lagrangian and Hamiltonian for the optimal controlproblem (24) In fact the Lagrangian of problem is given by

119871 (119878 119864 119868 119862 1199061 1199062) = 119860

1119878 (119905) + 119860

2119864 (119905) + 119860

3119868 (119905)

+ 1198604119862 (119905) +

1

2

(11986111199062

1(119905) + 119861

21199062

2(119905))

(33)

We are looking for the minimal value of the LagrangianTo accomplish this we define Hamiltonian119867 for the controlproblem as follows

119867(119878 119864 119868 119862 1199061 1199062 1205721 1205722 1205723 1205724)

= 119871 + 1205721(119905)

119889119878

119889119905

+ 1205722(119905)

119889119864

119889119905

+ 1205723(119905)

119889119868

119889119905

+ 1205724(119905)

119889119862

119889119905

(34)

where 120572119894(119905) for 119894 = 1 2 3 4 are the adjoint variables and can

be determined by solving the following system of differentialequations

1(119905) = minus

120597119867

120597119878

= minus [1198601+ 1205721((]1199012minus 1205824) minus 120588 (119868 + 120579119862) minus 119906

1minus ])

+1205722120588 (119868 + 120579119862)]

2(119905) = minus

120597119867

120597119864

= minus [1198602+ 1205721(]1199012minus 1205824) + 1205722(minus] minus 120582

1) + 12057231205821]

3(119905) = minus

120597119867

120597119868

= minus [1198603+ 1205721((]1199012minus 1205824) minus 120588119878) + 120572

2120588119878

+1205723(minus] minus 120582

2) + 120572411990131205822]

4(119905) = minus

120597119867

120597119862

= minus [1198604+ 1205721(]1199012minus 1205824minus ]1199011minus 120588120579119878)

+1205722120588120579119878 + 120572

4(]1199011minus ] + 120582

3+ 1199062)]

(35)

satisfying the transversality conditions and optimality condi-tions

1205721(119879) = 120572

2(119879) = 120572

3(119879) = 120572

4(119879) = 0

120597119867

1205971199061

= 11986111199061minus 1205721119878 = 0

120597119867

1205971199062

= 11986121199062minus 1205724119862 = 0

(36)

We now state and prove the following theorem

Theorem 6 There is an optimal control (119906lowast1 119906lowast

2) such that

119869 (119878 119864 119868 119878 119906lowast

1 119906lowast

2) = min 119869 (119878 119864 119868 119878 119906

1 1199062) (37)

subject to the system of differential equation (24)

Proof Here both the control and state variables are non-negative values In this minimized problem the necessaryconvexity of the objective functional in 119906

1and 1199062are satisfied

The set of all control variables 1199061(119905) 1199062(119905) is also convex and

closed by definition The optimal system is bounded whichdetermines the compactness needed for the existence of theoptimal control In addition the integrand in the functional

int

119879

0

[1198601119878 (119905) + 119860

2119864 (119905) + 119860

3119868 (119905) + 119860

4119862 (119905)

+

1

2

(11986111199062

1(119905) + 119861

21199062

2(119905))] 119889119905

(38)

is convex on the control set Hence the theorem (the proof iscomplete)

Computational and Mathematical Methods in Medicine 9

43 Necessary and Sufficient Conditions for Optimal ControlsApplying Pontryaginrsquos maximum principle [30] on the con-structed Hamiltonian 119867 and the theorem (24) we obtainthe optimal steady-state solution and corresponding controlas follows If taking 119883 = (119878 119864 119868 119862) and 119880 = (119906

1 1199062)

then (119883lowast 119880lowast) is an optimal solution of an optimal controlproblem then we now state the following theorem

Theorem 7 The optimal control pair (119906lowast1 119906lowast

2) which mini-

mizes J over the region Γ is given by

119906lowast

1= max[0119879]

0min (1(119905) 1)

119906lowast

2= max[0119879]

0min (2(119905) 1)

(39)

where 1= 11198781198611and 2= 41198621198612and let

1 2 3 4 be

the solution of system (35)

Proof With the help of the optimality conditions we have

120597119867

1205971199061

= 11986111199061minus 1205721119878 = 0 997904rArr 119906

1=

1119878

1198611

= (1)

120597119867

1205971199062

= 11986121199062minus 1205724119862 = 0 997904rArr 119906

2=

4119862

1198612

= (2)

(40)

Using the property of the control space Γ the two controlswhich are bounded with upper and lower bounds arerespectively 1 and 0 that is

119906lowast

1=

0 if 1le 0

1

if 0 lt 1lt 1

1 if 1ge 1

(41)

This can be rewritten in compact notation

119906lowast

1= max[0119879]

0min (1(119905) 1) (42)

and similarly

119906lowast

2=

0 if 2le 0

2

if 0 lt 2lt 1

1 if 2ge 1

(43)

This can be rewritten in compact notation

119906lowast

2= max[0119879]

0min (2(119905) 1) (44)

Hence for these pair of controls (119906lowast1 119906lowast

2) we get the optimum

value of the functional 119869 given by (24)

5 Numerical Examples

Numerical solutions to the optimality system (24) arediscussed in this section We make several interestingobservation by numerically simulating (2) in the rangeof parameter values We consider the parameter set

Δ = ] 120588 120579 1205821 1205822 1205823 1205824 1199011 1199012 1199013 1198601 1198602 1198603 1198604 1198611 1198612

some of the parameters are taken from the published articlesand some are assumed with feasible values Moreover thetime interval for which the optimal control is applied istaken as 70 years also consider Ψ = 119878

0 1198640 1198680 1198620 as initial

condition for simulation of the model The main parametervalues are listed in Table 1 We compare the results having nocontrols only vaccination control only treatment controland both vaccination and treatment controls

With the parameter values in Table 1 the system asymp-totically approaches towards the equilibrium 119864

lowastlowast

1(00858

03327 04003 05349) where the basic reproduction ratio1198770= 39810 (Figure 2) For the parameters set the system

(2) has two feasible equilibria one is disease free and theother is endemic and the endemic equilibrium is locallyasymptotically stable The effect of two control measures ondisease dynamics may be understood well if we considerFigure 2 It explains how control reproduction ratio 119877

0

evolves with different rates of 1199061and 119906

2 It is seen that both

vaccination and treatment reduce the value of 1198770effectively

But an integrated control works better than either of thecontrol measures

In this section we use an iterative method to obtainresults for an optimal control problem of the proposedmodel(24) We use Runge-Kuttarsquos fourth-order procedure [23] hereto solve the optimality system consisting of eight ordinarydifferential equations having four state equations and fouradjoint equations and boundary conditions Because stateequations have initial conditions and adjoint equations haveconditions at the final time an iterative program was createdto numerically simulate solutions Given an initial guessfor the controls to compute the optimal state values theprogram solves (24) with initial conditions (27) forward intime interval [0 70]using aRungeKuttamethod of the fourthorder Resulting state values are placed in adjoint equations(35) These adjoint equations with given final conditions arethen solved backwards in time Again a fourth order RungeKutta method is employed Both state and adjoint values areused to update the control using the characterization (39)and the entire process repeats itself This iterative processterminates when current state adjoint and control valuesare sufficiently close to successive values Then we use thebackward Runge-Kutta fourth-order procedure to solve theadjoint variables in the same time interval with the help ofthe solution of the state variables transversality conditions

We have plotted susceptible exposed acute infectionindividuals and carriers individuals with andwithout controlby considering values of parameters We simulate the systemat different values of rate of 119906

1and 119906

2

From Figures 3 4 and 5 we see that after 20 years thenumber of susceptible population decreases than when thereis no control In this case most of this population tends to theinfected class Again when only treatment control is appliedthen the number of susceptible population is not muchdifferent than the population in the case having no controlBut the susceptible population differs much from these twostrategies if we apply the strategies of only vaccination controland both vaccination and treatment controls At a high rateof vaccination the sensitive population density is reduced

10 Computational and Mathematical Methods in Medicine

Table 1 Parameter values used in numerical simulations

Parameter Description Values range Reference] Birth (and death) rate 00121 [17 18]120588 Transmission rate 08ndash2049 [17]120579 Infectiousness of carriers relative to acute infections 0-1 [17]1205821

Rate of moving from exposed to acute 6 per year [16]1205822

Rate at which individuals leave the acute infection class 4 per year [16]1205823

Rate of moving from carrier to recovery 0025 per year [16]1205824

Loss of recovery rate 003ndash006 [17]1198751

Probability of infected newborns 011 [19]1198752

Probability of immune newborns 01 [19]1198753

Proportion of acute infection individuals becoming carriers 005ndash09 [17]1198601

Weight factor for susceptible individuals 0091 [31]1198602

Weight factor for exposed individuals 001 [31]1198603

Weight factor for infected individuals 004 [31]1198604

Weight factor for carrier individuals 005 [31]1198611

Weight factor for the controls 1199061

15 [31]1198612

Weight factor for the controls 1199062

27 [31]1198780

Susceptible individuals 0493 [4]1198640

Exposed individuals 00035 [4]1198680

Acute infection individuals 00035 [4]1198620

Chronic HBV carriers 0007 [4]

0 10 20 30 40 50 60 70Time (year)

Susc

eptib

le p

opul

atio

n

NV-NTNV-T

V-NTV-T

10minus4

10minus3

10minus2

10minus1

100

Figure 3 The plot shows the changes in sensitive populations(without) vaccination and (without) treatment

to a very low level initially and then it takes longer time torestore the steady-state value

From Figures 6 7 and 8 we see that the number ofexposed population increase thanwhen there is no control Inthis case most of this population tends to the acute infectedclass Again when only treatment control is applied thenthe number of exposed population is not much different

0 10 20 30 40 50 60 70Time (year)

Susc

eptib

le p

opul

atio

n

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0365 u2 = 0

u1 = 045 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

Figure 4 The plot shows the sensitivity of sensitive populations fordifferent values of control 119906

1

than the population in the case having no control But theexposed population differs much from these two strategies ifwe apply the strategies of only vaccination control and bothvaccination and treatment controls

From Figures 9 10 and 11 we see that the number ofacute infected population increases than when there is nocontrol In this case most of this population tends to thecarrier class Again when only treatment control is appliedthen the number of acute infected population is not muchdifferent than the population in the case having no controlBut the acute infected population differsmuch from these two

Computational and Mathematical Methods in Medicine 11

0 10 20 30 40 50 60 70Time (year)

Susc

eptib

le p

opul

atio

n

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Figure 5 The plot shows the sensitivity of sensitive populations fordifferent values of control 119906

2

Expo

sed

popu

latio

n

NV-NTNV-T

V-NTV-T

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 6 The plot shows the changes in exposed populations(without) vaccination and (without) treatment

strategies if we apply the strategies of only vaccination controland both vaccination and treatment controls

Again from Figures 12 13 and 14 we see that thenumber of carrier population increases than when there isno control We see that the application of only vaccinationcontrol gives better result than the application of no controlAgain application of treatment control would give betterresult than the application of vaccination control since thetreatment control is better than the vaccination control whilethe application of both vaccination and treatment controlsgive the best result as in this case the number of carrierpopulation would be the least in number

Expo

sed

popu

latio

n

u1 = 0 u2 = 0

u1 = 03635 u2 = 0

u1 = 045 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 7 The plot shows the sensitivity of exposed populations fordifferent values of control 119906

1

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Expo

sed

popu

latio

n

Figure 8 The plot shows the sensitivity of exposed populations fordifferent values of control 119906

2

Finally the effect of two control measures on diseasedynamics may be understood well if we consider Figures 1ndash14 15 16 17 and 18 Since our main purpose is to reducethe number of sensitive exposed acute infection and carrierindividuals therefore numerical simulation results show thatoptimal combination of vaccination and treatment is themosteffective way to control hepatitis B virus infection

12 Computational and Mathematical Methods in MedicineAc

ute i

nfec

tion

popu

latio

n

NV-NTNV-T

V-NTV-T

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 9The plot shows the changes in acute infection populations(without) vaccination and (without) treatment

Acut

e inf

ectio

n po

pulat

ion

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0365 u2 = 0

u1 = 045 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

Figure 10 The plot shows the sensitivity of acute infection popula-tions for different values of control 119906

1

6 Conclusion and Suggestions

In this paper we propose an S-119864-119868-119862-119877model of hepatitis Bvirus infection with two controls vaccination and treatmentFirst we analyze the dynamic behavior of the system for con-stant controls In the constant controls case we calculate thebasic reproduction number and investigate the existence andstability of equilibriaThere are two nonnegative equilibria of

Acut

e inf

ectio

n po

pulat

ion

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Figure 11 The plot shows the sensitivity of acute infection popula-tions for different values of control 119906

2

Carr

ier p

opul

atio

n

NV-NTNV-T

V-NTV-T

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 12 The plot shows the changes in carrier populations(without) vaccination and (without) treatment

the system namely the disease-free and endemicWe see thatthe disease-free equilibriumwhich always exists and is locallyasymptotically stable if 119877

0lt 1 and endemic equilibrium

which exists and is locally asymptotically stable if 1198770gt 1

After investigating the dynamic behavior of the systemwith constant controls we formulate an optimal controlproblem if the controls become time dependent and solve itby using Pontryaginrsquos maximum principle Different possiblecombinations of controls are used and their effectiveness iscompared by simulation works Also from the numericalresults it is very clear that a combination of mixed controlmeasures respond better than any other independent control

Computational and Mathematical Methods in Medicine 13Ca

rrie

r pop

ulat

ion

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0365 u2 = 0

u1 = 046 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

Figure 13 The plot shows the sensitivity of carrier populations fordifferent values of control 119906

1

Carr

ier p

opul

atio

n

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Figure 14 The plot shows the sensitivity of carrier populations fordifferent values of control 119906

2

There is still a tremendous amount of work needed tobe done in this area Parameters are rarely constant becausethey depend on environmental conditions We do not knowhowever the detailed relationship between these parametersand environmental conditions There may be a time lag as asusceptible populationmay take some time to be infected andalso a susceptible population may take some time to immune

Popu

latio

n (N

V-N

T)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 15 The plot shows the changes in total populations withoutvaccination and without treatment

Popu

latio

n (V

-NT)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 16 The plot shows the changes in total populations withvaccination and without treatment

Popu

latio

n (N

V-T)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 17 The plot shows the changes in total populations withoutvaccination and with treatment

14 Computational and Mathematical Methods in MedicinePo

pulat

ion

(V-T

)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 18 The plot shows the changes in total populations withvaccination and treatment

after vaccination We leave all these possible extensions forthe future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] WHO Hepatitis B Fact Sheet No 204 The World HealthOrganisation Geneva Switzerland 2013 httpwwwwhointmediacentrefactsheetsfs204en

[2] CanadianCentre forOccupationalHealth and Safety ldquoHepatitisBrdquo httpwwwccohscaoshanswersdiseaseshepatitis bhtml

[3] ldquoHealthcare stumbling in RIrsquos Hepatitis fightrdquoThe Jakarta PostJanuary 2011

[4] G F Medley N A Lindop W J Edmunds and D JNokes ldquoHepatitis-B virus endemicity heterogeneity catas-trophic dynamics and controlrdquo Nature Medicine vol 7 no 5pp 619ndash624 2001

[5] J Mann and M Roberts ldquoModelling the epidemiology ofhepatitis B in New Zealandrdquo Journal of Theoretical Biology vol269 no 1 pp 266ndash272 2011

[6] ldquoHepatitis B (HBV)rdquo httpkidshealthorgteensexual healthstdsstd hepatitishtml

[7] CDC ldquoHepatitis B virus a comprehensive strategy for elim-inating transmission in the United States through universalchildhood vaccination Recommendations of the Immuniza-tion Practices Advisory Committee (ACIP)rdquo MMWR Recom-mendations and Reports vol 40(RR-13) pp 1ndash25 1991

[8] M K Libbus and L M Phillips ldquoPublic health management ofperinatal hepatitis B virusrdquo Public Health Nursing vol 26 no 4pp 353ndash361 2009

[9] F B Hollinger and D T Lau ldquoHepatitis B the pathway torecovery through treatmentrdquo Gastroenterology Clinics of NorthAmerica vol 35 no 4 pp 895ndash931 2006

[10] C-L Lai andM-F Yuen ldquoThe natural history and treatment ofchronic hepatitis B a critical evaluation of standard treatment

criteria and end pointsrdquoAnnals of InternalMedicine vol 147 no1 pp 58ndash61 2007

