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Hindawi Publishing CorporationBioMed Research InternationalVolume 2013 Article ID 150681 10 pageshttpdxdoiorg1011552013150681
Research ArticleTransmission Model of Hepatitis B Virus with theMigration Effect
Muhammad Altaf Khan1 Saeed Islam1 Muhammad Arif1 and Zahoor ul Haq2
1 Department of Mathematics Abdul Wali Khan University Mardan Khyber Pakhtunkhwa Pakistan2Department of Management Sciences Abdul Wali Khan University Mardan Khyber Pakhtunkhwa Pakistan
Correspondence should be addressed to Muhammad Altaf Khan altafdirgmailcom
Received 13 February 2013 Revised 25 March 2013 Accepted 25 March 2013
Academic Editor Kanury Rao
Copyright copy 2013 Muhammad Altaf Khan et alThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited
Hepatitis B is a globally infectious diseaseMathematicalmodeling ofHBV transmission is an interesting research area In this paperwe present characteristics of HBV virus transmission in the form of a mathematical model We analyzed the effect of immigrantsin the model to study the effect of immigrants for the host population We added the following flow parameters ldquothe transmissionbetweenmigrated and exposed classrdquo and ldquothe transmission betweenmigrated and acute classrdquoWith these new features we obtaineda compartment model of six differential equations First we find the basic threshold quantity Ro and then find the local asymptoticstability of disease-free equilibrium and endemic equilibrium Furthermore we find the global stability of the disease-free andendemic equilibria Previous similar publications have not added the kind of information about the numerical results of the modelIn our case from numerical simulation a detailed discussion of the parameters and their numerical results is presented We claimthat with these assumptions and by adding the migrated class the model informs policy for governments to be aware of theimmigrants and subject them to tests about the disease status Immigrants for short visits and students should be subjected totests to reduce the number of immigrants with disease
1 Introduction
According to World Health Organization about 350 millionpeople are infected with the hepatitis B virus (HBV)1 andabout 170 million people are chronically infected with thehepatitis C virus (HCV)2 The majority of those infected livein developing countries with few incidences inWestern coun-tries Migration is one of the defining issues of our time Forexample more than 5 million Canadians migrate out of thecountry each year and over 250000 new immigrants arrivein Canada each year Countries of the world are increasinglyconnected through travel andmigration and thus migrationhas health implications in one location for both local andglobal migrations since infectious diseases do not remainisolated geographically The UK Hepatitis Foundation esti-mated in 2007 that the number of hepatitis B cases in the UKdoubled in the previous 6 years chiefly due to immigrationof infected people many from the new member states of theEuropean Union where the prevalence of viral hepatitis ishigher The United Arab Emirates mandated that hepatitis C
testing is to be done at the time of residence visa renewalsand added hepatitis C to the list of diseases such as HIV andhepatitis B as diseases warranting deportation [1] In Chinahepatitis B virus infection is a major public health problemHepatitis B is the first one among the diseases with legalmanagement measures In China an estimated 93 millionpeople have been infected with the hepatitis B virus [2] Theseroepidemiological survey on HBV infection conducted in2006 showed that HBsAg carrier rate was 718 percent in theoverall dynamics of HBV In this paper we consider a systemof ordinary differential equations which describes the trans-mission ofHBV transmission in China Severalmathematicalmodels have been formulated on the HBV transmission inChina Medley and coauthors used a mathematical modeland developed the strategies to eliminate the HBV in NewZealand in 2008 [3 4] Anderson and May [5] used a mathe-matical model which illustrated the effects of carriers on thetransmission of HBV An age structure model was pro-posed by Zhao et al [6] to predict the dynamics of HBVtransmission and evaluate the long-term effectiveness of the
2 BioMed Research International
vaccination program in China Wang et al [7] proposed andanalyzed the hepatitis B virus infection in a diffusion modelconfined to a finite domain A hepatitis B virus (HBV) modelwith spatial diffusion and saturation response of the infectionrate is investigated by Xu and Ma [8] Also Zou et al [9]proposed a mathematical model to understand the transmis-sion dynamics and prevalence of HBV in mainland ChinaA model to describe waning of immunity after sometime hasbeen studied by a number of authors [10ndash16] In the context ofrapid global migration there is a potential for any disease tobe transferred faster than was previously possible Theseimplications concerning the movement of HBV and HCVmerit far more attention by countries and the internationalcommunity than they have given the problem to date This isespecially important given that the scope and speed ofmigration is expected to grow in coming years In the contextof rapid globalmigration there is a potential for any disease tobe transferred faster than was previously possible Theseimplications concerning the movement of HBV and HCVmerit far more attention by countries and the internationalcommunity than they have given the problem to date Thisis especially important given that the scope and speed ofmigration is expected to grow in coming years
In this paper we construct the compartmental model ofhepatitis B transmission We have categorized the model intosix compartments Susceptible-119878(119905) Exposed-119864(119905) Acute-119860(119905) Carrier-119862(119905) Vaccinated-119881(119905) and Migrated-119872(119905)individuals The migrated class of individuals comes fromdifferent parts of the world to the host country and theirinteraction occurs in the form of sexual interactions bloodtransportation and transfusion We modify the model fromPang et al [10] by adding some new transmission dynamicsand introduce the migrated class in the model Furthermoresome authors [11 13 15 17] show that acute hepatitis B couldbe found today in newborns of infected mothers Pang et al[10] added the vertical transmission term to exposed classfrom chronic carriers class on the basis of the characteristicsof HBV transmission In this paper we improved the modelof [10] with these new features by adding the migrated class119872(119905) and the following parameters
(i) 1205831 the transmission rate from migrated class to
exposed class
(ii) 1205832 the transmission rate frommigrated to acute class
(iii) 120575 the death rate at the migrated class