[11] R M Anderson and R M May Infectious Disease of HumansDynamics and Control Oxford University Press Oxford UK1991

[12] S Thornley C Bullen and M Roberts ldquoHepatitis B in ahigh prevalence New Zealand population a mathematicalmodel applied to infection control policyrdquo Journal ofTheoreticalBiology vol 254 no 3 pp 599ndash603 2008

[13] S-J Zhao Z-Y Xu and Y Lu ldquoA mathematical model ofhepatitis B virus transmission and its application for vaccinationstrategy inChinardquo International Journal of Epidemiology vol 29no 4 pp 744ndash752 2000

[14] K Wang W Wang and S Song ldquoDynamics of an HBV modelwith diffusion and delayrdquo Journal ofTheoretical Biology vol 253no 1 pp 36ndash44 2008

[15] R Xu andZMa ldquoAnHBVmodelwith diffusion and time delayrdquoJournal of Theoretical Biology vol 257 no 3 pp 499ndash509 2009

[16] L Zou W Zhang and S Ruan ldquoModeling the transmissiondynamics and control of hepatitis B virus in Chinardquo Journal ofTheoretical Biology vol 262 no 2 pp 330ndash338 2010

[17] J Pang J-A Cui and X Zhou ldquoDynamical behavior of ahepatitis B virus transmission model with vaccinationrdquo Journalof Theoretical Biology vol 265 no 4 pp 572ndash578 2010

[18] S Zhang and Y Zhou ldquoThe analysis and application of an HBVmodelrdquoAppliedMathematicalModelling vol 36 no 3 pp 1302ndash1312 2012

[19] S Bhattacharyya and S Ghosh ldquoOptimal control of verticallytransmitted diseaserdquoComputational andMathematicalMethodsin Medicine vol 11 no 4 pp 369ndash387 2010

[20] T K Kar andA Batabyal ldquoStability analysis and optimal controlof an SIR epidemic model with vaccinationrdquo Biosystems vol104 no 2-3 pp 127ndash135 2011

[21] T-K Kar and S Jana ldquoA theoretical study on mathematicalmodelling of an infectious disease with application of optimalcontrolrdquo Biosystems vol 111 no 1 pp 37ndash50 2013

[22] M J Keeling and P Rohani Modeling Infectious Diseases inHumans and Animals Princeton University Press PrincetonNJ USA 2008

[23] E Jung S Lenhart and Z Feng ldquoOptimal control of treatmentsin a two-strain tuberculosis modelrdquo Discrete and ContinuousDynamical Systems B vol 2 no 4 pp 473ndash482 2002

[24] G Birkhoff and G-C Rota Ordinary Differential EquationsJohn Wiley amp Sons New York NY USA 4th edition 1989

[25] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[26] T T Yusuf and F Benyah ldquoOptimal control of vaccination andtreatment for an SIR epidemiological modelrdquo World Journal ofModelling and Simulation vol 8 no 3 pp 194ndash204 2008

[27] H-R Joshi S Lenhart M Y Li and L Wang ldquoOptimalcontrol methods applied to disease modelsrdquo in MathematicalStudies on Human Disease Dynamics vol 410 of ContemporaryMathematics pp 187ndash207 American Mathematical SocietyProvidence RI USA 2006

[28] G Zaman Y H Kang and I H Jung ldquoStability analysis andoptimal vaccination of an SIR epidemic modelrdquo Biosystems vol93 no 3 pp 240ndash249 2008

[29] D Iacoviello and N Stasiob ldquoOptimal control for SIRCepidemic outbreakrdquo Computer Methods and Programs inBiomedicine vol 110 no 3 pp 333ndash342 2013

Computational and Mathematical Methods in Medicine 15

[30] L S Pontryagin V G Boltyanskii R V Gamkrelidze and EF Mishchenko The Mathematical Theory of Optimal ProcessesJohn Wiley amp Sons New York NY USA 1962

[31] WHO Hepatitis B Fact Sheet No 204 The World HealthOrganisation Geneva Switzerland 2000 httpwwwwhointmediacentrefactsheetsfs204en

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Disease Markers

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Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

Computational and Mathematical Methods in Medicine 7

It is easy to see that conditions (a) and (b) are satisfiedAfter computations we can prove that 119891

3(11989111198912minus 1198913) minus

1198912

11198914gt 0 is also validTheRouth-Hurwitz criterion and those

inequalities in (a)ndash(c) imply that the characteristic equationat 119864lowastlowast has only roots with negative real part which certifiesthe local stability of 119864lowastlowast

4 Optimal Control with Two Objectives

One of the early reasons for studying hepatitis B virus (HBV)infection is to improve the control variables and finally to putdown the infection of the population Optimal control is auseful mathematical tool that can be used to make decisionsin this case In the previous sections we have analyzed themodel with two control variables one is treatment and theother is vaccination and we consider their constant controlsthroughout the analysis But in fact these control variablesshould be time dependent In this section we consider thevaccination and treatment as time-dependent controls in acompact interval of time duration

Our goals here are to put down infection from the popu-lation by increasing the recovered individuals and reducingsusceptible exposed infected and carrier individuals in apopulation and to minimize the costs required to control thehepatitis B virus (HBV) infection by using vaccination andtreatment First of all we construct the objective functionalto be optimized as follows

119869 = int

119879

0

[1198601119878 (119905) + 119860

2119864 (119905) + 119860

3119868 (119905) + 119860

4119862 (119905)

+

1

2

(11986111199062

1(119905) + 119861

21199062

2(119905))] 119889119905

(23)

subject to

119878 (119905) = ] minus ]1199011119862 minus 120588 (119868 + 120579119862) 119878 minus ]119878 minus 119906

1119878

+ (1205824minus ]1199012) (1 minus 119878 minus 119864 minus 119868 minus 119862)

(119905) = 120588 (119868 + 120579119862) 119878 minus (] + 1205821) 119864

119868 (119905) = 1205821119864 minus (] + 120582

2) 119868

(119905) = ]1199011119862 + 119901

31205822119868 minus (] + 120582

3) 119862 minus 119906

2119862

(24)

Our object is to find (119906lowast1 119906lowast

2) such that

119869 (119906lowast

1 119906lowast

2) = min 119869 (119906

1 1199062) (25)

where

1199061(119905) 119906

2(119905) isin Γ

= 119906 (119905) | 0 le 119906 (119905) le 119906max (119905) le 1 0 le 119905 le 119879

119906 (119905) is Labesgue measurable

(26)

Here 119860119894 for 119894 = 1 4 are positive constants that

are represented to keep a balance in the size of 119878(119905) 119864(119905)119868(119905) and 119862(119905) respectively 119861

1and 119861

2 respectively are the

weights corresponding to the controls 1199061and 119906

2 119906max is the

maximum attainable value for controls (1199061max and 119906

2max)1199061max and 119906

2max will depend on the amount of resourcesavailable to implement each of the control measures [26]The 119861

1and 119861

2will depend on the relative importance of

each of the control measures in mitigating the spread of thedisease as well as the cost (human effort material resourcesetc) of implementing each of the control measures per unittime [26] Thus the terms 119861

11199062

1and 119861

21199062

2describe the costs

associated with vaccination and treatment respectively Thesquare of the control variables is taken here to remove theseverity of the side effects and overdoses of vaccination andtreatment [27]

41 Optimal Control Solution For existence of the solutionwe consider the control system (24) with initial condition

119878 (0) = 1198780 119864 (0) = 119864

0

119868 (0) = 1198680 119862 (0) = 119862

0

(27)

then we rewrite (2) in the following form

119889119909 (119905)

119889119905

= 119860119909 + 119865 (119909) (28)

where 119909(119905) = (119878(119905) 119864(119905) 119868(119905) 119862(119905)) is the vector of the statevariables and A and F(x) are defined as follows

A =

[[[[[[[

[

minus] minus 1199061minus 1205824+ ]1199012minus1205824+ ]1199012minus1205824+ ]1199012

minus1205824+ ]1199012

0 minus] minus 1205821

0 0

0 1205821

minus] minus 1205822

0

0 0 11990131205822

]1199011minus ] minus 120582

3minus 1199062

]]]]]]]

]

F (x) =[[[[[[

[

] + 120588 (119868 + 120579119862) 119878 + 1205824minus ]1199012

120588 (119868 + 120579119862) 119878

0

0

]]]]]]

]

(29)

8 Computational and Mathematical Methods in Medicine

We set

119866 (119909) = 119860119909 + 119865 (119909) (30)

The second term on the right-hand side of (30) satisfies

1003816100381610038161003816119865 (1199091) minus 119865 (119909

2)1003816100381610038161003816

le 119872 (10038161003816100381610038161198781minus 1198782

1003816100381610038161003816+10038161003816100381610038161198641minus 1198642

1003816100381610038161003816+10038161003816100381610038161198681minus 1198682

1003816100381610038161003816+10038161003816100381610038161198621minus 1198622

1003816100381610038161003816)

(31)

where the positive constant 119872 is independent of the statevariables 119909 and119872 le 1 Also we get

1003816100381610038161003816119866 (1199091) minus 119866 (119909

2)1003816100381610038161003816le 119871

10038161003816100381610038161199091minus 1199092

1003816100381610038161003816 (32)

where 119871 = max119872 119860 lt infin Therefore it follows thatthe function 119866 is uniformly Lipschitz continuous From thedefinition of the controls 119906

1(119905) and 119906

2(119905) and the restrictions

on the nonnegativeness of the state variables we see that asolution of the system (28) exists [24 28 29]

42The Lagrangian and Hamiltonian for the Control ProblemIn order to find an optimal solution pair first we shouldfind the Lagrangian and Hamiltonian for the optimal controlproblem (24) In fact the Lagrangian of problem is given by

119871 (119878 119864 119868 119862 1199061 1199062) = 119860

1119878 (119905) + 119860

2119864 (119905) + 119860

3119868 (119905)

+ 1198604119862 (119905) +

1

2

(11986111199062

1(119905) + 119861

21199062

2(119905))

(33)

We are looking for the minimal value of the LagrangianTo accomplish this we define Hamiltonian119867 for the controlproblem as follows

119867(119878 119864 119868 119862 1199061 1199062 1205721 1205722 1205723 1205724)

= 119871 + 1205721(119905)

119889119878

119889119905

+ 1205722(119905)

119889119864

119889119905

+ 1205723(119905)

119889119868

119889119905

+ 1205724(119905)

119889119862

119889119905

(34)

where 120572119894(119905) for 119894 = 1 2 3 4 are the adjoint variables and can

be determined by solving the following system of differentialequations

1(119905) = minus

120597119867

120597119878

= minus [1198601+ 1205721((]1199012minus 1205824) minus 120588 (119868 + 120579119862) minus 119906

1minus ])

+1205722120588 (119868 + 120579119862)]

2(119905) = minus

120597119867

120597119864

= minus [1198602+ 1205721(]1199012minus 1205824) + 1205722(minus] minus 120582

1) + 12057231205821]

3(119905) = minus

120597119867

120597119868

= minus [1198603+ 1205721((]1199012minus 1205824) minus 120588119878) + 120572

2120588119878

+1205723(minus] minus 120582

2) + 120572411990131205822]

4(119905) = minus

120597119867

120597119862

= minus [1198604+ 1205721(]1199012minus 1205824minus ]1199011minus 120588120579119878)

+1205722120588120579119878 + 120572

4(]1199011minus ] + 120582

3+ 1199062)]

(35)

satisfying the transversality conditions and optimality condi-tions

1205721(119879) = 120572

2(119879) = 120572

3(119879) = 120572

4(119879) = 0

120597119867

1205971199061

= 11986111199061minus 1205721119878 = 0

120597119867

1205971199062

= 11986121199062minus 1205724119862 = 0

(36)

We now state and prove the following theorem

Theorem 6 There is an optimal control (119906lowast1 119906lowast

2) such that

119869 (119878 119864 119868 119878 119906lowast

1 119906lowast

2) = min 119869 (119878 119864 119868 119878 119906

1 1199062) (37)

subject to the system of differential equation (24)

Proof Here both the control and state variables are non-negative values In this minimized problem the necessaryconvexity of the objective functional in 119906

1and 1199062are satisfied

The set of all control variables 1199061(119905) 1199062(119905) is also convex and

closed by definition The optimal system is bounded whichdetermines the compactness needed for the existence of theoptimal control In addition the integrand in the functional

int

119879

0

[1198601119878 (119905) + 119860

2119864 (119905) + 119860

3119868 (119905) + 119860

4119862 (119905)

+

1

2

(11986111199062

1(119905) + 119861

21199062

2(119905))] 119889119905

(38)

is convex on the control set Hence the theorem (the proof iscomplete)

Computational and Mathematical Methods in Medicine 9

43 Necessary and Sufficient Conditions for Optimal ControlsApplying Pontryaginrsquos maximum principle [30] on the con-structed Hamiltonian 119867 and the theorem (24) we obtainthe optimal steady-state solution and corresponding controlas follows If taking 119883 = (119878 119864 119868 119862) and 119880 = (119906

1 1199062)

then (119883lowast 119880lowast) is an optimal solution of an optimal controlproblem then we now state the following theorem

Theorem 7 The optimal control pair (119906lowast1 119906lowast

2) which mini-

mizes J over the region Γ is given by

119906lowast

1= max[0119879]

0min (1(119905) 1)

119906lowast

2= max[0119879]

0min (2(119905) 1)

(39)

where 1= 11198781198611and 2= 41198621198612and let

1 2 3 4 be

the solution of system (35)

Proof With the help of the optimality conditions we have

120597119867

1205971199061

= 11986111199061minus 1205721119878 = 0 997904rArr 119906

1=

1119878

1198611

= (1)

120597119867

1205971199062

= 11986121199062minus 1205724119862 = 0 997904rArr 119906

2=

4119862

1198612

= (2)

(40)

Using the property of the control space Γ the two controlswhich are bounded with upper and lower bounds arerespectively 1 and 0 that is

119906lowast

1=

0 if 1le 0

1

if 0 lt 1lt 1

1 if 1ge 1

(41)

This can be rewritten in compact notation

119906lowast

1= max[0119879]

0min (1(119905) 1) (42)

and similarly

119906lowast

2=

0 if 2le 0

2

if 0 lt 2lt 1

1 if 2ge 1

(43)

This can be rewritten in compact notation

119906lowast

2= max[0119879]

0min (2(119905) 1) (44)

Hence for these pair of controls (119906lowast1 119906lowast

2) we get the optimum

value of the functional 119869 given by (24)

5 Numerical Examples

Numerical solutions to the optimality system (24) arediscussed in this section We make several interestingobservation by numerically simulating (2) in the rangeof parameter values We consider the parameter set

Δ = ] 120588 120579 1205821 1205822 1205823 1205824 1199011 1199012 1199013 1198601 1198602 1198603 1198604 1198611 1198612

some of the parameters are taken from the published articlesand some are assumed with feasible values Moreover thetime interval for which the optimal control is applied istaken as 70 years also consider Ψ = 119878

0 1198640 1198680 1198620 as initial

condition for simulation of the model The main parametervalues are listed in Table 1 We compare the results having nocontrols only vaccination control only treatment controland both vaccination and treatment controls

With the parameter values in Table 1 the system asymp-totically approaches towards the equilibrium 119864

lowastlowast

1(00858

03327 04003 05349) where the basic reproduction ratio1198770= 39810 (Figure 2) For the parameters set the system

(2) has two feasible equilibria one is disease free and theother is endemic and the endemic equilibrium is locallyasymptotically stable The effect of two control measures ondisease dynamics may be understood well if we considerFigure 2 It explains how control reproduction ratio 119877

0

evolves with different rates of 1199061and 119906

2 It is seen that both

vaccination and treatment reduce the value of 1198770effectively

But an integrated control works better than either of thecontrol measures

In this section we use an iterative method to obtainresults for an optimal control problem of the proposedmodel(24) We use Runge-Kuttarsquos fourth-order procedure [23] hereto solve the optimality system consisting of eight ordinarydifferential equations having four state equations and fouradjoint equations and boundary conditions Because stateequations have initial conditions and adjoint equations haveconditions at the final time an iterative program was createdto numerically simulate solutions Given an initial guessfor the controls to compute the optimal state values theprogram solves (24) with initial conditions (27) forward intime interval [0 70]using aRungeKuttamethod of the fourthorder Resulting state values are placed in adjoint equations(35) These adjoint equations with given final conditions arethen solved backwards in time Again a fourth order RungeKutta method is employed Both state and adjoint values areused to update the control using the characterization (39)and the entire process repeats itself This iterative processterminates when current state adjoint and control valuesare sufficiently close to successive values Then we use thebackward Runge-Kutta fourth-order procedure to solve theadjoint variables in the same time interval with the help ofthe solution of the state variables transversality conditions