The paper is organized as follows Section 2 is devoted tothe mathematical formation of the model In Section 3 wefind the Basic Reproduction Number the disease-free equi-librium and endemic equilibrium of the proposed modelLocal asymptotic stability of the disease-free equilibrium andendemic equilibrium is discussed in Section 4 In Section 5we study the global asymptotic stability of disease-freeand endemic equilibria using the Lyapunov function InSection 6 we study the numerical results of the proposedmodel and present the results in the form of plots forillustrations The conclusion and references are presented inSection 7
120575119864
120575119881
119881
120575119862
120575119878119878
119862
120575119872
119872120575119860
(1 minus 119902)1205741119860
119860
1199021205742119860
1205741119864
119864
1205743119862
1205831119872
1205832119872
120575120587120578119862
120573(119860 + 120581119862)119878
120575(1 minus 120587)
1205750119881119901119878
120575120587 minus 120575120587120578119862
Figure 1 The complete flow diagram of hepatitis B virus transmis-sion model
2 Model Formulation
In this section we present the mathematical formulation ofthe compartmental model of hepatitis B which consists of asystem of differential equations The model is based on thecharacteristics of HBV transmissionWe divide the total pop-ulation into six compartments that is Susceptible individuals119878(119905) exposed 119864(119905) Acute 119860(119905) Carrier 119862(119905) Immunity class119881(119905) andMigrated119872(119905)The flow diagram (Figure 1) and thesystem is given in the following
119889119878 (119905)
119889119905= 120575120587 (1 minus 120578119862 (119905)) minus 120575119878 (119905) minus 120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905)
+ 120575119900119881 (119905) minus 119901119878 (119905)
119889119864 (119905)
119889119905= 120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905) minus 120575119864 (119905) + 120575120587120578119862 (119905)
minus 1205741119864 (119905) + 120583
1119872(119905)
119889119860 (119905)
119889119905= 1205741119864 (119905) minus 120575119860 (119905) minus 119902120574
2119860 (119905) minus (1 minus 119902) 120574
1119860 (119905)
+ 1205832119872(119905)
119889119862 (119905)
119889119905= 1199021205742119860 (119905) minus 120575119862 (119905) minus 120574
3119862 (119905)
119889119881 (119905)
119889119905= 1205743119862 (119905) + (1 minus 119902) 120574
1119860 (119905) minus 120575
119900119881 (119905) minus 120575119881 (119905)
+ 120575 (1 minus 120587) + 119901119878 (119905)
119889119872 (119905)
119889119905= minus (120583
1+ 1205832)119872 (119905) minus 120575119872 (119905)
(1)
BioMed Research International 3
Subject to the initial conditions
119878 (0) ge 0 119864 (0) ge 0 119860 (0) ge 0
119862 (0) ge 0 119881 (0) ge 0 119872 (0) ge 0
(2)
The proportion of failure immunization is shown by 120587120575 represent both the death and birth rate At the 120574
1rate the
exposed individuals become infectious andmove to theAcuteclass 120574
2is the rate at which the individuals move to the
carrier class 1205743is the flow of carrier to vaccinated class 120573
shows the transmission coefficient 120581 represents the carrierinfectiousness to acute infection 119902 is the proportion of acuteindividuals that become carrier 120575
119900represent the loss of
immunity rate and the individual become the susceptibleagain 119901 represents the vaccination of susceptible individuals1205831represents the rate of flow from migrated class to exposed
class and 1205742is the rate of transmission from migrated class
to acute class 120578 is the unimmunized children born to carriermothers 120575(1 minus 120587) measures the successful immunization ofnewborn babies and the term 120575120587(1 minus 120578119862(119905)) shows that thenewborns are unimmunized and become susceptible again
We assume that the total population size is equal to 1and just for simplifications 119878(119905) is the susceptible 119864(119905) theexposed 119860(119905) the acute 119862(119905) the carrier 119881(119905) the immunityand119872(119905) is themigrated class representing the state variablesin our proposed population model The sum of the totalpopulation is 119878(119905) + 119864(119905) + 119860(119905) + 119862(119905) + 119881(119905) + 119872(119905) = 1holds We just add (1) and we can easily get We ignore thefifth equation in system (1) so the new models become
119889119878 (119905)
119889119905= 120575120587 (1 minus 120578119862 (119905)) minus 120575119878 (119905) minus 120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905)
+ 120575119900(1minus(119878(119905)+119864(119905)+119860 (119905)+119862(119905)+119872(119905)))minus119901119878(119905)
119889119864 (119905)
119889119905= 120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905) minus 120575119864 (119905) + 120575120587120578119862 (119905)
minus 1205741119864 (119905) + 120583
1119872(119905)
119889119860 (119905)
119889119905= 1205741119864 (119905) minus 120575119860 (119905) minus 119902120574
2119860 (119905) minus (1 minus 119902) 120574
1119860 (119905)
+ 1205832119872(119905)
119889119862 (119905)
119889119905= 1199021205742119860 (119905) minus 120575119862 (119905) minus 120574
3119862 (119905)
119889119872 (119905)
119889119905= minus (120583
1+ 1205832)119872 (119905) minus 120575119872 (119905)
(3)
Let
Γ = (119878 119864 119860 119862119872) isin R5
+ | 119878 (119905) le
120575120587 + 120575119900
120575 + 120575119900+ 119901
(119878 + 119864 + 119860 + 119862 +119872 le120575120587 + 120575
119900
120575 + 120575119900
)
(4)
Here Γ is a positively invariant set All the solutions lie insideΓ which is our main focus
3 Basic ReproductionNumberThreshold Quantity
The basic reproduction number or the threshold quantityR0for the proposed model gives an average number of
secondary infection when an infection is introduced in apurely susceptible population We use the idea developed by[18] and also for detail see [19] We have
F = [
[
0 120573119878119900120573120581119878119900
0 0 0
0 0 0
]
]
119881 = [
[
120575 + 1205741
0 minus120575120587120578
minus1205741
120575 + 1199021205742+ (1 minus 119902) 120574
10
0 minus1199021205742
120575 + 1205743
]
]
119881minus1=
[
(120575+1199021205742+(1minus119902)1205741)(120575+1205743) 1199021205742120575120587120578 120575120587120578(120575+1199021205742+(1minus119902)1205741)
1205741(120575+1205743) (120575+1205741)(120575+1205743) 120575120587120578120574
1
11990212057411205742
1199021205742(120575+1205741) (120575+1205741)(120575+1199021205742+(1minus119902)1205741)
]
(120575 + 1205741) (120575 + 119902120574
2+ (1 minus 119902) 120574
1) (120575 + 120574
3) minus 120575120587120578119902120574
11205742
(5)
where 1198761= (120575 + 119902120574
2+ (1 minus 119902)120574
1) and
119865119881minus1=
[[[[[
[
0120573119878119900120575120587120578119902120574
2
(120575 + 1205741) (120575 + 119902120574
2+ (1 minus 119902) 120574
1) (120575 + 120574
3) minus 120575120587120578119902119903
11199032
120581120573119878119900(120575 + 120574
1(1 minus 119902) + 119902120574
2)
(120575 + 1205741) (120575 + 119902120574
2+ (1 minus 119902) 120574
1) (120575 + 120574