We have plotted susceptible exposed acute infectionindividuals and carriers individuals with andwithout controlby considering values of parameters We simulate the systemat different values of rate of 119906

1and 119906

2

From Figures 3 4 and 5 we see that after 20 years thenumber of susceptible population decreases than when thereis no control In this case most of this population tends to theinfected class Again when only treatment control is appliedthen the number of susceptible population is not muchdifferent than the population in the case having no controlBut the susceptible population differs much from these twostrategies if we apply the strategies of only vaccination controland both vaccination and treatment controls At a high rateof vaccination the sensitive population density is reduced

10 Computational and Mathematical Methods in Medicine

Table 1 Parameter values used in numerical simulations

Parameter Description Values range Reference] Birth (and death) rate 00121 [17 18]120588 Transmission rate 08ndash2049 [17]120579 Infectiousness of carriers relative to acute infections 0-1 [17]1205821

Rate of moving from exposed to acute 6 per year [16]1205822

Rate at which individuals leave the acute infection class 4 per year [16]1205823

Rate of moving from carrier to recovery 0025 per year [16]1205824

Loss of recovery rate 003ndash006 [17]1198751

Probability of infected newborns 011 [19]1198752

Probability of immune newborns 01 [19]1198753

Proportion of acute infection individuals becoming carriers 005ndash09 [17]1198601

Weight factor for susceptible individuals 0091 [31]1198602

Weight factor for exposed individuals 001 [31]1198603

Weight factor for infected individuals 004 [31]1198604

Weight factor for carrier individuals 005 [31]1198611

Weight factor for the controls 1199061

15 [31]1198612

Weight factor for the controls 1199062

27 [31]1198780

Susceptible individuals 0493 [4]1198640

Exposed individuals 00035 [4]1198680

Acute infection individuals 00035 [4]1198620

Chronic HBV carriers 0007 [4]

0 10 20 30 40 50 60 70Time (year)

Susc

eptib

le p

opul

atio

n

NV-NTNV-T

V-NTV-T

10minus4

10minus3

10minus2

10minus1

100

Figure 3 The plot shows the changes in sensitive populations(without) vaccination and (without) treatment

to a very low level initially and then it takes longer time torestore the steady-state value

From Figures 6 7 and 8 we see that the number ofexposed population increase thanwhen there is no control Inthis case most of this population tends to the acute infectedclass Again when only treatment control is applied thenthe number of exposed population is not much different

0 10 20 30 40 50 60 70Time (year)

Susc

eptib

le p

opul

atio

n

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0365 u2 = 0

u1 = 045 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

Figure 4 The plot shows the sensitivity of sensitive populations fordifferent values of control 119906

1

than the population in the case having no control But theexposed population differs much from these two strategies ifwe apply the strategies of only vaccination control and bothvaccination and treatment controls

From Figures 9 10 and 11 we see that the number ofacute infected population increases than when there is nocontrol In this case most of this population tends to thecarrier class Again when only treatment control is appliedthen the number of acute infected population is not muchdifferent than the population in the case having no controlBut the acute infected population differsmuch from these two

Computational and Mathematical Methods in Medicine 11

0 10 20 30 40 50 60 70Time (year)

Susc

eptib

le p

opul

atio

n

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Figure 5 The plot shows the sensitivity of sensitive populations fordifferent values of control 119906

2

Expo

sed

popu

latio

n

NV-NTNV-T

V-NTV-T

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 6 The plot shows the changes in exposed populations(without) vaccination and (without) treatment

strategies if we apply the strategies of only vaccination controland both vaccination and treatment controls

Again from Figures 12 13 and 14 we see that thenumber of carrier population increases than when there isno control We see that the application of only vaccinationcontrol gives better result than the application of no controlAgain application of treatment control would give betterresult than the application of vaccination control since thetreatment control is better than the vaccination control whilethe application of both vaccination and treatment controlsgive the best result as in this case the number of carrierpopulation would be the least in number

Expo

sed

popu

latio

n

u1 = 0 u2 = 0

u1 = 03635 u2 = 0

u1 = 045 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 7 The plot shows the sensitivity of exposed populations fordifferent values of control 119906

1

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Expo

sed

popu

latio

n

Figure 8 The plot shows the sensitivity of exposed populations fordifferent values of control 119906

2

Finally the effect of two control measures on diseasedynamics may be understood well if we consider Figures 1ndash14 15 16 17 and 18 Since our main purpose is to reducethe number of sensitive exposed acute infection and carrierindividuals therefore numerical simulation results show thatoptimal combination of vaccination and treatment is themosteffective way to control hepatitis B virus infection

12 Computational and Mathematical Methods in MedicineAc

ute i

nfec

tion

popu

latio

n

NV-NTNV-T

V-NTV-T

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 9The plot shows the changes in acute infection populations(without) vaccination and (without) treatment

Acut

e inf

ectio

n po

pulat

ion

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0365 u2 = 0

u1 = 045 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

Figure 10 The plot shows the sensitivity of acute infection popula-tions for different values of control 119906

1

6 Conclusion and Suggestions

In this paper we propose an S-119864-119868-119862-119877model of hepatitis Bvirus infection with two controls vaccination and treatmentFirst we analyze the dynamic behavior of the system for con-stant controls In the constant controls case we calculate thebasic reproduction number and investigate the existence andstability of equilibriaThere are two nonnegative equilibria of

Acut

e inf

ectio

n po

pulat

ion

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Figure 11 The plot shows the sensitivity of acute infection popula-tions for different values of control 119906

2

Carr

ier p

opul

atio

n

NV-NTNV-T

V-NTV-T

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 12 The plot shows the changes in carrier populations(without) vaccination and (without) treatment

the system namely the disease-free and endemicWe see thatthe disease-free equilibriumwhich always exists and is locallyasymptotically stable if 119877

0lt 1 and endemic equilibrium

which exists and is locally asymptotically stable if 1198770gt 1

After investigating the dynamic behavior of the systemwith constant controls we formulate an optimal controlproblem if the controls become time dependent and solve itby using Pontryaginrsquos maximum principle Different possiblecombinations of controls are used and their effectiveness iscompared by simulation works Also from the numericalresults it is very clear that a combination of mixed controlmeasures respond better than any other independent control

Computational and Mathematical Methods in Medicine 13Ca

rrie

r pop

ulat

ion

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0365 u2 = 0

u1 = 046 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

Figure 13 The plot shows the sensitivity of carrier populations fordifferent values of control 119906

1

Carr

ier p

opul

atio

n

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Figure 14 The plot shows the sensitivity of carrier populations fordifferent values of control 119906

2

There is still a tremendous amount of work needed tobe done in this area Parameters are rarely constant becausethey depend on environmental conditions We do not knowhowever the detailed relationship between these parametersand environmental conditions There may be a time lag as asusceptible populationmay take some time to be infected andalso a susceptible population may take some time to immune

Popu

latio

n (N

V-N

T)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 15 The plot shows the changes in total populations withoutvaccination and without treatment

Popu

latio

n (V

-NT)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 16 The plot shows the changes in total populations withvaccination and without treatment

Popu

latio

n (N

V-T)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 17 The plot shows the changes in total populations withoutvaccination and with treatment

14 Computational and Mathematical Methods in MedicinePo

pulat

ion

(V-T

)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 18 The plot shows the changes in total populations withvaccination and treatment

after vaccination We leave all these possible extensions forthe future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] WHO Hepatitis B Fact Sheet No 204 The World HealthOrganisation Geneva Switzerland 2013 httpwwwwhointmediacentrefactsheetsfs204en

[2] CanadianCentre forOccupationalHealth and Safety ldquoHepatitisBrdquo httpwwwccohscaoshanswersdiseaseshepatitis bhtml

[3] ldquoHealthcare stumbling in RIrsquos Hepatitis fightrdquoThe Jakarta PostJanuary 2011

[4] G F Medley N A Lindop W J Edmunds and D JNokes ldquoHepatitis-B virus endemicity heterogeneity catas-trophic dynamics and controlrdquo Nature Medicine vol 7 no 5pp 619ndash624 2001

[5] J Mann and M Roberts ldquoModelling the epidemiology ofhepatitis B in New Zealandrdquo Journal of Theoretical Biology vol269 no 1 pp 266ndash272 2011

[6] ldquoHepatitis B (HBV)rdquo httpkidshealthorgteensexual healthstdsstd hepatitishtml

[7] CDC ldquoHepatitis B virus a comprehensive strategy for elim-inating transmission in the United States through universalchildhood vaccination Recommendations of the Immuniza-tion Practices Advisory Committee (ACIP)rdquo MMWR Recom-mendations and Reports vol 40(RR-13) pp 1ndash25 1991

[8] M K Libbus and L M Phillips ldquoPublic health management ofperinatal hepatitis B virusrdquo Public Health Nursing vol 26 no 4pp 353ndash361 2009

[9] F B Hollinger and D T Lau ldquoHepatitis B the pathway torecovery through treatmentrdquo Gastroenterology Clinics of NorthAmerica vol 35 no 4 pp 895ndash931 2006

[10] C-L Lai andM-F Yuen ldquoThe natural history and treatment ofchronic hepatitis B a critical evaluation of standard treatment

criteria and end pointsrdquoAnnals of InternalMedicine vol 147 no1 pp 58ndash61 2007

[11] R M Anderson and R M May Infectious Disease of HumansDynamics and Control Oxford University Press Oxford UK1991

[12] S Thornley C Bullen and M Roberts ldquoHepatitis B in ahigh prevalence New Zealand population a mathematicalmodel applied to infection control policyrdquo Journal ofTheoreticalBiology vol 254 no 3 pp 599ndash603 2008

[13] S-J Zhao Z-Y Xu and Y Lu ldquoA mathematical model ofhepatitis B virus transmission and its application for vaccinationstrategy inChinardquo International Journal of Epidemiology vol 29no 4 pp 744ndash752 2000

[14] K Wang W Wang and S Song ldquoDynamics of an HBV modelwith diffusion and delayrdquo Journal ofTheoretical Biology vol 253no 1 pp 36ndash44 2008

[15] R Xu andZMa ldquoAnHBVmodelwith diffusion and time delayrdquoJournal of Theoretical Biology vol 257 no 3 pp 499ndash509 2009

[16] L Zou W Zhang and S Ruan ldquoModeling the transmissiondynamics and control of hepatitis B virus in Chinardquo Journal ofTheoretical Biology vol 262 no 2 pp 330ndash338 2010

[17] J Pang J-A Cui and X Zhou ldquoDynamical behavior of ahepatitis B virus transmission model with vaccinationrdquo Journalof Theoretical Biology vol 265 no 4 pp 572ndash578 2010

[18] S Zhang and Y Zhou ldquoThe analysis and application of an HBVmodelrdquoAppliedMathematicalModelling vol 36 no 3 pp 1302ndash1312 2012

[19] S Bhattacharyya and S Ghosh ldquoOptimal control of verticallytransmitted diseaserdquoComputational andMathematicalMethodsin Medicine vol 11 no 4 pp 369ndash387 2010

[20] T K Kar andA Batabyal ldquoStability analysis and optimal controlof an SIR epidemic model with vaccinationrdquo Biosystems vol104 no 2-3 pp 127ndash135 2011

[21] T-K Kar and S Jana ldquoA theoretical study on mathematicalmodelling of an infectious disease with application of optimalcontrolrdquo Biosystems vol 111 no 1 pp 37ndash50 2013

[22] M J Keeling and P Rohani Modeling Infectious Diseases inHumans and Animals Princeton University Press PrincetonNJ USA 2008

[23] E Jung S Lenhart and Z Feng ldquoOptimal control of treatmentsin a two-strain tuberculosis modelrdquo Discrete and ContinuousDynamical Systems B vol 2 no 4 pp 473ndash482 2002

[24] G Birkhoff and G-C Rota Ordinary Differential EquationsJohn Wiley amp Sons New York NY USA 4th edition 1989

[25] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[26] T T Yusuf and F Benyah ldquoOptimal control of vaccination andtreatment for an SIR epidemiological modelrdquo World Journal ofModelling and Simulation vol 8 no 3 pp 194ndash204 2008

[27] H-R Joshi S Lenhart M Y Li and L Wang ldquoOptimalcontrol methods applied to disease modelsrdquo in MathematicalStudies on Human Disease Dynamics vol 410 of ContemporaryMathematics pp 187ndash207 American Mathematical SocietyProvidence RI USA 2006

[28] G Zaman Y H Kang and I H Jung ldquoStability analysis andoptimal vaccination of an SIR epidemic modelrdquo Biosystems vol93 no 3 pp 240ndash249 2008

[29] D Iacoviello and N Stasiob ldquoOptimal control for SIRCepidemic outbreakrdquo Computer Methods and Programs inBiomedicine vol 110 no 3 pp 333ndash342 2013

Computational and Mathematical Methods in Medicine 15

[30] L S Pontryagin V G Boltyanskii R V Gamkrelidze and EF Mishchenko The Mathematical Theory of Optimal ProcessesJohn Wiley amp Sons New York NY USA 1962

[31] WHO Hepatitis B Fact Sheet No 204 The World HealthOrganisation Geneva Switzerland 2000 httpwwwwhointmediacentrefactsheetsfs204en

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

8 Computational and Mathematical Methods in Medicine

We set

119866 (119909) = 119860119909 + 119865 (119909) (30)

The second term on the right-hand side of (30) satisfies

1003816100381610038161003816119865 (1199091) minus 119865 (119909

2)1003816100381610038161003816

le 119872 (10038161003816100381610038161198781minus 1198782

1003816100381610038161003816+10038161003816100381610038161198641minus 1198642

1003816100381610038161003816+10038161003816100381610038161198681minus 1198682

1003816100381610038161003816+10038161003816100381610038161198621minus 1198622

1003816100381610038161003816)

(31)

where the positive constant 119872 is independent of the statevariables 119909 and119872 le 1 Also we get

1003816100381610038161003816119866 (1199091) minus 119866 (119909

2)1003816100381610038161003816le 119871

10038161003816100381610038161199091minus 1199092

1003816100381610038161003816 (32)

where 119871 = max119872 119860 lt infin Therefore it follows thatthe function 119866 is uniformly Lipschitz continuous From thedefinition of the controls 119906

1(119905) and 119906

2(119905) and the restrictions

on the nonnegativeness of the state variables we see that asolution of the system (28) exists [24 28 29]

42The Lagrangian and Hamiltonian for the Control ProblemIn order to find an optimal solution pair first we shouldfind the Lagrangian and Hamiltonian for the optimal controlproblem (24) In fact the Lagrangian of problem is given by

119871 (119878 119864 119868 119862 1199061 1199062) = 119860

1119878 (119905) + 119860

2119864 (119905) + 119860

3119868 (119905)

+ 1198604119862 (119905) +

1

2

(11986111199062

1(119905) + 119861

21199062

2(119905))

(33)

We are looking for the minimal value of the LagrangianTo accomplish this we define Hamiltonian119867 for the controlproblem as follows

119867(119878 119864 119868 119862 1199061 1199062 1205721 1205722 1205723 1205724)

= 119871 + 1205721(119905)

119889119878

119889119905

+ 1205722(119905)

119889119864

119889119905

+ 1205723(119905)

119889119868

119889119905

+ 1205724(119905)

119889119862

119889119905

(34)

where 120572119894(119905) for 119894 = 1 2 3 4 are the adjoint variables and can

be determined by solving the following system of differentialequations

1(119905) = minus

120597119867

120597119878

= minus [1198601+ 1205721((]1199012minus 1205824) minus 120588 (119868 + 120579119862) minus 119906

1minus ])

+1205722120588 (119868 + 120579119862)]

2(119905) = minus

120597119867

120597119864

= minus [1198602+ 1205721(]1199012minus 1205824) + 1205722(minus] minus 120582

1) + 12057231205821]

3(119905) = minus

120597119867

120597119868

= minus [1198603+ 1205721((]1199012minus 1205824) minus 120588119878) + 120572

2120588119878

+1205723(minus] minus 120582

2) + 120572411990131205822]

4(119905) = minus

120597119867

120597119862

= minus [1198604+ 1205721(]1199012minus 1205824minus ]1199011minus 120588120579119878)

+1205722120588120579119878 + 120572

4(]1199011minus ] + 120582

3+ 1199062)]

(35)

satisfying the transversality conditions and optimality condi-tions

1205721(119879) = 120572

2(119879) = 120572

3(119879) = 120572

4(119879) = 0

120597119867

1205971199061

= 11986111199061minus 1205721119878 = 0

120597119867

1205971199062

= 11986121199062minus 1205724119862 = 0

(36)

We now state and prove the following theorem

Theorem 6 There is an optimal control (119906lowast1 119906lowast

2) such that

119869 (119878 119864 119868 119878 119906lowast

1 119906lowast

2) = min 119869 (119878 119864 119868 119878 119906

1 1199062) (37)

subject to the system of differential equation (24)

Proof Here both the control and state variables are non-negative values In this minimized problem the necessaryconvexity of the objective functional in 119906

1and 1199062are satisfied

The set of all control variables 1199061(119905) 1199062(119905) is also convex and

closed by definition The optimal system is bounded whichdetermines the compactness needed for the existence of theoptimal control In addition the integrand in the functional

int

119879

0

[1198601119878 (119905) + 119860

2119864 (119905) + 119860

3119868 (119905) + 119860

4119862 (119905)

+

1

2

(11986111199062

1(119905) + 119861

21199062

2(119905))] 119889119905

(38)

is convex on the control set Hence the theorem (the proof iscomplete)

Computational and Mathematical Methods in Medicine 9

43 Necessary and Sufficient Conditions for Optimal ControlsApplying Pontryaginrsquos maximum principle [30] on the con-structed Hamiltonian 119867 and the theorem (24) we obtainthe optimal steady-state solution and corresponding controlas follows If taking 119883 = (119878 119864 119868 119862) and 119880 = (119906

1 1199062)

then (119883lowast 119880lowast) is an optimal solution of an optimal controlproblem then we now state the following theorem

Theorem 7 The optimal control pair (119906lowast1 119906lowast

2) which mini-

mizes J over the region Γ is given by

119906lowast

1= max[0119879]

0min (1(119905) 1)

119906lowast

2= max[0119879]

0min (2(119905) 1)

(39)

where 1= 11198781198611and 2= 41198621198612and let

1 2 3 4 be

the solution of system (35)

Proof With the help of the optimality conditions we have

120597119867

1205971199061

= 11986111199061minus 1205721119878 = 0 997904rArr 119906

1=

1119878

1198611

= (1)

120597119867

1205971199062

= 11986121199062minus 1205724119862 = 0 997904rArr 119906

2=

4119862

1198612

= (2)

(40)

Using the property of the control space Γ the two controlswhich are bounded with upper and lower bounds arerespectively 1 and 0 that is

119906lowast

1=

0 if 1le 0

1

if 0 lt 1lt 1

1 if 1ge 1

(41)

This can be rewritten in compact notation

119906lowast

1= max[0119879]

0min (1(119905) 1) (42)

and similarly

119906lowast

2=

0 if 2le 0

2

if 0 lt 2lt 1

1 if 2ge 1

(43)

This can be rewritten in compact notation

119906lowast

2= max[0119879]

0min (2(119905) 1) (44)

Hence for these pair of controls (119906lowast1 119906lowast

2) we get the optimum

value of the functional 119869 given by (24)

5 Numerical Examples

Numerical solutions to the optimality system (24) arediscussed in this section We make several interestingobservation by numerically simulating (2) in the rangeof parameter values We consider the parameter set

Δ = ] 120588 120579 1205821 1205822 1205823 1205824 1199011 1199012 1199013 1198601 1198602 1198603 1198604 1198611 1198612

some of the parameters are taken from the published articlesand some are assumed with feasible values Moreover thetime interval for which the optimal control is applied istaken as 70 years also consider Ψ = 119878

0 1198640 1198680 1198620 as initial

condition for simulation of the model The main parametervalues are listed in Table 1 We compare the results having nocontrols only vaccination control only treatment controland both vaccination and treatment controls

With the parameter values in Table 1 the system asymp-totically approaches towards the equilibrium 119864

lowastlowast

1(00858

03327 04003 05349) where the basic reproduction ratio1198770= 39810 (Figure 2) For the parameters set the system

(2) has two feasible equilibria one is disease free and theother is endemic and the endemic equilibrium is locallyasymptotically stable The effect of two control measures ondisease dynamics may be understood well if we considerFigure 2 It explains how control reproduction ratio 119877

0

evolves with different rates of 1199061and 119906

2 It is seen that both

vaccination and treatment reduce the value of 1198770effectively

But an integrated control works better than either of thecontrol measures

In this section we use an iterative method to obtainresults for an optimal control problem of the proposedmodel(24) We use Runge-Kuttarsquos fourth-order procedure [23] hereto solve the optimality system consisting of eight ordinarydifferential equations having four state equations and fouradjoint equations and boundary conditions Because stateequations have initial conditions and adjoint equations haveconditions at the final time an iterative program was createdto numerically simulate solutions Given an initial guessfor the controls to compute the optimal state values theprogram solves (24) with initial conditions (27) forward intime interval [0 70]using aRungeKuttamethod of the fourthorder Resulting state values are placed in adjoint equations(35) These adjoint equations with given final conditions arethen solved backwards in time Again a fourth order RungeKutta method is employed Both state and adjoint values areused to update the control using the characterization (39)and the entire process repeats itself This iterative processterminates when current state adjoint and control valuesare sufficiently close to successive values Then we use thebackward Runge-Kutta fourth-order procedure to solve theadjoint variables in the same time interval with the help ofthe solution of the state variables transversality conditions

We have plotted susceptible exposed acute infectionindividuals and carriers individuals with andwithout controlby considering values of parameters We simulate the systemat different values of rate of 119906

1and 119906

2

From Figures 3 4 and 5 we see that after 20 years thenumber of susceptible population decreases than when thereis no control In this case most of this population tends to theinfected class Again when only treatment control is appliedthen the number of susceptible population is not muchdifferent than the population in the case having no controlBut the susceptible population differs much from these twostrategies if we apply the strategies of only vaccination controland both vaccination and treatment controls At a high rateof vaccination the sensitive population density is reduced

10 Computational and Mathematical Methods in Medicine

Table 1 Parameter values used in numerical simulations

Parameter Description Values range Reference] Birth (and death) rate 00121 [17 18]120588 Transmission rate 08ndash2049 [17]120579 Infectiousness of carriers relative to acute infections 0-1 [17]1205821

Rate of moving from exposed to acute 6 per year [16]1205822

Rate at which individuals leave the acute infection class 4 per year [16]1205823

Rate of moving from carrier to recovery 0025 per year [16]1205824

Loss of recovery rate 003ndash006 [17]1198751

Probability of infected newborns 011 [19]1198752

Probability of immune newborns 01 [19]1198753

Proportion of acute infection individuals becoming carriers 005ndash09 [17]1198601

Weight factor for susceptible individuals 0091 [31]1198602

Weight factor for exposed individuals 001 [31]1198603

Weight factor for infected individuals 004 [31]1198604

Weight factor for carrier individuals 005 [31]1198611

Weight factor for the controls 1199061

15 [31]1198612

Weight factor for the controls 1199062

27 [31]1198780

Susceptible individuals 0493 [4]1198640

Exposed individuals 00035 [4]1198680

Acute infection individuals 00035 [4]1198620

Chronic HBV carriers 0007 [4]

0 10 20 30 40 50 60 70Time (year)

Susc

eptib

le p

opul

atio

n

NV-NTNV-T

V-NTV-T

10minus4

10minus3

10minus2

10minus1

100

Figure 3 The plot shows the changes in sensitive populations(without) vaccination and (without) treatment

to a very low level initially and then it takes longer time torestore the steady-state value

From Figures 6 7 and 8 we see that the number ofexposed population increase thanwhen there is no control Inthis case most of this population tends to the acute infectedclass Again when only treatment control is applied thenthe number of exposed population is not much different

0 10 20 30 40 50 60 70Time (year)

Susc

eptib

le p

opul

atio

n

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0365 u2 = 0

u1 = 045 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

Figure 4 The plot shows the sensitivity of sensitive populations fordifferent values of control 119906

1

than the population in the case having no control But theexposed population differs much from these two strategies ifwe apply the strategies of only vaccination control and bothvaccination and treatment controls

From Figures 9 10 and 11 we see that the number ofacute infected population increases than when there is nocontrol In this case most of this population tends to thecarrier class Again when only treatment control is appliedthen the number of acute infected population is not muchdifferent than the population in the case having no controlBut the acute infected population differsmuch from these two

Computational and Mathematical Methods in Medicine 11

0 10 20 30 40 50 60 70Time (year)

Susc

eptib

le p

opul

atio

n

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Figure 5 The plot shows the sensitivity of sensitive populations fordifferent values of control 119906

2

Expo

sed

popu

latio

n

NV-NTNV-T

V-NTV-T

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 6 The plot shows the changes in exposed populations(without) vaccination and (without) treatment

strategies if we apply the strategies of only vaccination controland both vaccination and treatment controls

Again from Figures 12 13 and 14 we see that thenumber of carrier population increases than when there isno control We see that the application of only vaccinationcontrol gives better result than the application of no controlAgain application of treatment control would give betterresult than the application of vaccination control since thetreatment control is better than the vaccination control whilethe application of both vaccination and treatment controlsgive the best result as in this case the number of carrierpopulation would be the least in number

Expo

sed

popu

latio

n

u1 = 0 u2 = 0

u1 = 03635 u2 = 0

u1 = 045 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 7 The plot shows the sensitivity of exposed populations fordifferent values of control 119906

1

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Expo

sed

popu

latio

n

Figure 8 The plot shows the sensitivity of exposed populations fordifferent values of control 119906

2

Finally the effect of two control measures on diseasedynamics may be understood well if we consider Figures 1ndash14 15 16 17 and 18 Since our main purpose is to reducethe number of sensitive exposed acute infection and carrierindividuals therefore numerical simulation results show thatoptimal combination of vaccination and treatment is themosteffective way to control hepatitis B virus infection

12 Computational and Mathematical Methods in MedicineAc

ute i

nfec

tion

popu

latio

n

NV-NTNV-T

V-NTV-T

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 9The plot shows the changes in acute infection populations(without) vaccination and (without) treatment

Acut

e inf

ectio

n po

pulat

ion

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0365 u2 = 0

u1 = 045 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

Figure 10 The plot shows the sensitivity of acute infection popula-tions for different values of control 119906

1

6 Conclusion and Suggestions

In this paper we propose an S-119864-119868-119862-119877model of hepatitis Bvirus infection with two controls vaccination and treatmentFirst we analyze the dynamic behavior of the system for con-stant controls In the constant controls case we calculate thebasic reproduction number and investigate the existence andstability of equilibriaThere are two nonnegative equilibria of

Acut

e inf

ectio

n po

pulat

ion

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Figure 11 The plot shows the sensitivity of acute infection popula-tions for different values of control 119906

2

Carr

ier p

opul

atio

n

NV-NTNV-T

V-NTV-T

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 12 The plot shows the changes in carrier populations(without) vaccination and (without) treatment

the system namely the disease-free and endemicWe see thatthe disease-free equilibriumwhich always exists and is locallyasymptotically stable if 119877

0lt 1 and endemic equilibrium

which exists and is locally asymptotically stable if 1198770gt 1

After investigating the dynamic behavior of the systemwith constant controls we formulate an optimal controlproblem if the controls become time dependent and solve itby using Pontryaginrsquos maximum principle Different possiblecombinations of controls are used and their effectiveness iscompared by simulation works Also from the numericalresults it is very clear that a combination of mixed controlmeasures respond better than any other independent control

Computational and Mathematical Methods in Medicine 13Ca

rrie

r pop

ulat

ion

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0365 u2 = 0

u1 = 046 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

Figure 13 The plot shows the sensitivity of carrier populations fordifferent values of control 119906

1

Carr

ier p

opul

atio

n

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Figure 14 The plot shows the sensitivity of carrier populations fordifferent values of control 119906

2

There is still a tremendous amount of work needed tobe done in this area Parameters are rarely constant becausethey depend on environmental conditions We do not knowhowever the detailed relationship between these parametersand environmental conditions There may be a time lag as asusceptible populationmay take some time to be infected andalso a susceptible population may take some time to immune

Popu

latio

n (N

V-N

T)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 15 The plot shows the changes in total populations withoutvaccination and without treatment

Popu

latio

n (V

-NT)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 16 The plot shows the changes in total populations withvaccination and without treatment

Popu

latio

n (N

V-T)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 17 The plot shows the changes in total populations withoutvaccination and with treatment

14 Computational and Mathematical Methods in MedicinePo

pulat

ion

(V-T

)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 18 The plot shows the changes in total populations withvaccination and treatment

after vaccination We leave all these possible extensions forthe future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] WHO Hepatitis B Fact Sheet No 204 The World HealthOrganisation Geneva Switzerland 2013 httpwwwwhointmediacentrefactsheetsfs204en

[2] CanadianCentre forOccupationalHealth and Safety ldquoHepatitisBrdquo httpwwwccohscaoshanswersdiseaseshepatitis bhtml

[3] ldquoHealthcare stumbling in RIrsquos Hepatitis fightrdquoThe Jakarta PostJanuary 2011

[4] G F Medley N A Lindop W J Edmunds and D JNokes ldquoHepatitis-B virus endemicity heterogeneity catas-trophic dynamics and controlrdquo Nature Medicine vol 7 no 5pp 619ndash624 2001

[5] J Mann and M Roberts ldquoModelling the epidemiology ofhepatitis B in New Zealandrdquo Journal of Theoretical Biology vol269 no 1 pp 266ndash272 2011

[6] ldquoHepatitis B (HBV)rdquo httpkidshealthorgteensexual healthstdsstd hepatitishtml

[7] CDC ldquoHepatitis B virus a comprehensive strategy for elim-inating transmission in the United States through universalchildhood vaccination Recommendations of the Immuniza-tion Practices Advisory Committee (ACIP)rdquo MMWR Recom-mendations and Reports vol 40(RR-13) pp 1ndash25 1991

[8] M K Libbus and L M Phillips ldquoPublic health management ofperinatal hepatitis B virusrdquo Public Health Nursing vol 26 no 4pp 353ndash361 2009

[9] F B Hollinger and D T Lau ldquoHepatitis B the pathway torecovery through treatmentrdquo Gastroenterology Clinics of NorthAmerica vol 35 no 4 pp 895ndash931 2006

[10] C-L Lai andM-F Yuen ldquoThe natural history and treatment ofchronic hepatitis B a critical evaluation of standard treatment

criteria and end pointsrdquoAnnals of InternalMedicine vol 147 no1 pp 58ndash61 2007

[11] R M Anderson and R M May Infectious Disease of HumansDynamics and Control Oxford University Press Oxford UK1991

[12] S Thornley C Bullen and M Roberts ldquoHepatitis B in ahigh prevalence New Zealand population a mathematicalmodel applied to infection control policyrdquo Journal ofTheoreticalBiology vol 254 no 3 pp 599ndash603 2008

[13] S-J Zhao Z-Y Xu and Y Lu ldquoA mathematical model ofhepatitis B virus transmission and its application for vaccinationstrategy inChinardquo International Journal of Epidemiology vol 29no 4 pp 744ndash752 2000

[14] K Wang W Wang and S Song ldquoDynamics of an HBV modelwith diffusion and delayrdquo Journal ofTheoretical Biology vol 253no 1 pp 36ndash44 2008

[15] R Xu andZMa ldquoAnHBVmodelwith diffusion and time delayrdquoJournal of Theoretical Biology vol 257 no 3 pp 499ndash509 2009

[16] L Zou W Zhang and S Ruan ldquoModeling the transmissiondynamics and control of hepatitis B virus in Chinardquo Journal ofTheoretical Biology vol 262 no 2 pp 330ndash338 2010

[17] J Pang J-A Cui and X Zhou ldquoDynamical behavior of ahepatitis B virus transmission model with vaccinationrdquo Journalof Theoretical Biology vol 265 no 4 pp 572ndash578 2010