3) minus 120575120587120578119902119903
11199032
0 0 0
0 0 0
]]]]]
]
(6)
4 BioMed Research International
So the reproduction number given by 120588(119865119881minus1) is
R119900=120581120573119878119900(120575 + 120574
1(1 minus 119902) + 119902120574
2) + 120575120587120578119902119903
11199032
(120575 + 1205741) (120575 + 119902120574
2+ (1 minus 119902) 120574
1) (120575 + 120574
3) (7)
Here 119878119900 shows the disease-free equilibrium (DFE) and119863119900=
(119878119900 0 0 0 0) giving 119878119900 = (120575120587+120575
119900)(120575+120575
119900+119901) The endemic
equilibrium point 119879lowast = (119878lowast 119864lowast 119860lowast 119862lowast119872lowast) for system
(3) whose endemic equilibrium is given in the followingsubsection
Endemic Equilibria To find the endemic equilibria of thesystem (3) by setting 119878 = 119878
lowast 119864 = 119864lowast 119860 = 119860
lowast 119862 = 119862lowast
and 119872 = 119872lowast equating left side of the system (3) equal to
zero we obtained
119878lowast=1198761(120575 + 120574
1) (120575 + 120574
3) (1 minus 119877
0) + 120581120573119878
1199001198761
1205741120573 ((120575 + 120574
3) + 120581119902120574
2)
minus(120575 + 120574
3) [(120575 + 120574
1) + 12058311205741]119872lowast
1205741120573 ((120575 + 120574
3) + 120581119902120574
2) 119860lowast
119864lowast=1198761119860lowastminus 1205832119872lowast
1205741
119862lowast=
1199021205742
(120575 + 1205743)119860lowast
(8)
4 Local Stability Analysis
In this section we find the local stability of disease-free andendemic equilibria First we show the local asymptoticalstability of DFE equilibrium and then we find the localasymptotical stability of endemic equilibrium Now we showthe local stability of DFE about the point 119863
119900= (119878119900 0 0 0 0)
in the following theorem
Theorem 1 For 1198770le 1 the disease-free equilibrium of the
system (3) about an equilibrium point 119863119900= (119878119900 0 0 0 0) is
locally asymptotically stable if ((1198761(120575 + 120575
119900+ 119901)(120575 + 120574
1) gt
1205731205741(120575120587 + 120575
119900)) otherwise the disease-free equilibrium of the
system (3) is unstable for 1198770gt 1
Proof To show the local stability of the system (3) about thepoint119863
119900 we set the left-hand side of the system (3) equating
to zero and we obtain the following Jacobian matrix 119869119900(120577)
119869119900(120577) = (
minus120575 minus 120575119900minus 119901 120575
119900minus120573119878119900minus 120575119900minus120575120587120578 minus 120573120581119878
119900minus 120575119900
0
0 minus (120575 + 1205741) 120573119878
119900120573120581119878119900+ 120575120587120578 120583
1
0 1205741
minus1198761
0 1205832
0 0 1199021205742
minus (120575 + 1205743) 0
0 0 0 0 minus (1205831+ 1205832+ 120575)
) (9)
By the elementary row operation we get the followingmatrix
119869119900(120577) = (
minus(120575 + 120575119900+ 119901) minus120575
119900minus120573119878119900minus 120575119900
minus120575120587120578 minus 120573120581119878119900
0
0 minus (120575 + 1205741) 120573119878
119900120573120581119878119900+ 120575120587120578 120583
1
0 0 minus1198761(120575 + 120574
1) + 12057411205731198781199001205741(120573120581119878119900+ 120575120587120578) (120575 + 120574
1) 1205832+ 12058311205741
0 0 0 1198791
0
0 0 0 0 minus (1205831+ 1205832+ 120575)
) (10)
where 1198761= (120575 + 119902120574
2+ (1 minus 119902)120574
1) and 119879
1= minus(minus119876
1(120575 + 120574
1) +
1205741120573119878119900) minus 119902120574
11205742(120573120581119878119900+ 120575120587120578) The characteristic equation to
the previous Jacobian matrix is given by
(120582+120575 + 120575119900+119901) (120582 + 120583
1+1205832+ 120575) (120582
3+ 11988611205822+1198862120582+1198863) = 0
(11)
The first two eigenvalues minus(120575+120575119900+119901) and minus(120583
1+1205832+120575) have
negative real parts For the rest of the eigenvalues we get
1198861= (120575 + 120575
119900+ 119901) (119876
1(120575 + 120574
3) + 1 + 119876
1) (1 minus 119877
0)
+ 120573 (120575120587 + 120575119900) ((1205741(120575 + 120574
3) + 119902120574
11205742) minus (119876
1120581 + 1205741))
times (120575 + 120575119900+ 119901)minus1
1198862= (1198761(120575 + 120574
1) (120575 + 120575
119900+ 119901) minus 120573120574
1(120575120587 + 120575
119900))
times (120575 + 120575119900+ 119901) [(120575 + 120574
1) + (120575 + 120574
1)2
]
+ (120575 + 1205741) [1198761(120575 + 120574
1) (120575 + 120575
119900+ 119901) minus 120573120574
1(120575120587 + 120575
119900)]
+ 11990212057411205742(120575 + 120574
1) [120573120581 (120575120587 + 120575
119900) + 120575120587120578 (120575 + 120575 + 119901)]
times (120575 + 120575119900+ 119901)minus2
+ 11990212057411205742(120575 + 120574
1) (120573120581 (120575120587 + 120575
119900) + 120575120587120578 (120575 + 120575
119900+ 119901))
times (120575 + 120575119900+ 119901)minus1
BioMed Research International 5
1198863=
(120575 + 1205741)
(120575 + 120575119900+ 119901)
(1198761(120575 + 120575
119900+ 119901) (120575 + 120574
1) minus 120573120574
1(120575120587 + 120575
119900))
times [11990212057411205742(120573120581 (120575120587 + 120575
119900) + 120575120587120578)
+ (1198761(120575 + 120574
1) (120575 + 120575
119900+ 119901) minus 120573120574
1(120575120587 + 120575
119900))]
(12)
By the Routh-Hurwitz criteria 1198861gt 0 119886
3gt 0 and 119886
11198862gt 1198863
Here 1198861gt 0 when 119877
0le 1 and ((119876
1(120575 + 120575
119900+ 119901)(120575 + 120574
1) gt
1205731205741(120575120587 + 120575
119900)) Also 119886
2gt 0 and 119886
3gt 0 and then 119886
11198862gt
1198863 So according to the Routh-Hurwitz criteria the Jacobian
matrix has negative real parts if and only if 1198770le 1 Thus by
Routh-Hurwitz criteria the DFE of the system (3) is locallyasymptotically stable about the point119863
119900= (119878119900 0 0 0 0) The
proof is completed
The stability of the disease-free equilibrium for 1198770le 1
means that the disease dies out from the population Next weshow that the endemic equilibrium of the system (3) is locallyasymptotically stable for 119877
0gt 1 When the disease-free
equilibrium is locally asymptotically stable for 1198770le 1 then
the endemic equilibrium does not exist but we are interestedto know about the properties of an endemic equilibrium for1198770gt 1
41 Stability of Endemic Equilibrium (EE) In this subsectionwe find the local asymptotic stability of EE about 119863lowast =
(119878lowast 119864lowast 119860lowast 119862lowast119872lowast) and we prove the local stability of
endemic equilibrium in the following
Theorem2 For1198770gt 1 the endemic equilibrium119863lowast of system
(3) is locally asymptotical stable if the following conditionshold
12057411198852120575119900119878lowast2
gt 12057411205732120581
1198661gt 1198662
(13)
otherwise the system is unstable
Proof Here we prove the that the system (3) about theequilibrium point119863lowast is locally stable and for this we obtainthe Jacobian matrix 119869