[18] S Zhang and Y Zhou ldquoThe analysis and application of an HBVmodelrdquoAppliedMathematicalModelling vol 36 no 3 pp 1302ndash1312 2012

[19] S Bhattacharyya and S Ghosh ldquoOptimal control of verticallytransmitted diseaserdquoComputational andMathematicalMethodsin Medicine vol 11 no 4 pp 369ndash387 2010

[20] T K Kar andA Batabyal ldquoStability analysis and optimal controlof an SIR epidemic model with vaccinationrdquo Biosystems vol104 no 2-3 pp 127ndash135 2011

[21] T-K Kar and S Jana ldquoA theoretical study on mathematicalmodelling of an infectious disease with application of optimalcontrolrdquo Biosystems vol 111 no 1 pp 37ndash50 2013

[22] M J Keeling and P Rohani Modeling Infectious Diseases inHumans and Animals Princeton University Press PrincetonNJ USA 2008

[23] E Jung S Lenhart and Z Feng ldquoOptimal control of treatmentsin a two-strain tuberculosis modelrdquo Discrete and ContinuousDynamical Systems B vol 2 no 4 pp 473ndash482 2002

[24] G Birkhoff and G-C Rota Ordinary Differential EquationsJohn Wiley amp Sons New York NY USA 4th edition 1989

[25] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[26] T T Yusuf and F Benyah ldquoOptimal control of vaccination andtreatment for an SIR epidemiological modelrdquo World Journal ofModelling and Simulation vol 8 no 3 pp 194ndash204 2008

[27] H-R Joshi S Lenhart M Y Li and L Wang ldquoOptimalcontrol methods applied to disease modelsrdquo in MathematicalStudies on Human Disease Dynamics vol 410 of ContemporaryMathematics pp 187ndash207 American Mathematical SocietyProvidence RI USA 2006

[28] G Zaman Y H Kang and I H Jung ldquoStability analysis andoptimal vaccination of an SIR epidemic modelrdquo Biosystems vol93 no 3 pp 240ndash249 2008

[29] D Iacoviello and N Stasiob ldquoOptimal control for SIRCepidemic outbreakrdquo Computer Methods and Programs inBiomedicine vol 110 no 3 pp 333ndash342 2013

Computational and Mathematical Methods in Medicine 15

[30] L S Pontryagin V G Boltyanskii R V Gamkrelidze and EF Mishchenko The Mathematical Theory of Optimal ProcessesJohn Wiley amp Sons New York NY USA 1962

[31] WHO Hepatitis B Fact Sheet No 204 The World HealthOrganisation Geneva Switzerland 2000 httpwwwwhointmediacentrefactsheetsfs204en

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

Computational and Mathematical Methods in Medicine 9

43 Necessary and Sufficient Conditions for Optimal ControlsApplying Pontryaginrsquos maximum principle [30] on the con-structed Hamiltonian 119867 and the theorem (24) we obtainthe optimal steady-state solution and corresponding controlas follows If taking 119883 = (119878 119864 119868 119862) and 119880 = (119906

1 1199062)

then (119883lowast 119880lowast) is an optimal solution of an optimal controlproblem then we now state the following theorem

Theorem 7 The optimal control pair (119906lowast1 119906lowast

2) which mini-

mizes J over the region Γ is given by

119906lowast

1= max[0119879]

0min (1(119905) 1)

119906lowast

2= max[0119879]

0min (2(119905) 1)

(39)

where 1= 11198781198611and 2= 41198621198612and let

1 2 3 4 be

the solution of system (35)

Proof With the help of the optimality conditions we have

120597119867

1205971199061

= 11986111199061minus 1205721119878 = 0 997904rArr 119906

1=

1119878

1198611

= (1)

120597119867

1205971199062

= 11986121199062minus 1205724119862 = 0 997904rArr 119906

2=

4119862

1198612

= (2)

(40)

Using the property of the control space Γ the two controlswhich are bounded with upper and lower bounds arerespectively 1 and 0 that is

119906lowast

1=

0 if 1le 0

1

if 0 lt 1lt 1

1 if 1ge 1

(41)

This can be rewritten in compact notation

119906lowast

1= max[0119879]

0min (1(119905) 1) (42)

and similarly

119906lowast

2=

0 if 2le 0

2

if 0 lt 2lt 1

1 if 2ge 1

(43)

This can be rewritten in compact notation

119906lowast

2= max[0119879]

0min (2(119905) 1) (44)

Hence for these pair of controls (119906lowast1 119906lowast

2) we get the optimum

value of the functional 119869 given by (24)

5 Numerical Examples

Numerical solutions to the optimality system (24) arediscussed in this section We make several interestingobservation by numerically simulating (2) in the rangeof parameter values We consider the parameter set

Δ = ] 120588 120579 1205821 1205822 1205823 1205824 1199011 1199012 1199013 1198601 1198602 1198603 1198604 1198611 1198612

some of the parameters are taken from the published articlesand some are assumed with feasible values Moreover thetime interval for which the optimal control is applied istaken as 70 years also consider Ψ = 119878

0 1198640 1198680 1198620 as initial

condition for simulation of the model The main parametervalues are listed in Table 1 We compare the results having nocontrols only vaccination control only treatment controland both vaccination and treatment controls

With the parameter values in Table 1 the system asymp-totically approaches towards the equilibrium 119864

lowastlowast

1(00858

03327 04003 05349) where the basic reproduction ratio1198770= 39810 (Figure 2) For the parameters set the system

(2) has two feasible equilibria one is disease free and theother is endemic and the endemic equilibrium is locallyasymptotically stable The effect of two control measures ondisease dynamics may be understood well if we considerFigure 2 It explains how control reproduction ratio 119877

0

evolves with different rates of 1199061and 119906

2 It is seen that both

vaccination and treatment reduce the value of 1198770effectively

But an integrated control works better than either of thecontrol measures

In this section we use an iterative method to obtainresults for an optimal control problem of the proposedmodel(24) We use Runge-Kuttarsquos fourth-order procedure [23] hereto solve the optimality system consisting of eight ordinarydifferential equations having four state equations and fouradjoint equations and boundary conditions Because stateequations have initial conditions and adjoint equations haveconditions at the final time an iterative program was createdto numerically simulate solutions Given an initial guessfor the controls to compute the optimal state values theprogram solves (24) with initial conditions (27) forward intime interval [0 70]using aRungeKuttamethod of the fourthorder Resulting state values are placed in adjoint equations(35) These adjoint equations with given final conditions arethen solved backwards in time Again a fourth order RungeKutta method is employed Both state and adjoint values areused to update the control using the characterization (39)and the entire process repeats itself This iterative processterminates when current state adjoint and control valuesare sufficiently close to successive values Then we use thebackward Runge-Kutta fourth-order procedure to solve theadjoint variables in the same time interval with the help ofthe solution of the state variables transversality conditions

We have plotted susceptible exposed acute infectionindividuals and carriers individuals with andwithout controlby considering values of parameters We simulate the systemat different values of rate of 119906

1and 119906

2

From Figures 3 4 and 5 we see that after 20 years thenumber of susceptible population decreases than when thereis no control In this case most of this population tends to theinfected class Again when only treatment control is appliedthen the number of susceptible population is not muchdifferent than the population in the case having no controlBut the susceptible population differs much from these twostrategies if we apply the strategies of only vaccination controland both vaccination and treatment controls At a high rateof vaccination the sensitive population density is reduced

10 Computational and Mathematical Methods in Medicine

Table 1 Parameter values used in numerical simulations

Parameter Description Values range Reference] Birth (and death) rate 00121 [17 18]120588 Transmission rate 08ndash2049 [17]120579 Infectiousness of carriers relative to acute infections 0-1 [17]1205821

Rate of moving from exposed to acute 6 per year [16]1205822

Rate at which individuals leave the acute infection class 4 per year [16]1205823

Rate of moving from carrier to recovery 0025 per year [16]1205824

Loss of recovery rate 003ndash006 [17]1198751

Probability of infected newborns 011 [19]1198752

Probability of immune newborns 01 [19]1198753

Proportion of acute infection individuals becoming carriers 005ndash09 [17]1198601

Weight factor for susceptible individuals 0091 [31]1198602

Weight factor for exposed individuals 001 [31]1198603

Weight factor for infected individuals 004 [31]1198604

Weight factor for carrier individuals 005 [31]1198611

Weight factor for the controls 1199061

15 [31]1198612

Weight factor for the controls 1199062

27 [31]1198780

Susceptible individuals 0493 [4]1198640

Exposed individuals 00035 [4]1198680

Acute infection individuals 00035 [4]1198620

Chronic HBV carriers 0007 [4]

0 10 20 30 40 50 60 70Time (year)

Susc

eptib

le p

opul

atio

n

NV-NTNV-T

V-NTV-T

10minus4

10minus3

10minus2

10minus1

100

Figure 3 The plot shows the changes in sensitive populations(without) vaccination and (without) treatment

to a very low level initially and then it takes longer time torestore the steady-state value

From Figures 6 7 and 8 we see that the number ofexposed population increase thanwhen there is no control Inthis case most of this population tends to the acute infectedclass Again when only treatment control is applied thenthe number of exposed population is not much different

0 10 20 30 40 50 60 70Time (year)

Susc

eptib

le p

opul

atio

n

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0365 u2 = 0

u1 = 045 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

Figure 4 The plot shows the sensitivity of sensitive populations fordifferent values of control 119906

1

than the population in the case having no control But theexposed population differs much from these two strategies ifwe apply the strategies of only vaccination control and bothvaccination and treatment controls

From Figures 9 10 and 11 we see that the number ofacute infected population increases than when there is nocontrol In this case most of this population tends to thecarrier class Again when only treatment control is appliedthen the number of acute infected population is not muchdifferent than the population in the case having no controlBut the acute infected population differsmuch from these two

Computational and Mathematical Methods in Medicine 11

0 10 20 30 40 50 60 70Time (year)

Susc

eptib

le p

opul

atio

n

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Figure 5 The plot shows the sensitivity of sensitive populations fordifferent values of control 119906

2

Expo

sed

popu

latio

n

NV-NTNV-T

V-NTV-T

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 6 The plot shows the changes in exposed populations(without) vaccination and (without) treatment

strategies if we apply the strategies of only vaccination controland both vaccination and treatment controls

Again from Figures 12 13 and 14 we see that thenumber of carrier population increases than when there isno control We see that the application of only vaccinationcontrol gives better result than the application of no controlAgain application of treatment control would give betterresult than the application of vaccination control since thetreatment control is better than the vaccination control whilethe application of both vaccination and treatment controlsgive the best result as in this case the number of carrierpopulation would be the least in number

Expo

sed

popu

latio

n

u1 = 0 u2 = 0

u1 = 03635 u2 = 0

u1 = 045 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 7 The plot shows the sensitivity of exposed populations fordifferent values of control 119906

1

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Expo

sed

popu

latio

n

Figure 8 The plot shows the sensitivity of exposed populations fordifferent values of control 119906

2

Finally the effect of two control measures on diseasedynamics may be understood well if we consider Figures 1ndash14 15 16 17 and 18 Since our main purpose is to reducethe number of sensitive exposed acute infection and carrierindividuals therefore numerical simulation results show thatoptimal combination of vaccination and treatment is themosteffective way to control hepatitis B virus infection

12 Computational and Mathematical Methods in MedicineAc

ute i

nfec

tion

popu

latio

n

NV-NTNV-T

V-NTV-T

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 9The plot shows the changes in acute infection populations(without) vaccination and (without) treatment

Acut

e inf

ectio

n po

pulat

ion

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0365 u2 = 0

u1 = 045 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

Figure 10 The plot shows the sensitivity of acute infection popula-tions for different values of control 119906

1

6 Conclusion and Suggestions

In this paper we propose an S-119864-119868-119862-119877model of hepatitis Bvirus infection with two controls vaccination and treatmentFirst we analyze the dynamic behavior of the system for con-stant controls In the constant controls case we calculate thebasic reproduction number and investigate the existence andstability of equilibriaThere are two nonnegative equilibria of

Acut

e inf

ectio

n po

pulat

ion

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Figure 11 The plot shows the sensitivity of acute infection popula-tions for different values of control 119906

2

Carr

ier p

opul

atio

n

NV-NTNV-T

V-NTV-T

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 12 The plot shows the changes in carrier populations(without) vaccination and (without) treatment

the system namely the disease-free and endemicWe see thatthe disease-free equilibriumwhich always exists and is locallyasymptotically stable if 119877

0lt 1 and endemic equilibrium

which exists and is locally asymptotically stable if 1198770gt 1

After investigating the dynamic behavior of the systemwith constant controls we formulate an optimal controlproblem if the controls become time dependent and solve itby using Pontryaginrsquos maximum principle Different possiblecombinations of controls are used and their effectiveness iscompared by simulation works Also from the numericalresults it is very clear that a combination of mixed controlmeasures respond better than any other independent control

Computational and Mathematical Methods in Medicine 13Ca

rrie

r pop

ulat

ion

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0365 u2 = 0

u1 = 046 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

Figure 13 The plot shows the sensitivity of carrier populations fordifferent values of control 119906

1

Carr

ier p

opul

atio

n

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Figure 14 The plot shows the sensitivity of carrier populations fordifferent values of control 119906

2

There is still a tremendous amount of work needed tobe done in this area Parameters are rarely constant becausethey depend on environmental conditions We do not knowhowever the detailed relationship between these parametersand environmental conditions There may be a time lag as asusceptible populationmay take some time to be infected andalso a susceptible population may take some time to immune

Popu

latio

n (N

V-N

T)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 15 The plot shows the changes in total populations withoutvaccination and without treatment

Popu

latio

n (V

-NT)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 16 The plot shows the changes in total populations withvaccination and without treatment

Popu

latio

n (N

V-T)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 17 The plot shows the changes in total populations withoutvaccination and with treatment

14 Computational and Mathematical Methods in MedicinePo

pulat

ion

(V-T

)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 18 The plot shows the changes in total populations withvaccination and treatment

after vaccination We leave all these possible extensions forthe future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] WHO Hepatitis B Fact Sheet No 204 The World HealthOrganisation Geneva Switzerland 2013 httpwwwwhointmediacentrefactsheetsfs204en

[2] CanadianCentre forOccupationalHealth and Safety ldquoHepatitisBrdquo httpwwwccohscaoshanswersdiseaseshepatitis bhtml

[3] ldquoHealthcare stumbling in RIrsquos Hepatitis fightrdquoThe Jakarta PostJanuary 2011

[4] G F Medley N A Lindop W J Edmunds and D JNokes ldquoHepatitis-B virus endemicity heterogeneity catas-trophic dynamics and controlrdquo Nature Medicine vol 7 no 5pp 619ndash624 2001

[5] J Mann and M Roberts ldquoModelling the epidemiology ofhepatitis B in New Zealandrdquo Journal of Theoretical Biology vol269 no 1 pp 266ndash272 2011

[6] ldquoHepatitis B (HBV)rdquo httpkidshealthorgteensexual healthstdsstd hepatitishtml

[7] CDC ldquoHepatitis B virus a comprehensive strategy for elim-inating transmission in the United States through universalchildhood vaccination Recommendations of the Immuniza-tion Practices Advisory Committee (ACIP)rdquo MMWR Recom-mendations and Reports vol 40(RR-13) pp 1ndash25 1991

[8] M K Libbus and L M Phillips ldquoPublic health management ofperinatal hepatitis B virusrdquo Public Health Nursing vol 26 no 4pp 353ndash361 2009

[9] F B Hollinger and D T Lau ldquoHepatitis B the pathway torecovery through treatmentrdquo Gastroenterology Clinics of NorthAmerica vol 35 no 4 pp 895ndash931 2006

[10] C-L Lai andM-F Yuen ldquoThe natural history and treatment ofchronic hepatitis B a critical evaluation of standard treatment

criteria and end pointsrdquoAnnals of InternalMedicine vol 147 no1 pp 58ndash61 2007

[11] R M Anderson and R M May Infectious Disease of HumansDynamics and Control Oxford University Press Oxford UK1991

[12] S Thornley C Bullen and M Roberts ldquoHepatitis B in ahigh prevalence New Zealand population a mathematicalmodel applied to infection control policyrdquo Journal ofTheoreticalBiology vol 254 no 3 pp 599ndash603 2008

[13] S-J Zhao Z-Y Xu and Y Lu ldquoA mathematical model ofhepatitis B virus transmission and its application for vaccinationstrategy inChinardquo International Journal of Epidemiology vol 29no 4 pp 744ndash752 2000

[14] K Wang W Wang and S Song ldquoDynamics of an HBV modelwith diffusion and delayrdquo Journal ofTheoretical Biology vol 253no 1 pp 36ndash44 2008

[15] R Xu andZMa ldquoAnHBVmodelwith diffusion and time delayrdquoJournal of Theoretical Biology vol 257 no 3 pp 499ndash509 2009

[16] L Zou W Zhang and S Ruan ldquoModeling the transmissiondynamics and control of hepatitis B virus in Chinardquo Journal ofTheoretical Biology vol 262 no 2 pp 330ndash338 2010

[17] J Pang J-A Cui and X Zhou ldquoDynamical behavior of ahepatitis B virus transmission model with vaccinationrdquo Journalof Theoretical Biology vol 265 no 4 pp 572ndash578 2010

[18] S Zhang and Y Zhou ldquoThe analysis and application of an HBVmodelrdquoAppliedMathematicalModelling vol 36 no 3 pp 1302ndash1312 2012

[19] S Bhattacharyya and S Ghosh ldquoOptimal control of verticallytransmitted diseaserdquoComputational andMathematicalMethodsin Medicine vol 11 no 4 pp 369ndash387 2010

[20] T K Kar andA Batabyal ldquoStability analysis and optimal controlof an SIR epidemic model with vaccinationrdquo Biosystems vol104 no 2-3 pp 127ndash135 2011

[21] T-K Kar and S Jana ldquoA theoretical study on mathematicalmodelling of an infectious disease with application of optimalcontrolrdquo Biosystems vol 111 no 1 pp 37ndash50 2013

[22] M J Keeling and P Rohani Modeling Infectious Diseases inHumans and Animals Princeton University Press PrincetonNJ USA 2008

[23] E Jung S Lenhart and Z Feng ldquoOptimal control of treatmentsin a two-strain tuberculosis modelrdquo Discrete and ContinuousDynamical Systems B vol 2 no 4 pp 473ndash482 2002

[24] G Birkhoff and G-C Rota Ordinary Differential EquationsJohn Wiley amp Sons New York NY USA 4th edition 1989

[25] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[26] T T Yusuf and F Benyah ldquoOptimal control of vaccination andtreatment for an SIR epidemiological modelrdquo World Journal ofModelling and Simulation vol 8 no 3 pp 194ndash204 2008

[27] H-R Joshi S Lenhart M Y Li and L Wang ldquoOptimalcontrol methods applied to disease modelsrdquo in MathematicalStudies on Human Disease Dynamics vol 410 of ContemporaryMathematics pp 187ndash207 American Mathematical SocietyProvidence RI USA 2006

[28] G Zaman Y H Kang and I H Jung ldquoStability analysis andoptimal vaccination of an SIR epidemic modelrdquo Biosystems vol93 no 3 pp 240ndash249 2008

[29] D Iacoviello and N Stasiob ldquoOptimal control for SIRCepidemic outbreakrdquo Computer Methods and Programs inBiomedicine vol 110 no 3 pp 333ndash342 2013

Computational and Mathematical Methods in Medicine 15

[30] L S Pontryagin V G Boltyanskii R V Gamkrelidze and EF Mishchenko The Mathematical Theory of Optimal ProcessesJohn Wiley amp Sons New York NY USA 1962

[31] WHO Hepatitis B Fact Sheet No 204 The World HealthOrganisation Geneva Switzerland 2000 httpwwwwhointmediacentrefactsheetsfs204en

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

10 Computational and Mathematical Methods in Medicine

Table 1 Parameter values used in numerical simulations

Parameter Description Values range Reference] Birth (and death) rate 00121 [17 18]120588 Transmission rate 08ndash2049 [17]120579 Infectiousness of carriers relative to acute infections 0-1 [17]1205821

Rate of moving from exposed to acute 6 per year [16]1205822

Rate at which individuals leave the acute infection class 4 per year [16]1205823

Rate of moving from carrier to recovery 0025 per year [16]1205824

Loss of recovery rate 003ndash006 [17]1198751

Probability of infected newborns 011 [19]1198752

Probability of immune newborns 01 [19]1198753

Proportion of acute infection individuals becoming carriers 005ndash09 [17]1198601

Weight factor for susceptible individuals 0091 [31]1198602

Weight factor for exposed individuals 001 [31]1198603

Weight factor for infected individuals 004 [31]1198604

Weight factor for carrier individuals 005 [31]1198611

Weight factor for the controls 1199061

15 [31]1198612

Weight factor for the controls 1199062

27 [31]1198780

Susceptible individuals 0493 [4]1198640

Exposed individuals 00035 [4]1198680

Acute infection individuals 00035 [4]1198620

Chronic HBV carriers 0007 [4]

0 10 20 30 40 50 60 70Time (year)

Susc

eptib

le p

opul

atio

n

NV-NTNV-T

V-NTV-T

10minus4

10minus3

10minus2

10minus1

100

Figure 3 The plot shows the changes in sensitive populations(without) vaccination and (without) treatment

to a very low level initially and then it takes longer time torestore the steady-state value

From Figures 6 7 and 8 we see that the number ofexposed population increase thanwhen there is no control Inthis case most of this population tends to the acute infectedclass Again when only treatment control is applied thenthe number of exposed population is not much different

0 10 20 30 40 50 60 70Time (year)

Susc

eptib

le p

opul

atio

n

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0365 u2 = 0

u1 = 045 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

Figure 4 The plot shows the sensitivity of sensitive populations fordifferent values of control 119906

1

than the population in the case having no control But theexposed population differs much from these two strategies ifwe apply the strategies of only vaccination control and bothvaccination and treatment controls

From Figures 9 10 and 11 we see that the number ofacute infected population increases than when there is nocontrol In this case most of this population tends to thecarrier class Again when only treatment control is appliedthen the number of acute infected population is not muchdifferent than the population in the case having no controlBut the acute infected population differsmuch from these two

Computational and Mathematical Methods in Medicine 11

0 10 20 30 40 50 60 70Time (year)

Susc

eptib

le p

opul

atio

n

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Figure 5 The plot shows the sensitivity of sensitive populations fordifferent values of control 119906

2

Expo

sed

popu

latio

n

NV-NTNV-T

V-NTV-T

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 6 The plot shows the changes in exposed populations(without) vaccination and (without) treatment

strategies if we apply the strategies of only vaccination controland both vaccination and treatment controls

Again from Figures 12 13 and 14 we see that thenumber of carrier population increases than when there isno control We see that the application of only vaccinationcontrol gives better result than the application of no controlAgain application of treatment control would give betterresult than the application of vaccination control since thetreatment control is better than the vaccination control whilethe application of both vaccination and treatment controlsgive the best result as in this case the number of carrierpopulation would be the least in number

Expo

sed

popu

latio

n

u1 = 0 u2 = 0

u1 = 03635 u2 = 0

u1 = 045 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 7 The plot shows the sensitivity of exposed populations fordifferent values of control 119906

1

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Expo

sed

popu

latio

n

Figure 8 The plot shows the sensitivity of exposed populations fordifferent values of control 119906

2

Finally the effect of two control measures on diseasedynamics may be understood well if we consider Figures 1ndash14 15 16 17 and 18 Since our main purpose is to reducethe number of sensitive exposed acute infection and carrierindividuals therefore numerical simulation results show thatoptimal combination of vaccination and treatment is themosteffective way to control hepatitis B virus infection

12 Computational and Mathematical Methods in MedicineAc

ute i

nfec

tion

popu

latio

n

NV-NTNV-T

V-NTV-T

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 9The plot shows the changes in acute infection populations(without) vaccination and (without) treatment

Acut

e inf

ectio

n po

pulat

ion

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0365 u2 = 0

u1 = 045 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

Figure 10 The plot shows the sensitivity of acute infection popula-tions for different values of control 119906

1

6 Conclusion and Suggestions

In this paper we propose an S-119864-119868-119862-119877model of hepatitis Bvirus infection with two controls vaccination and treatmentFirst we analyze the dynamic behavior of the system for con-stant controls In the constant controls case we calculate thebasic reproduction number and investigate the existence andstability of equilibriaThere are two nonnegative equilibria of

Acut

e inf

ectio

n po

pulat

ion

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Figure 11 The plot shows the sensitivity of acute infection popula-tions for different values of control 119906

2

Carr

ier p

opul

atio

n

NV-NTNV-T

V-NTV-T

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 12 The plot shows the changes in carrier populations(without) vaccination and (without) treatment

the system namely the disease-free and endemicWe see thatthe disease-free equilibriumwhich always exists and is locallyasymptotically stable if 119877

0lt 1 and endemic equilibrium

which exists and is locally asymptotically stable if 1198770gt 1

After investigating the dynamic behavior of the systemwith constant controls we formulate an optimal controlproblem if the controls become time dependent and solve itby using Pontryaginrsquos maximum principle Different possiblecombinations of controls are used and their effectiveness iscompared by simulation works Also from the numericalresults it is very clear that a combination of mixed controlmeasures respond better than any other independent control

Computational and Mathematical Methods in Medicine 13Ca

rrie

r pop

ulat

ion

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0365 u2 = 0

u1 = 046 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

Figure 13 The plot shows the sensitivity of carrier populations fordifferent values of control 119906

1

Carr

ier p

opul

atio

n

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Figure 14 The plot shows the sensitivity of carrier populations fordifferent values of control 119906

2

There is still a tremendous amount of work needed tobe done in this area Parameters are rarely constant becausethey depend on environmental conditions We do not knowhowever the detailed relationship between these parametersand environmental conditions There may be a time lag as asusceptible populationmay take some time to be infected andalso a susceptible population may take some time to immune

Popu

latio

n (N

V-N

T)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 15 The plot shows the changes in total populations withoutvaccination and without treatment

Popu

latio

n (V

-NT)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 16 The plot shows the changes in total populations withvaccination and without treatment

Popu

latio

n (N

V-T)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 17 The plot shows the changes in total populations withoutvaccination and with treatment

14 Computational and Mathematical Methods in MedicinePo

pulat

ion

(V-T

)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 18 The plot shows the changes in total populations withvaccination and treatment

after vaccination We leave all these possible extensions forthe future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] WHO Hepatitis B Fact Sheet No 204 The World HealthOrganisation Geneva Switzerland 2013 httpwwwwhointmediacentrefactsheetsfs204en

[2] CanadianCentre forOccupationalHealth and Safety ldquoHepatitisBrdquo httpwwwccohscaoshanswersdiseaseshepatitis bhtml

[3] ldquoHealthcare stumbling in RIrsquos Hepatitis fightrdquoThe Jakarta PostJanuary 2011

[4] G F Medley N A Lindop W J Edmunds and D JNokes ldquoHepatitis-B virus endemicity heterogeneity catas-trophic dynamics and controlrdquo Nature Medicine vol 7 no 5pp 619ndash624 2001

[5] J Mann and M Roberts ldquoModelling the epidemiology ofhepatitis B in New Zealandrdquo Journal of Theoretical Biology vol269 no 1 pp 266ndash272 2011

[6] ldquoHepatitis B (HBV)rdquo httpkidshealthorgteensexual healthstdsstd hepatitishtml

[7] CDC ldquoHepatitis B virus a comprehensive strategy for elim-inating transmission in the United States through universalchildhood vaccination Recommendations of the Immuniza-tion Practices Advisory Committee (ACIP)rdquo MMWR Recom-mendations and Reports vol 40(RR-13) pp 1ndash25 1991

[8] M K Libbus and L M Phillips ldquoPublic health management ofperinatal hepatitis B virusrdquo Public Health Nursing vol 26 no 4pp 353ndash361 2009

[9] F B Hollinger and D T Lau ldquoHepatitis B the pathway torecovery through treatmentrdquo Gastroenterology Clinics of NorthAmerica vol 35 no 4 pp 895ndash931 2006

[10] C-L Lai andM-F Yuen ldquoThe natural history and treatment ofchronic hepatitis B a critical evaluation of standard treatment

criteria and end pointsrdquoAnnals of InternalMedicine vol 147 no1 pp 58ndash61 2007

[11] R M Anderson and R M May Infectious Disease of HumansDynamics and Control Oxford University Press Oxford UK1991

[12] S Thornley C Bullen and M Roberts ldquoHepatitis B in ahigh prevalence New Zealand population a mathematicalmodel applied to infection control policyrdquo Journal ofTheoreticalBiology vol 254 no 3 pp 599ndash603 2008

[13] S-J Zhao Z-Y Xu and Y Lu ldquoA mathematical model ofhepatitis B virus transmission and its application for vaccinationstrategy inChinardquo International Journal of Epidemiology vol 29no 4 pp 744ndash752 2000

[14] K Wang W Wang and S Song ldquoDynamics of an HBV modelwith diffusion and delayrdquo Journal ofTheoretical Biology vol 253no 1 pp 36ndash44 2008

[15] R Xu andZMa ldquoAnHBVmodelwith diffusion and time delayrdquoJournal of Theoretical Biology vol 257 no 3 pp 499ndash509 2009

[16] L Zou W Zhang and S Ruan ldquoModeling the transmissiondynamics and control of hepatitis B virus in Chinardquo Journal ofTheoretical Biology vol 262 no 2 pp 330ndash338 2010

[17] J Pang J-A Cui and X Zhou ldquoDynamical behavior of ahepatitis B virus transmission model with vaccinationrdquo Journalof Theoretical Biology vol 265 no 4 pp 572ndash578 2010

[18] S Zhang and Y Zhou ldquoThe analysis and application of an HBVmodelrdquoAppliedMathematicalModelling vol 36 no 3 pp 1302ndash1312 2012

[19] S Bhattacharyya and S Ghosh ldquoOptimal control of verticallytransmitted diseaserdquoComputational andMathematicalMethodsin Medicine vol 11 no 4 pp 369ndash387 2010

[20] T K Kar andA Batabyal ldquoStability analysis and optimal controlof an SIR epidemic model with vaccinationrdquo Biosystems vol104 no 2-3 pp 127ndash135 2011

[21] T-K Kar and S Jana ldquoA theoretical study on mathematicalmodelling of an infectious disease with application of optimalcontrolrdquo Biosystems vol 111 no 1 pp 37ndash50 2013

[22] M J Keeling and P Rohani Modeling Infectious Diseases inHumans and Animals Princeton University Press PrincetonNJ USA 2008

[23] E Jung S Lenhart and Z Feng ldquoOptimal control of treatmentsin a two-strain tuberculosis modelrdquo Discrete and ContinuousDynamical Systems B vol 2 no 4 pp 473ndash482 2002

[24] G Birkhoff and G-C Rota Ordinary Differential EquationsJohn Wiley amp Sons New York NY USA 4th edition 1989

[25] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[26] T T Yusuf and F Benyah ldquoOptimal control of vaccination andtreatment for an SIR epidemiological modelrdquo World Journal ofModelling and Simulation vol 8 no 3 pp 194ndash204 2008

[27] H-R Joshi S Lenhart M Y Li and L Wang ldquoOptimalcontrol methods applied to disease modelsrdquo in MathematicalStudies on Human Disease Dynamics vol 410 of ContemporaryMathematics pp 187ndash207 American Mathematical SocietyProvidence RI USA 2006

[28] G Zaman Y H Kang and I H Jung ldquoStability analysis andoptimal vaccination of an SIR epidemic modelrdquo Biosystems vol93 no 3 pp 240ndash249 2008

[29] D Iacoviello and N Stasiob ldquoOptimal control for SIRCepidemic outbreakrdquo Computer Methods and Programs inBiomedicine vol 110 no 3 pp 333ndash342 2013

Computational and Mathematical Methods in Medicine 15

[30] L S Pontryagin V G Boltyanskii R V Gamkrelidze and EF Mishchenko The Mathematical Theory of Optimal ProcessesJohn Wiley amp Sons New York NY USA 1962

[31] WHO Hepatitis B Fact Sheet No 204 The World HealthOrganisation Geneva Switzerland 2000 httpwwwwhointmediacentrefactsheetsfs204en

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

Computational and Mathematical Methods in Medicine 11

0 10 20 30 40 50 60 70Time (year)

Susc

eptib

le p

opul

atio

n

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Figure 5 The plot shows the sensitivity of sensitive populations fordifferent values of control 119906