lowast(120577) of the system (3) in the following
119869lowast(120577) = (
minus(120575 + 120575119900+ 119901 + 120573 (119860
lowast+ 120581119862lowast)) 120575
119900minus120573119878lowastminus 120575119900minus120575120587120578 minus 120573120581119878
lowastminus 120575119900
minus120575119900
120573 (119860lowast+ 120581119862lowast) minus (120575 + 120574
1) 120573119878
lowast120573120581119878lowast+ 120575120587120578 120583
1
0 1205741
minus1198761
0 1205832
0 0 1199021205742
minus (120575 + 1205743) 0
0 0 0 0 minus (120575 + 1205831+ 1205832)
) (14)
By elementary row operation and after simplification we getthe following Jacobian matrix
119869lowast(120577) = (
minus1198851
minus120575119900
minus120573119878lowastminus 120575119900
1198853
minus120575119900
0 minus1198851(120575 + 120574
1) minus 11988521205751199001198851120573119878lowastminus 1198852(120573119878lowast+ 120575119900) 119885
11198854+ 11988521198853
11988511205831minus 1198852120575119900
0 0 1198855
1205741(11988511198854+ 11988521198853) 119885
6
0 0 0 1198857
0
0 0 0 0 minus (1205831+ 1205832+ 120575)
) (15)
where
1198851= 120575 + 120575
119900+ 119901 + 120573 (119860
lowast+ 120581119862lowast)
1198852= 120573 (119860
lowast+ 120581119862lowast)
1198853= minus (120575120587120578 + 120573120581119878
lowast+ 120575119900)
1198854= (120573120581119878
lowast+ 120575120587120578)
1198855= minus119876
1(minus1198851(120575 + 120574
1) minus 1198852120575119900)
+ 1205741(1198854120573119878lowastminus 1198852(120573119878lowast+ 120575119900))
1198856= minus1205832(minus1198851(120575 + 120574
1) minus 1198852120575119900) + 1205741(11988511205831minus 1198852120575119900)
1198857= minus1198855(120575 + 120574
3) minus 11988551205741(11988511198854+ 11988521198853)
(16)
The eigenvalue 1205821= minus(120583
1+ 1205832+ 120575) lt 0 120582
2= minus1198851 and using
the value of 1198851and further 119862lowast we get 120582
2= minus(120575 + 120575
119900+ 119901 +
120573((120575 + 1205743) + 120581119902120574
2)119860lowast) lt 0 120582
3= minus(119885
1(120575 + 120574
1) + 1198852120575119900) lt 0
as 1198851gt 0 and 119885
2gt 0 120582
4= 1198855 1205824lt 0 if and only if 119885
5lt 0
After the simplifications we get
120573lowast119860lowast(12057411198852120575119900minus 12057411205732120581119878lowast2
)
+ 1205741120573120575120587120578120573
lowastlowastlowast119872lowast
minus [(1205741120573120575120587120578120573
lowastlowast+ 120573lowast1198761) 1198851(120575 + 120574
1)
+11987611198852120575119900120573lowast] gt 0
(17)
where 120573lowast = 1205741120573((120575 + 120574
3) + 120581119902120574
2) 120573lowastlowast = 119876
1(120575 + 120574
1)(120575 +
1205743)(1minus119877
0)+120581120573119878
1199001198761 and 120574
1120573120575120587120578120573
lowastlowastlowast119872lowastgt [(1205741120573120575120587120578120573
lowastlowast+
120573lowast1198761)1198851(120575+1205741)+11987611198852120575119900120573lowast] say119866
1= 1205741120573120575120587120578120573
lowastlowastlowast119872lowast and
6 BioMed Research International
1198662= [(1205741120573120575120587120578120573
lowastlowast+120573lowast1198761)1198851(120575+1205741)+11987611198852120575119900120573lowast]120573lowastlowastlowast
= (120575+
1205743)[(120575 + 120574
1) + 12057411205831] So 120582
4has negative real part if (120574
11198852120575119900gt
12057411205732120581119878lowast2) For 120582
5= 1198857 we obtained negative real parts
and by using the 1198855which is positive under the conditions
described in 1205824 and 119885
1gt 0 119885
2gt 0 we get the negative
real partsThus all the eigenvalues have negative real parts soby the Routh-Hurwitz criteria the endemic equilibrium point119863lowast is locally asymptotically stable when 119877
0gt 1 The proof is
completed
5 Global Stability of DFE
In this section we present the global stability of disease-freeequilibrium DFE of the system (3) For different biologicalmodel the Lyapunov function was used by [20 21] for theglobal stability For our model we define and constructLyapunov function in the following for the global stability ofDFE Further the global stability of endemic equilibrium weuse the Lyapunov function and find its global asymptoticalstability
Theorem 3 For 1198770le 1 the disease-free equilibrium of the
system (3) is stable globally asymptotically if 119878 = 119878119900 and
unstable for 1198770gt 1
Proof Here we define the Lyapunov function for the globalstability of disease-free equilibrium given by
119881 (119905) = [1198891(119878 minus 119878
119900) + 1198892119864 + 1198893119860 + 119889
4119862 + 119889
5119872] (18)
Differentiating the previous function with respect to 119905 andusing the system (3)
1198811015840(119905) = 119889
11198781015840+ 11988921198641015840+ 11988931198601015840+ 11988941198621015840+ 11988951198721015840
1198811015840(119905)
= 1198891[120575120587 (1 minus 120578119862 (119905)) minus 120575119878 (119905) minus 120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905)
+ 120575119900(1minus(119878 (119905) + 119864 (119905) +119860 (119905) + 119862 (119905) +119872 (119905)))minus119901119878 (119905)]
+ 1198892[120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905) minus 120575119864 (119905) + 120575120587120578119862 (119905)
minus1205741119864 (119905) + 120583
1119872(119905)]
+ 1198893[1205741119864 (119905) minus 120575119860 (119905) minus 119902120574
2119860 (119905) minus (1 minus 119902) 120574
1119860 (119905)
+1205832119872(119905)]
+ 1198894[1199021205742119860 (119905) minus 120575119862 (119905) minus 120574
3119862 (119905)]
+ 1198895[minus (1205831+ 1205832)119872 (119905) minus 120575119872 (119905)]
(19)
where 119889119894 119894 = 1 2 5 are some positive constants to be
chosen later
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Time (year)
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 2 The plot shows the HBV transmission model of hepatitisB with 120583
1= 090 and 120583
2= 090
Time (year)0 1 2 3 4 5 6 7 8 9 10
0
10
20
30
40
50
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 3 The plot shows the HBV transmission model of hepatitisB with 120583
1= 080 and 120583
2= 090
After the arrangement we obtain
1198811015840(119905) = [119889
2minus 1198891] (120573 (119860 + 120581119862)) 119878 + [119889
2minus 1198891] 120575120587120578119862
+ [11988931205741minus 1198892(120575 + 120574
1)] 119864 + [119889
21205831minus 1198895120575]119872
+ [11988941199021205742minus 11988931198761] 119860 + 119889
1120575120587 minus 119889
1120575119878 + 119889
1120575119900
minus 1198891120575119900(119878 + 119864 + 119860 + 119862 +119872) minus 119889
1119901119878
minus 1198894(120575 + 120574
3) 119862
(20)
BioMed Research International 7
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Time (year)
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 4 The plot shows the HBV transmission model of hepatitisB with 120583
1= 070 and 120583
2= 080
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Time (year)
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 5 The plot shows the HBV transmission model of hepatitisB with 120583
1= 050 and 120583
2= 060
Choosing the constants 1198891= 1198892= 1205741 1198893= (120575 + 120574
1) 1198894=
1198761(120575 + 120574
1)1199021205742 and 119889
5= 12057411205831120575
After the simplification we get
1198811015840(119905) = minus (119878 minus 119878
119900) minus 1205741120575119900119864 minus 1205741120575119900119860
minus (1205741120575119900+(120575 + 120574
3) (120575 + 120574
1) 1198761
1199021205742
)119862
minus (1205741120575119900+ (120575119900+ 1205741) 1205832)119872
(21)
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Time (year)
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 6 The plot shows the HBV transmission model of hepatitisB with 120583
1= 020 and 120583
2= 030
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Time (year)
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 7 The plot shows the HBV transmission model of hepatitisB with 120583
1= 020 and 120583
2= 040
where 119878119900 = (120575120587 + 120575119900)(120575 + 120575
119900+ 119901) 1198811015840(119905) = 0 if and only if
119878 = 119878119900 and 119864 = 119860 = 119862 = 119872 = 0 Also 1198811015840(119905) is negative for
(119878 gt 119878119900) So by [22] the DFE is globally asymptotically stable
in Γ The proof is completed
51 Global Stability of Endemic Equilibrium In this subsec-tion we show the global asymptotical stability of the system(3) To do this we state and prove the following theorem forthe global stability of endemic equilibrium
8 BioMed Research International
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Time (year)
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 8 The plot shows the HBV transmission model of hepatitisB with 120583
1= 010 and 120583
2= 020
Theorem 4 For 1198770gt 1 system (3) is globally asymptotically
stable if 119878 = 119878lowast and 120575119900gt 1205831 and unstable for 119877
0le 1
Proof To prove that system (3) is globally asymptoticallystable we define the Lyapunov in the following
119871 (119905) =(1205831+ 1205832+ 1205833) 1205741
1205741+ 120575
(119878 minus 119878lowast) +
(1205831+ 1205832+ 1205833) 1205741
1205741+ 120575
119864
+ (1205831+ 1205832+ 1205833) 119860 +
(1205831+ 1205832+ 1205833) 1198761
1199021205742
119862 + 1205832119872
(22)
Taking the derivative with respect to time 119905 using the system(3)
1198711015840(119905) =
(1205831+ 1205832+ 1205833) 1205741
1205741+ 120575
times [120575120587 (1 minus 120578119862 (119905)) minus 120575119878 (119905)minus120573 (119860 (119905)+120581119862 (119905)) 119878 (119905)
+ 120575119900(1 minus (119878 (119905) + 119864 (119905) + 119860 (119905) + 119862 (119905) + 119872 (119905)))
minus119901119878 (119905)]
+(1205831+ 1205832+ 1205833) 1205741
1205741+ 120575
[120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905) minus 120575119864 (119905)
+120575120587120578119862 (119905) minus 1205741119864 (119905) + 120583
1119872(119905)]
+ (1205831+ 1205832+ 1205833) [1205741119864 (119905) minus 120575119860 (119905) minus 119902120574
2119860 (119905)
minus (1 minus 119902) 1205741119860 (119905) + 120583
2119872(119905)]
+(1205831+ 1205832+ 1205833) 1198761
1199021205742
[1199021205742119860 (119905) minus 120575119862 (119905) minus 120574
3119862 (119905)]
+ 1205832[minus (1205831+ 1205832+ 120575)119872]
(23)Simplifying we obtained
1198711015840(119905) = minus
(1205831+ 1205832+ 1205833) 1205741
(1205741+ 120575) (120575 + 120575
119900+ 119901)
(119878 minus 119878lowast)
minus(1205831+ 1205832+ 1205833) 1205741120575119900
(1205741+ 120575)
[119864 + 119860 + 119862]
minus(1205831+ 1205832+ 1205833) 1205741
(1205741+ 120575)
[120575119900minus 1205831]119872
(24)
The endemic equilibrium of the system (3) is globally asymp-totically stable for 119877
0gt 1 if 119878 = 119878
lowast and 120575119900gt 1205831
So the endemic equilibrium of the system (3) is globallyasymptotically stable The proof is completed
6 Numerical Simulations
In this section we present the numerical simulation of theproposed model (3) by using the Runge-Kutta order fourscheme For different values of the parameters the numericalresults are presented in Figures 2 3 4 5 6 7 and 8 Thevariation of migration parameters 120583
1and 120583
2 with different
values is presented In the numerical solution of the modelthe parameters and their values are presented in Table 1In our simulation the susceptible individuals are shown bydashed line the exposed individuals by dotted dashed theacute individuals by bold line the carrier by dashed lineand the migrated individuals by red bold line Figures 2to 8 represent the compartmental population of hepatitis Bindividuals with migration effect The values presented inTable 1 are fixed except for120583
1and1205832 In Figure 2 by the values
for 1205831= 09 and 120583
2= 09 we see that the population of
exposed and acute individuals is increasing In Figure 3 weset 1205831= 08 and 120583
2= 09 and the population of exposed
and acute individuals decreased Decreasing the values of 1205831
and 1205832 we obtain different results see Figures 2 to 8 When
we decrease the population of immigrants who have the HBVvirus we see the decrease in the population of exposed andacute individuals The parameters presented in Table 1 wereused by different authors for example the natural deathrate equally birth rate by [4] the rate at which the latentindividuals become infectious by [3] the rate at which theindividuals move to the carrier class by [3] the rate at whichthe individuals move to the carrier class [3] the transmissioncoefficient 120573 by [24] 120575
119900the loss of immunity rate by [13] the
value of 120578 by [13] the value of 120581 by [10] and120587 and 119902 from [23]We assume the values for the parameters 120574
3 119901 1205831 and 120583
2in
our simulation
7 Conclusion
A compartmental model of HBV transmission virus hasbeen presented A mathematical model has been obtained
BioMed Research International 9
Table 1 Parameter values used in numerical simulations
Notation Parameter description Range Source120575 Natural death rate equally birth rate 00143 [4]120587 The failure immunization 0-1 [23]1205741
The rate at which the latent individuals become infectious 6 [3]1205742
The rate at which the individuals move to the carrier class 4 [3]1205743
The rate of flow from carrier to the vaccinated class 034 Assumed120573 The transmission coefficient 08 [24]119902 The proportion of individuals become carrier 0005 [23]120575119900
Represent the loss of immunity 006ndash003 [13]119901 Represent the vaccination of susceptible 03 Assumed1205831
The rate of flow fromMigrated class to exposed class 023 Assumed1205832
The rate of flow fromMigrated class to acute class 056 Assumed120578 Unimmunized children born to carrier mothers 07 [13]120581 The infectiousness of carriers related to acute infection 0-1 [10]
by adding (1) the migrated class (2) HBV transmission ratebetween migrated class and exposed class (3) transmissionrate between migrated class and acute class and (4) deathrate of individuals in the migrated class By adding thesenew features we have obtained a compartmental model ofHBVwithmigration effect First we obtained the basic repro-duction number for the proposed model The disease-freeequilibrium is locally as well as globally asymptotically stablefor 1198770le 1 We obtained the local and global asymptotical
stability for the endemic equilibrium For the reproductionnumber 119877
0gt 1 the endemic equilibrium is locally as
well as globally asymptotically stable Furthermore we havesolved the compartment model numerically and the resultsare presented in Figures 2 to 8 By changing the values of 120583
1
and 1205832 different results have been obtained It is concluded
that when the value of 1205831and 120583
2decreases the population
of (exposed acute and carrier) individuals also decreaseThe proportion of infected individuals decreases when theproportion ofmigrated individuals (who have theHBVvirus)decreases So the number of infected individuals is directlyproportional to the number of migrated individuals
Last yet not the least the authors of this work have agreedto devise in a course of time a more advanced model onrestraining HBV transmission through migration
Acknowledgment
Theauthors would like to thank the Kanury V S Rao for theircareful reading of the original manuscripts and their manyvaluable comments and suggestions that greatly improve thepresentation of this work
References
[1] F Ahmed and G R Foster ldquoGlobal hepatitis migration and itsimpact onWestern healthcarerdquoGut vol 59 no 8 pp 1009ndash10112010
[2] ldquoChinese center for disease control and prevention (CCDC)rdquo 2010httpwwwchinacdccnn272442n272530n3479265n347930337095html
[3] 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
[4] 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
[5] R M Anderson and R M May Infectious Disease of HumansDynamics and Control Oxford University Press Oxford UK1991
[6] S Zhao Z Xu and Y Lu ldquoA mathematical model of hepatitis Bvirus transmission and its application for vaccination strategyin Chinardquo International Journal of Epidemiology vol 29 no 4pp 744ndash752 2000
[7] K Wang W Wang and S Song ldquoDynamics of an HBV modelwith diffusion and delayrdquo Journal ofTheoretical Biology vol 253no 1 pp 36ndash44 2008
[8] R Xu andZMa ldquoAnHBVmodelwith diffusion and time delayrdquoJournal of Theoretical Biology vol 257 no 3 pp 499ndash509 2009
[9] 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
[10] 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
[11] K P Maier Hepatitis-Hepatitisfolgen Georg Thieme VerlagStuttgart Germany 2000
[12] G L Mandell R G Douglas and J E Bennett Principles andPractice of Infectious Diseases A Wiley Medical PublicationJohn Wiley and Sons New York NY USA 1979
[13] CW Shepard E P Simard L Finelli A E Fiore and B P BellldquoHepatitis B virus infection epidemiology and vaccinationrdquoEpidemiologic Reviews vol 28 no 1 pp 112ndash125 2006
[14] X Weng and Y Zhang Infectious Diseases Fudan UniversityPress Shanghai China 2003
[15] J Wu and Y Luo Infectious Diseases Central South UniversityPress Changsha China 2004
[16] J BWyngaarden L H Smith and J C BennettCecil Text Bookof Medicine WB Saunders Philadelphia Pa USA 19th edition1992
10 BioMed Research International
[17] B JMcMahonW LM Alward andD BHall ldquoAcute hepatitisB virus infection relation of age to the clinical expressionof disease and subsequent development of the carrier staterdquoJournal of Infectious Diseases vol 151 no 4 pp 599ndash603 1985
[18] P Van Den Driessche and J Watmough ldquoReproduction num-bers and sub-threshold endemic equilibria for compartmentalmodels of disease transmissionrdquoMathematical Biosciences vol180 pp 29ndash48 2002
[19] O Diekmann and J A P HeesterbeekMathematical Epideniol-ogy of Infectious Diseases 2000
[20] A Mwasa and J M Tchuenche ldquoMathematical analysis of acholera model with public health interventionsrdquo BioSystemsvol 105 no 3 pp 190ndash200 2011
[21] J Li and Y Yang ldquoSIR-SVS epidemic models with continuousand impulsive vaccination strategiesrdquo Journal of TheoreticalBiology vol 280 no 1 pp 108ndash116 2011
[22] J P LaSalle ldquoStability of nonautonomous systemsrdquo NonlinearAnalysis vol 1 no 1 pp 83ndash90 1976
[23] World Health Organization ldquoHepatitis B WHOCDSCSRLYO20022Hepatitis Brdquo httpwwwwhointcsrdiseasehepa-titiswhocdscsrlyo20022en
[24] W J Edmunds G FMedley andD J Nokes ldquoThe transmissiondynamics and control of hepatitis B virus in the GambiardquoStatistics in Medicine vol 15 pp 2215ndash2233 2002
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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International Journal of
Microbiology
2 BioMed Research International
vaccination program in China Wang et al [7] proposed andanalyzed the hepatitis B virus infection in a diffusion modelconfined to a finite domain A hepatitis B virus (HBV) modelwith spatial diffusion and saturation response of the infectionrate is investigated by Xu and Ma [8] Also Zou et al [9]proposed a mathematical model to understand the transmis-sion dynamics and prevalence of HBV in mainland ChinaA model to describe waning of immunity after sometime hasbeen studied by a number of authors [10ndash16] In the context ofrapid global migration there is a potential for any disease tobe transferred faster than was previously possible Theseimplications concerning the movement of HBV and HCVmerit far more attention by countries and the internationalcommunity than they have given the problem to date This isespecially important given that the scope and speed ofmigration is expected to grow in coming years In the contextof rapid globalmigration there is a potential for any disease tobe transferred faster than was previously possible Theseimplications concerning the movement of HBV and HCVmerit far more attention by countries and the internationalcommunity than they have given the problem to date Thisis especially important given that the scope and speed ofmigration is expected to grow in coming years
In this paper we construct the compartmental model ofhepatitis B transmission We have categorized the model intosix compartments Susceptible-119878(119905) Exposed-119864(119905) Acute-119860(119905) Carrier-119862(119905) Vaccinated-119881(119905) and Migrated-119872(119905)individuals The migrated class of individuals comes fromdifferent parts of the world to the host country and theirinteraction occurs in the form of sexual interactions bloodtransportation and transfusion We modify the model fromPang et al [10] by adding some new transmission dynamicsand introduce the migrated class in the model Furthermoresome authors [11 13 15 17] show that acute hepatitis B couldbe found today in newborns of infected mothers Pang et al[10] added the vertical transmission term to exposed classfrom chronic carriers class on the basis of the characteristicsof HBV transmission In this paper we improved the modelof [10] with these new features by adding the migrated class119872(119905) and the following parameters
(i) 1205831 the transmission rate from migrated class to
exposed class
(ii) 1205832 the transmission rate frommigrated to acute class
(iii) 120575 the death rate at the migrated class
The paper is organized as follows Section 2 is devoted tothe mathematical formation of the model In Section 3 wefind the Basic Reproduction Number the disease-free equi-librium and endemic equilibrium of the proposed modelLocal asymptotic stability of the disease-free equilibrium andendemic equilibrium is discussed in Section 4 In Section 5we study the global asymptotic stability of disease-freeand endemic equilibria using the Lyapunov function InSection 6 we study the numerical results of the proposedmodel and present the results in the form of plots forillustrations The conclusion and references are presented inSection 7
120575119864
120575119881
119881
120575119862
120575119878119878
119862
120575119872
119872120575119860
(1 minus 119902)1205741119860
119860
1199021205742119860
1205741119864
119864
1205743119862
1205831119872
1205832119872
120575120587120578119862
120573(119860 + 120581119862)119878
120575(1 minus 120587)
1205750119881119901119878
120575120587 minus 120575120587120578119862
Figure 1 The complete flow diagram of hepatitis B virus transmis-sion model
2 Model Formulation
In this section we present the mathematical formulation ofthe compartmental model of hepatitis B which consists of asystem of differential equations The model is based on thecharacteristics of HBV transmissionWe divide the total pop-ulation into six compartments that is Susceptible individuals119878(119905) exposed 119864(119905) Acute 119860(119905) Carrier 119862(119905) Immunity class119881(119905) andMigrated119872(119905)The flow diagram (Figure 1) and thesystem is given in the following
119889119878 (119905)
119889119905= 120575120587 (1 minus 120578119862 (119905)) minus 120575119878 (119905) minus 120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905)
+ 120575119900119881 (119905) minus 119901119878 (119905)
119889119864 (119905)
119889119905= 120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905) minus 120575119864 (119905) + 120575120587120578119862 (119905)
minus 1205741119864 (119905) + 120583
1119872(119905)
119889119860 (119905)
119889119905= 1205741119864 (119905) minus 120575119860 (119905) minus 119902120574
2119860 (119905) minus (1 minus 119902) 120574
1119860 (119905)
+ 1205832119872(119905)
119889119862 (119905)
119889119905= 1199021205742119860 (119905) minus 120575119862 (119905) minus 120574
3119862 (119905)
119889119881 (119905)
119889119905= 1205743119862 (119905) + (1 minus 119902) 120574
1119860 (119905) minus 120575
119900119881 (119905) minus 120575119881 (119905)
+ 120575 (1 minus 120587) + 119901119878 (119905)
119889119872 (119905)
119889119905= minus (120583
1+ 1205832)119872 (119905) minus 120575119872 (119905)
(1)
BioMed Research International 3
Subject to the initial conditions
119878 (0) ge 0 119864 (0) ge 0 119860 (0) ge 0
119862 (0) ge 0 119881 (0) ge 0 119872 (0) ge 0
(2)
The proportion of failure immunization is shown by 120587120575 represent both the death and birth rate At the 120574
1rate the
exposed individuals become infectious andmove to theAcuteclass 120574
2is the rate at which the individuals move to the
carrier class 1205743is the flow of carrier to vaccinated class 120573
shows the transmission coefficient 120581 represents the carrierinfectiousness to acute infection 119902 is the proportion of acuteindividuals that become carrier 120575
119900represent the loss of
immunity rate and the individual become the susceptibleagain 119901 represents the vaccination of susceptible individuals1205831represents the rate of flow from migrated class to exposed
class and 1205742is the rate of transmission from migrated class
to acute class 120578 is the unimmunized children born to carriermothers 120575(1 minus 120587) measures the successful immunization ofnewborn babies and the term 120575120587(1 minus 120578119862(119905)) shows that thenewborns are unimmunized and become susceptible again
We assume that the total population size is equal to 1and just for simplifications 119878(119905) is the susceptible 119864(119905) theexposed 119860(119905) the acute 119862(119905) the carrier 119881(119905) the immunityand119872(119905) is themigrated class representing the state variablesin our proposed population model The sum of the totalpopulation is 119878(119905) + 119864(119905) + 119860(119905) + 119862(119905) + 119881(119905) + 119872(119905) = 1holds We just add (1) and we can easily get We ignore thefifth equation in system (1) so the new models become
119889119878 (119905)
119889119905= 120575120587 (1 minus 120578119862 (119905)) minus 120575119878 (119905) minus 120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905)
+ 120575119900(1minus(119878(119905)+119864(119905)+119860 (119905)+119862(119905)+119872(119905)))minus119901119878(119905)
119889119864 (119905)
119889119905= 120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905) minus 120575119864 (119905) + 120575120587120578119862 (119905)
minus 1205741119864 (119905) + 120583
1119872(119905)
119889119860 (119905)
119889119905= 1205741119864 (119905) minus 120575119860 (119905) minus 119902120574
2119860 (119905) minus (1 minus 119902) 120574
1119860 (119905)
+ 1205832119872(119905)
119889119862 (119905)
119889119905= 1199021205742119860 (119905) minus 120575119862 (119905) minus 120574
3119862 (119905)
119889119872 (119905)
119889119905= minus (120583
1+ 1205832)119872 (119905) minus 120575119872 (119905)
(3)
Let
Γ = (119878 119864 119860 119862119872) isin R5
+ | 119878 (119905) le
120575120587 + 120575119900
120575 + 120575119900+ 119901
(119878 + 119864 + 119860 + 119862 +119872 le120575120587 + 120575
119900
120575 + 120575119900
)
(4)
Here Γ is a positively invariant set All the solutions lie insideΓ which is our main focus
3 Basic ReproductionNumberThreshold Quantity
The basic reproduction number or the threshold quantityR0for the proposed model gives an average number of
secondary infection when an infection is introduced in apurely susceptible population We use the idea developed by[18] and also for detail see [19] We have
F = [
[
0 120573119878119900120573120581119878119900
0 0 0
0 0 0
]
]
119881 = [
[
120575 + 1205741
0 minus120575120587120578
minus1205741
120575 + 1199021205742+ (1 minus 119902) 120574
10
0 minus1199021205742
120575 + 1205743
]
]
119881minus1=
[
(120575+1199021205742+(1minus119902)1205741)(120575+1205743) 1199021205742120575120587120578 120575120587120578(120575+1199021205742+(1minus119902)1205741)
1205741(120575+1205743) (120575+1205741)(120575+1205743) 120575120587120578120574
1
11990212057411205742
1199021205742(120575+1205741) (120575+1205741)(120575+1199021205742+(1minus119902)1205741)
]
(120575 + 1205741) (120575 + 119902120574
2+ (1 minus 119902) 120574
1) (120575 + 120574
3) minus 120575120587120578119902120574
11205742
(5)
where 1198761= (120575 + 119902120574
2+ (1 minus 119902)120574
1) and
119865119881minus1=
[[[[[
[
0120573119878119900120575120587120578119902120574
2
(120575 + 1205741) (120575 + 119902120574
2+ (1 minus 119902) 120574
1) (120575 + 120574
3) minus 120575120587120578119902119903
11199032
120581120573119878119900(120575 + 120574
1(1 minus 119902) + 119902120574
2)
(120575 + 1205741) (120575 + 119902120574
2+ (1 minus 119902) 120574
1) (120575 + 120574
3) minus 120575120587120578119902119903
11199032
0 0 0
0 0 0
]]]]]
]
(6)
4 BioMed Research International
So the reproduction number given by 120588(119865119881minus1) is
R119900=120581120573119878119900(120575 + 120574
1(1 minus 119902) + 119902120574
2) + 120575120587120578119902119903
11199032
(120575 + 1205741) (120575 + 119902120574
2+ (1 minus 119902) 120574
1) (120575 + 120574
3) (7)
Here 119878119900 shows the disease-free equilibrium (DFE) and119863119900=
(119878119900 0 0 0 0) giving 119878119900 = (120575120587+120575
119900)(120575+120575
119900+119901) The endemic
equilibrium point 119879lowast = (119878lowast 119864lowast 119860lowast 119862lowast119872lowast) for system
(3) whose endemic equilibrium is given in the followingsubsection
Endemic Equilibria To find the endemic equilibria of thesystem (3) by setting 119878 = 119878
lowast 119864 = 119864lowast 119860 = 119860
lowast 119862 = 119862lowast
and 119872 = 119872lowast equating left side of the system (3) equal to
zero we obtained
119878lowast=1198761(120575 + 120574
1) (120575 + 120574
3) (1 minus 119877
0) + 120581120573119878
1199001198761
1205741120573 ((120575 + 120574
3) + 120581119902120574
2)
minus(120575 + 120574
3) [(120575 + 120574
1) + 12058311205741]119872lowast
1205741120573 ((120575 + 120574
3) + 120581119902120574
2) 119860lowast
119864lowast=1198761119860lowastminus 1205832119872lowast
1205741
119862lowast=
1199021205742
(120575 + 1205743)119860lowast
(8)
4 Local Stability Analysis
In this section we find the local stability of disease-free andendemic equilibria First we show the local asymptoticalstability of DFE equilibrium and then we find the localasymptotical stability of endemic equilibrium Now we showthe local stability of DFE about the point 119863
119900= (119878119900 0 0 0 0)
in the following theorem
Theorem 1 For 1198770le 1 the disease-free equilibrium of the
system (3) about an equilibrium point 119863119900= (119878119900 0 0 0 0) is
locally asymptotically stable if ((1198761(120575 + 120575
119900+ 119901)(120575 + 120574
1) gt
1205731205741(120575120587 + 120575
119900)) otherwise the disease-free equilibrium of the
system (3) is unstable for 1198770gt 1
Proof To show the local stability of the system (3) about thepoint119863
119900 we set the left-hand side of the system (3) equating
to zero and we obtain the following Jacobian matrix 119869119900(120577)
119869119900(120577) = (
minus120575 minus 120575119900minus 119901 120575
119900minus120573119878119900minus 120575119900minus120575120587120578 minus 120573120581119878
119900minus 120575119900
0
0 minus (120575 + 1205741) 120573119878
119900120573120581119878119900+ 120575120587120578 120583
1
0 1205741
minus1198761
0 1205832
0 0 1199021205742
minus (120575 + 1205743) 0
0 0 0 0 minus (1205831+ 1205832+ 120575)
) (9)
By the elementary row operation we get the followingmatrix
119869119900(120577) = (
minus(120575 + 120575119900+ 119901) minus120575
119900minus120573119878119900minus 120575119900
minus120575120587120578 minus 120573120581119878119900
0
0 minus (120575 + 1205741) 120573119878
119900120573120581119878119900+ 120575120587120578 120583
1
0 0 minus1198761(120575 + 120574
1) + 12057411205731198781199001205741(120573120581119878119900+ 120575120587120578) (120575 + 120574
1) 1205832+ 12058311205741
0 0 0 1198791
0
0 0 0 0 minus (1205831+ 1205832+ 120575)
) (10)
where 1198761= (120575 + 119902120574
2+ (1 minus 119902)120574
1) and 119879
1= minus(minus119876
1(120575 + 120574
1) +
1205741120573119878119900) minus 119902120574
11205742(120573120581119878119900+ 120575120587120578) The characteristic equation to
the previous Jacobian matrix is given by
(120582+120575 + 120575119900+119901) (120582 + 120583
1+1205832+ 120575) (120582
3+ 11988611205822+1198862120582+1198863) = 0
(11)
The first two eigenvalues minus(120575+120575119900+119901) and minus(120583
1+1205832+120575) have
negative real parts For the rest of the eigenvalues we get
1198861= (120575 + 120575
119900+ 119901) (119876
1(120575 + 120574
3) + 1 + 119876
1) (1 minus 119877
0)
+ 120573 (120575120587 + 120575119900) ((1205741(120575 + 120574
3) + 119902120574
11205742) minus (119876
1120581 + 1205741))
times (120575 + 120575119900+ 119901)minus1
1198862= (1198761(120575 + 120574
1) (120575 + 120575
119900+ 119901) minus 120573120574
1(120575120587 + 120575
119900))
times (120575 + 120575119900+ 119901) [(120575 + 120574
1) + (120575 + 120574
1)2
]
+ (120575 + 1205741) [1198761(120575 + 120574
1) (120575 + 120575
119900+ 119901) minus 120573120574
1(120575120587 + 120575
119900)]
+ 11990212057411205742(120575 + 120574
1) [120573120581 (120575120587 + 120575
119900) + 120575120587120578 (120575 + 120575 + 119901)]
times (120575 + 120575119900+ 119901)minus2
+ 11990212057411205742(120575 + 120574
1) (120573120581 (120575120587 + 120575
119900) + 120575120587120578 (120575 + 120575
119900+ 119901))
times (120575 + 120575119900+ 119901)minus1
BioMed Research International 5
1198863=
(120575 + 1205741)
(120575 + 120575119900+ 119901)
(1198761(120575 + 120575
119900+ 119901) (120575 + 120574
1) minus 120573120574
1(120575120587 + 120575
119900))
times [11990212057411205742(120573120581 (120575120587 + 120575
119900) + 120575120587120578)
+ (1198761(120575 + 120574
1) (120575 + 120575
119900+ 119901) minus 120573120574
1(120575120587 + 120575
119900))]
(12)
By the Routh-Hurwitz criteria 1198861gt 0 119886
3gt 0 and 119886
11198862gt 1198863
Here 1198861gt 0 when 119877
0le 1 and ((119876
1(120575 + 120575
119900+ 119901)(120575 + 120574
1) gt
1205731205741(120575120587 + 120575
119900)) Also 119886
2gt 0 and 119886
3gt 0 and then 119886
11198862gt
1198863 So according to the Routh-Hurwitz criteria the Jacobian
matrix has negative real parts if and only if 1198770le 1 Thus by
Routh-Hurwitz criteria the DFE of the system (3) is locallyasymptotically stable about the point119863
119900= (119878119900 0 0 0 0) The
proof is completed
The stability of the disease-free equilibrium for 1198770le 1
means that the disease dies out from the population Next weshow that the endemic equilibrium of the system (3) is locallyasymptotically stable for 119877
0gt 1 When the disease-free
equilibrium is locally asymptotically stable for 1198770le 1 then
the endemic equilibrium does not exist but we are interestedto know about the properties of an endemic equilibrium for1198770gt 1
41 Stability of Endemic Equilibrium (EE) In this subsectionwe find the local asymptotic stability of EE about 119863lowast =
(