2

Expo

sed

popu

latio

n

NV-NTNV-T

V-NTV-T

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 6 The plot shows the changes in exposed populations(without) vaccination and (without) treatment

strategies if we apply the strategies of only vaccination controland both vaccination and treatment controls

Again from Figures 12 13 and 14 we see that thenumber of carrier population increases than when there isno control We see that the application of only vaccinationcontrol gives better result than the application of no controlAgain application of treatment control would give betterresult than the application of vaccination control since thetreatment control is better than the vaccination control whilethe application of both vaccination and treatment controlsgive the best result as in this case the number of carrierpopulation would be the least in number

Expo

sed

popu

latio

n

u1 = 0 u2 = 0

u1 = 03635 u2 = 0

u1 = 045 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 7 The plot shows the sensitivity of exposed populations fordifferent values of control 119906

1

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Expo

sed

popu

latio

n

Figure 8 The plot shows the sensitivity of exposed populations fordifferent values of control 119906

2

Finally the effect of two control measures on diseasedynamics may be understood well if we consider Figures 1ndash14 15 16 17 and 18 Since our main purpose is to reducethe number of sensitive exposed acute infection and carrierindividuals therefore numerical simulation results show thatoptimal combination of vaccination and treatment is themosteffective way to control hepatitis B virus infection

12 Computational and Mathematical Methods in MedicineAc

ute i

nfec

tion

popu

latio

n

NV-NTNV-T

V-NTV-T

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 9The plot shows the changes in acute infection populations(without) vaccination and (without) treatment

Acut

e inf

ectio

n po

pulat

ion

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0365 u2 = 0

u1 = 045 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

Figure 10 The plot shows the sensitivity of acute infection popula-tions for different values of control 119906

1

6 Conclusion and Suggestions

In this paper we propose an S-119864-119868-119862-119877model of hepatitis Bvirus infection with two controls vaccination and treatmentFirst we analyze the dynamic behavior of the system for con-stant controls In the constant controls case we calculate thebasic reproduction number and investigate the existence andstability of equilibriaThere are two nonnegative equilibria of

Acut

e inf

ectio

n po

pulat

ion

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Figure 11 The plot shows the sensitivity of acute infection popula-tions for different values of control 119906

2

Carr

ier p

opul

atio

n

NV-NTNV-T

V-NTV-T

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 12 The plot shows the changes in carrier populations(without) vaccination and (without) treatment

the system namely the disease-free and endemicWe see thatthe disease-free equilibriumwhich always exists and is locallyasymptotically stable if 119877

0lt 1 and endemic equilibrium

which exists and is locally asymptotically stable if 1198770gt 1

After investigating the dynamic behavior of the systemwith constant controls we formulate an optimal controlproblem if the controls become time dependent and solve itby using Pontryaginrsquos maximum principle Different possiblecombinations of controls are used and their effectiveness iscompared by simulation works Also from the numericalresults it is very clear that a combination of mixed controlmeasures respond better than any other independent control

Computational and Mathematical Methods in Medicine 13Ca

rrie

r pop

ulat

ion

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0365 u2 = 0

u1 = 046 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

Figure 13 The plot shows the sensitivity of carrier populations fordifferent values of control 119906

1

Carr

ier p

opul

atio

n

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Figure 14 The plot shows the sensitivity of carrier populations fordifferent values of control 119906

2

There is still a tremendous amount of work needed tobe done in this area Parameters are rarely constant becausethey depend on environmental conditions We do not knowhowever the detailed relationship between these parametersand environmental conditions There may be a time lag as asusceptible populationmay take some time to be infected andalso a susceptible population may take some time to immune

Popu

latio

n (N

V-N

T)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 15 The plot shows the changes in total populations withoutvaccination and without treatment

Popu

latio

n (V

-NT)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 16 The plot shows the changes in total populations withvaccination and without treatment

Popu

latio

n (N

V-T)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 17 The plot shows the changes in total populations withoutvaccination and with treatment

14 Computational and Mathematical Methods in MedicinePo

pulat

ion

(V-T

)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 18 The plot shows the changes in total populations withvaccination and treatment

after vaccination We leave all these possible extensions forthe future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] WHO Hepatitis B Fact Sheet No 204 The World HealthOrganisation Geneva Switzerland 2013 httpwwwwhointmediacentrefactsheetsfs204en

[2] CanadianCentre forOccupationalHealth and Safety ldquoHepatitisBrdquo httpwwwccohscaoshanswersdiseaseshepatitis bhtml

[3] ldquoHealthcare stumbling in RIrsquos Hepatitis fightrdquoThe Jakarta PostJanuary 2011

[4] G F Medley N A Lindop W J Edmunds and D JNokes ldquoHepatitis-B virus endemicity heterogeneity catas-trophic dynamics and controlrdquo Nature Medicine vol 7 no 5pp 619ndash624 2001

[5] J Mann and M Roberts ldquoModelling the epidemiology ofhepatitis B in New Zealandrdquo Journal of Theoretical Biology vol269 no 1 pp 266ndash272 2011

[6] ldquoHepatitis B (HBV)rdquo httpkidshealthorgteensexual healthstdsstd hepatitishtml

[7] CDC ldquoHepatitis B virus a comprehensive strategy for elim-inating transmission in the United States through universalchildhood vaccination Recommendations of the Immuniza-tion Practices Advisory Committee (ACIP)rdquo MMWR Recom-mendations and Reports vol 40(RR-13) pp 1ndash25 1991

[8] M K Libbus and L M Phillips ldquoPublic health management ofperinatal hepatitis B virusrdquo Public Health Nursing vol 26 no 4pp 353ndash361 2009

[9] F B Hollinger and D T Lau ldquoHepatitis B the pathway torecovery through treatmentrdquo Gastroenterology Clinics of NorthAmerica vol 35 no 4 pp 895ndash931 2006

[10] C-L Lai andM-F Yuen ldquoThe natural history and treatment ofchronic hepatitis B a critical evaluation of standard treatment

criteria and end pointsrdquoAnnals of InternalMedicine vol 147 no1 pp 58ndash61 2007

[11] R M Anderson and R M May Infectious Disease of HumansDynamics and Control Oxford University Press Oxford UK1991

[12] S Thornley C Bullen and M Roberts ldquoHepatitis B in ahigh prevalence New Zealand population a mathematicalmodel applied to infection control policyrdquo Journal ofTheoreticalBiology vol 254 no 3 pp 599ndash603 2008

[13] S-J Zhao Z-Y Xu and Y Lu ldquoA mathematical model ofhepatitis B virus transmission and its application for vaccinationstrategy inChinardquo International Journal of Epidemiology vol 29no 4 pp 744ndash752 2000

[14] K Wang W Wang and S Song ldquoDynamics of an HBV modelwith diffusion and delayrdquo Journal ofTheoretical Biology vol 253no 1 pp 36ndash44 2008

[15] R Xu andZMa ldquoAnHBVmodelwith diffusion and time delayrdquoJournal of Theoretical Biology vol 257 no 3 pp 499ndash509 2009

[16] L Zou W Zhang and S Ruan ldquoModeling the transmissiondynamics and control of hepatitis B virus in Chinardquo Journal ofTheoretical Biology vol 262 no 2 pp 330ndash338 2010

[17] J Pang J-A Cui and X Zhou ldquoDynamical behavior of ahepatitis B virus transmission model with vaccinationrdquo Journalof Theoretical Biology vol 265 no 4 pp 572ndash578 2010

[18] S Zhang and Y Zhou ldquoThe analysis and application of an HBVmodelrdquoAppliedMathematicalModelling vol 36 no 3 pp 1302ndash1312 2012

[19] S Bhattacharyya and S Ghosh ldquoOptimal control of verticallytransmitted diseaserdquoComputational andMathematicalMethodsin Medicine vol 11 no 4 pp 369ndash387 2010

[20] T K Kar andA Batabyal ldquoStability analysis and optimal controlof an SIR epidemic model with vaccinationrdquo Biosystems vol104 no 2-3 pp 127ndash135 2011

[21] T-K Kar and S Jana ldquoA theoretical study on mathematicalmodelling of an infectious disease with application of optimalcontrolrdquo Biosystems vol 111 no 1 pp 37ndash50 2013

[22] M J Keeling and P Rohani Modeling Infectious Diseases inHumans and Animals Princeton University Press PrincetonNJ USA 2008

[23] E Jung S Lenhart and Z Feng ldquoOptimal control of treatmentsin a two-strain tuberculosis modelrdquo Discrete and ContinuousDynamical Systems B vol 2 no 4 pp 473ndash482 2002

[24] G Birkhoff and G-C Rota Ordinary Differential EquationsJohn Wiley amp Sons New York NY USA 4th edition 1989

[25] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[26] T T Yusuf and F Benyah ldquoOptimal control of vaccination andtreatment for an SIR epidemiological modelrdquo World Journal ofModelling and Simulation vol 8 no 3 pp 194ndash204 2008

[27] H-R Joshi S Lenhart M Y Li and L Wang ldquoOptimalcontrol methods applied to disease modelsrdquo in MathematicalStudies on Human Disease Dynamics vol 410 of ContemporaryMathematics pp 187ndash207 American Mathematical SocietyProvidence RI USA 2006

[28] G Zaman Y H Kang and I H Jung ldquoStability analysis andoptimal vaccination of an SIR epidemic modelrdquo Biosystems vol93 no 3 pp 240ndash249 2008

[29] D Iacoviello and N Stasiob ldquoOptimal control for SIRCepidemic outbreakrdquo Computer Methods and Programs inBiomedicine vol 110 no 3 pp 333ndash342 2013

Computational and Mathematical Methods in Medicine 15

[30] L S Pontryagin V G Boltyanskii R V Gamkrelidze and EF Mishchenko The Mathematical Theory of Optimal ProcessesJohn Wiley amp Sons New York NY USA 1962

[31] WHO Hepatitis B Fact Sheet No 204 The World HealthOrganisation Geneva Switzerland 2000 httpwwwwhointmediacentrefactsheetsfs204en

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

12 Computational and Mathematical Methods in MedicineAc

ute i

nfec

tion

popu

latio

n

NV-NTNV-T

V-NTV-T

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 9The plot shows the changes in acute infection populations(without) vaccination and (without) treatment

Acut

e inf

ectio

n po

pulat

ion

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0365 u2 = 0

u1 = 045 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

Figure 10 The plot shows the sensitivity of acute infection popula-tions for different values of control 119906

1

6 Conclusion and Suggestions

In this paper we propose an S-119864-119868-119862-119877model of hepatitis Bvirus infection with two controls vaccination and treatmentFirst we analyze the dynamic behavior of the system for con-stant controls In the constant controls case we calculate thebasic reproduction number and investigate the existence andstability of equilibriaThere are two nonnegative equilibria of

Acut

e inf

ectio

n po

pulat

ion

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Figure 11 The plot shows the sensitivity of acute infection popula-tions for different values of control 119906

2

Carr

ier p

opul

atio

n

NV-NTNV-T

V-NTV-T

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 12 The plot shows the changes in carrier populations(without) vaccination and (without) treatment

the system namely the disease-free and endemicWe see thatthe disease-free equilibriumwhich always exists and is locallyasymptotically stable if 119877

0lt 1 and endemic equilibrium

which exists and is locally asymptotically stable if 1198770gt 1

After investigating the dynamic behavior of the systemwith constant controls we formulate an optimal controlproblem if the controls become time dependent and solve itby using Pontryaginrsquos maximum principle Different possiblecombinations of controls are used and their effectiveness iscompared by simulation works Also from the numericalresults it is very clear that a combination of mixed controlmeasures respond better than any other independent control

Computational and Mathematical Methods in Medicine 13Ca

rrie

r pop

ulat

ion

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0365 u2 = 0

u1 = 046 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

Figure 13 The plot shows the sensitivity of carrier populations fordifferent values of control 119906

1

Carr

ier p

opul

atio

n

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Figure 14 The plot shows the sensitivity of carrier populations fordifferent values of control 119906

2

There is still a tremendous amount of work needed tobe done in this area Parameters are rarely constant becausethey depend on environmental conditions We do not knowhowever the detailed relationship between these parametersand environmental conditions There may be a time lag as asusceptible populationmay take some time to be infected andalso a susceptible population may take some time to immune

Popu

latio

n (N

V-N

T)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 15 The plot shows the changes in total populations withoutvaccination and without treatment

Popu

latio

n (V

-NT)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 16 The plot shows the changes in total populations withvaccination and without treatment

Popu

latio

n (N

V-T)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 17 The plot shows the changes in total populations withoutvaccination and with treatment

14 Computational and Mathematical Methods in MedicinePo

pulat

ion

(V-T

)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 18 The plot shows the changes in total populations withvaccination and treatment

after vaccination We leave all these possible extensions forthe future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] WHO Hepatitis B Fact Sheet No 204 The World HealthOrganisation Geneva Switzerland 2013 httpwwwwhointmediacentrefactsheetsfs204en

[2] CanadianCentre forOccupationalHealth and Safety ldquoHepatitisBrdquo httpwwwccohscaoshanswersdiseaseshepatitis bhtml

[3] ldquoHealthcare stumbling in RIrsquos Hepatitis fightrdquoThe Jakarta PostJanuary 2011

[4] G F Medley N A Lindop W J Edmunds and D JNokes ldquoHepatitis-B virus endemicity heterogeneity catas-trophic dynamics and controlrdquo Nature Medicine vol 7 no 5pp 619ndash624 2001

[5] J Mann and M Roberts ldquoModelling the epidemiology ofhepatitis B in New Zealandrdquo Journal of Theoretical Biology vol269 no 1 pp 266ndash272 2011

[6] ldquoHepatitis B (HBV)rdquo httpkidshealthorgteensexual healthstdsstd hepatitishtml

[7] CDC ldquoHepatitis B virus a comprehensive strategy for elim-inating transmission in the United States through universalchildhood vaccination Recommendations of the Immuniza-tion Practices Advisory Committee (ACIP)rdquo MMWR Recom-mendations and Reports vol 40(RR-13) pp 1ndash25 1991

[8] M K Libbus and L M Phillips ldquoPublic health management ofperinatal hepatitis B virusrdquo Public Health Nursing vol 26 no 4pp 353ndash361 2009

[9] F B Hollinger and D T Lau ldquoHepatitis B the pathway torecovery through treatmentrdquo Gastroenterology Clinics of NorthAmerica vol 35 no 4 pp 895ndash931 2006

[10] C-L Lai andM-F Yuen ldquoThe natural history and treatment ofchronic hepatitis B a critical evaluation of standard treatment

criteria and end pointsrdquoAnnals of InternalMedicine vol 147 no1 pp 58ndash61 2007

[11] R M Anderson and R M May Infectious Disease of HumansDynamics and Control Oxford University Press Oxford UK1991

[12] S Thornley C Bullen and M Roberts ldquoHepatitis B in ahigh prevalence New Zealand population a mathematicalmodel applied to infection control policyrdquo Journal ofTheoreticalBiology vol 254 no 3 pp 599ndash603 2008

[13] S-J Zhao Z-Y Xu and Y Lu ldquoA mathematical model ofhepatitis B virus transmission and its application for vaccinationstrategy inChinardquo International Journal of Epidemiology vol 29no 4 pp 744ndash752 2000

[14] K Wang W Wang and S Song ldquoDynamics of an HBV modelwith diffusion and delayrdquo Journal ofTheoretical Biology vol 253no 1 pp 36ndash44 2008

[15] R Xu andZMa ldquoAnHBVmodelwith diffusion and time delayrdquoJournal of Theoretical Biology vol 257 no 3 pp 499ndash509 2009

[16] L Zou W Zhang and S Ruan ldquoModeling the transmissiondynamics and control of hepatitis B virus in Chinardquo Journal ofTheoretical Biology vol 262 no 2 pp 330ndash338 2010

[17] J Pang J-A Cui and X Zhou ldquoDynamical behavior of ahepatitis B virus transmission model with vaccinationrdquo Journalof Theoretical Biology vol 265 no 4 pp 572ndash578 2010

[18] S Zhang and Y Zhou ldquoThe analysis and application of an HBVmodelrdquoAppliedMathematicalModelling vol 36 no 3 pp 1302ndash1312 2012

[19] S Bhattacharyya and S Ghosh ldquoOptimal control of verticallytransmitted diseaserdquoComputational andMathematicalMethodsin Medicine vol 11 no 4 pp 369ndash387 2010

[20] T K Kar andA Batabyal ldquoStability analysis and optimal controlof an SIR epidemic model with vaccinationrdquo Biosystems vol104 no 2-3 pp 127ndash135 2011

[21] T-K Kar and S Jana ldquoA theoretical study on mathematicalmodelling of an infectious disease with application of optimalcontrolrdquo Biosystems vol 111 no 1 pp 37ndash50 2013

[22] M J Keeling and P Rohani Modeling Infectious Diseases inHumans and Animals Princeton University Press PrincetonNJ USA 2008

[23] E Jung S Lenhart and Z Feng ldquoOptimal control of treatmentsin a two-strain tuberculosis modelrdquo Discrete and ContinuousDynamical Systems B vol 2 no 4 pp 473ndash482 2002

[24] G Birkhoff and G-C Rota Ordinary Differential EquationsJohn Wiley amp Sons New York NY USA 4th edition 1989

[25] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[26] T T Yusuf and F Benyah ldquoOptimal control of vaccination andtreatment for an SIR epidemiological modelrdquo World Journal ofModelling and Simulation vol 8 no 3 pp 194ndash204 2008

[27] H-R Joshi S Lenhart M Y Li and L Wang ldquoOptimalcontrol methods applied to disease modelsrdquo in MathematicalStudies on Human Disease Dynamics vol 410 of ContemporaryMathematics pp 187ndash207 American Mathematical SocietyProvidence RI USA 2006

[28] G Zaman Y H Kang and I H Jung ldquoStability analysis andoptimal vaccination of an SIR epidemic modelrdquo Biosystems vol93 no 3 pp 240ndash249 2008

[29] D Iacoviello and N Stasiob ldquoOptimal control for SIRCepidemic outbreakrdquo Computer Methods and Programs inBiomedicine vol 110 no 3 pp 333ndash342 2013

Computational and Mathematical Methods in Medicine 15

[30] L S Pontryagin V G Boltyanskii R V Gamkrelidze and EF Mishchenko The Mathematical Theory of Optimal ProcessesJohn Wiley amp Sons New York NY USA 1962

[31] WHO Hepatitis B Fact Sheet No 204 The World HealthOrganisation Geneva Switzerland 2000 httpwwwwhointmediacentrefactsheetsfs204en

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

Computational and Mathematical Methods in Medicine 13Ca

rrie

r pop

ulat

ion

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0365 u2 = 0

u1 = 046 u2 = 0

u1 = 073 u2 = 0

u1 = 09 u2 = 0

u1 = 09 u2 = 09

Figure 13 The plot shows the sensitivity of carrier populations fordifferent values of control 119906

1

Carr

ier p

opul

atio

n

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

u1 = 0 u2 = 0

u1 = 0 u2 = 0365

u1 = 0 u2 = 045

u1 = 0 u2 = 073

u1 = 0 u2 = 09

u1 = 09 u2 = 09

Figure 14 The plot shows the sensitivity of carrier populations fordifferent values of control 119906

2

There is still a tremendous amount of work needed tobe done in this area Parameters are rarely constant becausethey depend on environmental conditions We do not knowhowever the detailed relationship between these parametersand environmental conditions There may be a time lag as asusceptible populationmay take some time to be infected andalso a susceptible population may take some time to immune

Popu

latio

n (N

V-N

T)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 15 The plot shows the changes in total populations withoutvaccination and without treatment

Popu

latio

n (V

-NT)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 16 The plot shows the changes in total populations withvaccination and without treatment

Popu

latio

n (N

V-T)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 17 The plot shows the changes in total populations withoutvaccination and with treatment

14 Computational and Mathematical Methods in MedicinePo

pulat

ion

(V-T

)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 18 The plot shows the changes in total populations withvaccination and treatment

after vaccination We leave all these possible extensions forthe future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] WHO Hepatitis B Fact Sheet No 204 The World HealthOrganisation Geneva Switzerland 2013 httpwwwwhointmediacentrefactsheetsfs204en

[2] CanadianCentre forOccupationalHealth and Safety ldquoHepatitisBrdquo httpwwwccohscaoshanswersdiseaseshepatitis bhtml

[3] ldquoHealthcare stumbling in RIrsquos Hepatitis fightrdquoThe Jakarta PostJanuary 2011

[4] G F Medley N A Lindop W J Edmunds and D JNokes ldquoHepatitis-B virus endemicity heterogeneity catas-trophic dynamics and controlrdquo Nature Medicine vol 7 no 5pp 619ndash624 2001

[5] J Mann and M Roberts ldquoModelling the epidemiology ofhepatitis B in New Zealandrdquo Journal of Theoretical Biology vol269 no 1 pp 266ndash272 2011

[6] ldquoHepatitis B (HBV)rdquo httpkidshealthorgteensexual healthstdsstd hepatitishtml

[7] CDC ldquoHepatitis B virus a comprehensive strategy for elim-inating transmission in the United States through universalchildhood vaccination Recommendations of the Immuniza-tion Practices Advisory Committee (ACIP)rdquo MMWR Recom-mendations and Reports vol 40(RR-13) pp 1ndash25 1991

[8] M K Libbus and L M Phillips ldquoPublic health management ofperinatal hepatitis B virusrdquo Public Health Nursing vol 26 no 4pp 353ndash361 2009

[9] F B Hollinger and D T Lau ldquoHepatitis B the pathway torecovery through treatmentrdquo Gastroenterology Clinics of NorthAmerica vol 35 no 4 pp 895ndash931 2006

[10] C-L Lai andM-F Yuen ldquoThe natural history and treatment ofchronic hepatitis B a critical evaluation of standard treatment

criteria and end pointsrdquoAnnals of InternalMedicine vol 147 no1 pp 58ndash61 2007

[11] R M Anderson and R M May Infectious Disease of HumansDynamics and Control Oxford University Press Oxford UK1991

[12] S Thornley C Bullen and M Roberts ldquoHepatitis B in ahigh prevalence New Zealand population a mathematicalmodel applied to infection control policyrdquo Journal ofTheoreticalBiology vol 254 no 3 pp 599ndash603 2008

[13] S-J Zhao Z-Y Xu and Y Lu ldquoA mathematical model ofhepatitis B virus transmission and its application for vaccinationstrategy inChinardquo International Journal of Epidemiology vol 29no 4 pp 744ndash752 2000

[14] K Wang W Wang and S Song ldquoDynamics of an HBV modelwith diffusion and delayrdquo Journal ofTheoretical Biology vol 253no 1 pp 36ndash44 2008

[15] R Xu andZMa ldquoAnHBVmodelwith diffusion and time delayrdquoJournal of Theoretical Biology vol 257 no 3 pp 499ndash509 2009

[16] L Zou W Zhang and S Ruan ldquoModeling the transmissiondynamics and control of hepatitis B virus in Chinardquo Journal ofTheoretical Biology vol 262 no 2 pp 330ndash338 2010

[17] J Pang J-A Cui and X Zhou ldquoDynamical behavior of ahepatitis B virus transmission model with vaccinationrdquo Journalof Theoretical Biology vol 265 no 4 pp 572ndash578 2010

[18] S Zhang and Y Zhou ldquoThe analysis and application of an HBVmodelrdquoAppliedMathematicalModelling vol 36 no 3 pp 1302ndash1312 2012

[19] S Bhattacharyya and S Ghosh ldquoOptimal control of verticallytransmitted diseaserdquoComputational andMathematicalMethodsin Medicine vol 11 no 4 pp 369ndash387 2010

[20] T K Kar andA Batabyal ldquoStability analysis and optimal controlof an SIR epidemic model with vaccinationrdquo Biosystems vol104 no 2-3 pp 127ndash135 2011

[21] T-K Kar and S Jana ldquoA theoretical study on mathematicalmodelling of an infectious disease with application of optimalcontrolrdquo Biosystems vol 111 no 1 pp 37ndash50 2013

[22] M J Keeling and P Rohani Modeling Infectious Diseases inHumans and Animals Princeton University Press PrincetonNJ USA 2008

[23] E Jung S Lenhart and Z Feng ldquoOptimal control of treatmentsin a two-strain tuberculosis modelrdquo Discrete and ContinuousDynamical Systems B vol 2 no 4 pp 473ndash482 2002

[24] G Birkhoff and G-C Rota Ordinary Differential EquationsJohn Wiley amp Sons New York NY USA 4th edition 1989

[25] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[26] T T Yusuf and F Benyah ldquoOptimal control of vaccination andtreatment for an SIR epidemiological modelrdquo World Journal ofModelling and Simulation vol 8 no 3 pp 194ndash204 2008

[27] H-R Joshi S Lenhart M Y Li and L Wang ldquoOptimalcontrol methods applied to disease modelsrdquo in MathematicalStudies on Human Disease Dynamics vol 410 of ContemporaryMathematics pp 187ndash207 American Mathematical SocietyProvidence RI USA 2006

[28] G Zaman Y H Kang and I H Jung ldquoStability analysis andoptimal vaccination of an SIR epidemic modelrdquo Biosystems vol93 no 3 pp 240ndash249 2008

[29] D Iacoviello and N Stasiob ldquoOptimal control for SIRCepidemic outbreakrdquo Computer Methods and Programs inBiomedicine vol 110 no 3 pp 333ndash342 2013

Computational and Mathematical Methods in Medicine 15

[30] L S Pontryagin V G Boltyanskii R V Gamkrelidze and EF Mishchenko The Mathematical Theory of Optimal ProcessesJohn Wiley amp Sons New York NY USA 1962

[31] WHO Hepatitis B Fact Sheet No 204 The World HealthOrganisation Geneva Switzerland 2000 httpwwwwhointmediacentrefactsheetsfs204en

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

14 Computational and Mathematical Methods in MedicinePo

pulat

ion

(V-T

)

S

E

I

C

0 10 20 30 40 50 60 70Time (year)

10minus4

10minus3

10minus2

10minus1

100

Figure 18 The plot shows the changes in total populations withvaccination and treatment

after vaccination We leave all these possible extensions forthe future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] WHO Hepatitis B Fact Sheet No 204 The World HealthOrganisation Geneva Switzerland 2013 httpwwwwhointmediacentrefactsheetsfs204en

[2] CanadianCentre forOccupationalHealth and Safety ldquoHepatitisBrdquo httpwwwccohscaoshanswersdiseaseshepatitis bhtml

[3] ldquoHealthcare stumbling in RIrsquos Hepatitis fightrdquoThe Jakarta PostJanuary 2011

[4] G F Medley N A Lindop W J Edmunds and D JNokes ldquoHepatitis-B virus endemicity heterogeneity catas-trophic dynamics and controlrdquo Nature Medicine vol 7 no 5pp 619ndash624 2001

[5] J Mann and M Roberts ldquoModelling the epidemiology ofhepatitis B in New Zealandrdquo Journal of Theoretical Biology vol269 no 1 pp 266ndash272 2011

[6] ldquoHepatitis B (HBV)rdquo httpkidshealthorgteensexual healthstdsstd hepatitishtml

[7] CDC ldquoHepatitis B virus a comprehensive strategy for elim-inating transmission in the United States through universalchildhood vaccination Recommendations of the Immuniza-tion Practices Advisory Committee (ACIP)rdquo MMWR Recom-mendations and Reports vol 40(RR-13) pp 1ndash25 1991

[8] M K Libbus and L M Phillips ldquoPublic health management ofperinatal hepatitis B virusrdquo Public Health Nursing vol 26 no 4pp 353ndash361 2009

[9] F B Hollinger and D T Lau ldquoHepatitis B the pathway torecovery through treatmentrdquo Gastroenterology Clinics of NorthAmerica vol 35 no 4 pp 895ndash931 2006

[10] C-L Lai andM-F Yuen ldquoThe natural history and treatment ofchronic hepatitis B a critical evaluation of standard treatment

criteria and end pointsrdquoAnnals of InternalMedicine vol 147 no1 pp 58ndash61 2007

[11] R M Anderson and R M May Infectious Disease of HumansDynamics and Control Oxford University Press Oxford UK1991

[12] S Thornley C Bullen and M Roberts ldquoHepatitis B in ahigh prevalence New Zealand population a mathematicalmodel applied to infection control policyrdquo Journal ofTheoreticalBiology vol 254 no 3 pp 599ndash603 2008

[13] S-J Zhao Z-Y Xu and Y Lu ldquoA mathematical model ofhepatitis B virus transmission and its application for vaccinationstrategy inChinardquo International Journal of Epidemiology vol 29no 4 pp 744ndash752 2000

[14] K Wang W Wang and S Song ldquoDynamics of an HBV modelwith diffusion and delayrdquo Journal ofTheoretical Biology vol 253no 1 pp 36ndash44 2008

[15] R Xu andZMa ldquoAnHBVmodelwith diffusion and time delayrdquoJournal of Theoretical Biology vol 257 no 3 pp 499ndash509 2009

[16] L Zou W Zhang and S Ruan ldquoModeling the transmissiondynamics and control of hepatitis B virus in Chinardquo Journal ofTheoretical Biology vol 262 no 2 pp 330ndash338 2010

[17] J Pang J-A Cui and X Zhou ldquoDynamical behavior of ahepatitis B virus transmission model with vaccinationrdquo Journalof Theoretical Biology vol 265 no 4 pp 572ndash578 2010

[18] S Zhang and Y Zhou ldquoThe analysis and application of an HBVmodelrdquoAppliedMathematicalModelling vol 36 no 3 pp 1302ndash1312 2012

[19] S Bhattacharyya and S Ghosh ldquoOptimal control of verticallytransmitted diseaserdquoComputational andMathematicalMethodsin Medicine vol 11 no 4 pp 369ndash387 2010

[20] T K Kar andA Batabyal ldquoStability analysis and optimal controlof an SIR epidemic model with vaccinationrdquo Biosystems vol104 no 2-3 pp 127ndash135 2011

[21] T-K Kar and S Jana ldquoA theoretical study on mathematicalmodelling of an infectious disease with application of optimalcontrolrdquo Biosystems vol 111 no 1 pp 37ndash50 2013

[22] M J Keeling and P Rohani Modeling Infectious Diseases inHumans and Animals Princeton University Press PrincetonNJ USA 2008

[23] E Jung S Lenhart and Z Feng ldquoOptimal control of treatmentsin a two-strain tuberculosis modelrdquo Discrete and ContinuousDynamical Systems B vol 2 no 4 pp 473ndash482 2002

[24] G Birkhoff and G-C Rota Ordinary Differential EquationsJohn Wiley amp Sons New York NY USA 4th edition 1989

[25] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[26] T T Yusuf and F Benyah ldquoOptimal control of vaccination andtreatment for an SIR epidemiological modelrdquo World Journal ofModelling and Simulation vol 8 no 3 pp 194ndash204 2008

[27] H-R Joshi S Lenhart M Y Li and L Wang ldquoOptimalcontrol methods applied to disease modelsrdquo in MathematicalStudies on Human Disease Dynamics vol 410 of ContemporaryMathematics pp 187ndash207 American Mathematical SocietyProvidence RI USA 2006

[28] G Zaman Y H Kang and I H Jung ldquoStability analysis andoptimal vaccination of an SIR epidemic modelrdquo Biosystems vol93 no 3 pp 240ndash249 2008

[29] D Iacoviello and N Stasiob ldquoOptimal control for SIRCepidemic outbreakrdquo Computer Methods and Programs inBiomedicine vol 110 no 3 pp 333ndash342 2013

Computational and Mathematical Methods in Medicine 15

[30] L S Pontryagin V G Boltyanskii R V Gamkrelidze and EF Mishchenko The Mathematical Theory of Optimal ProcessesJohn Wiley amp Sons New York NY USA 1962

[31] WHO Hepatitis B Fact Sheet No 204 The World HealthOrganisation Geneva Switzerland 2000 httpwwwwhointmediacentrefactsheetsfs204en

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

Computational and Mathematical Methods in Medicine 15

[30] L S Pontryagin V G Boltyanskii R V Gamkrelidze and EF Mishchenko The Mathematical Theory of Optimal ProcessesJohn Wiley amp Sons New York NY USA 1962

[31] WHO Hepatitis B Fact Sheet No 204 The World HealthOrganisation Geneva Switzerland 2000 httpwwwwhointmediacentrefactsheetsfs204en

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Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

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Disease Markers

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BioMed Research International

OncologyJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

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Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom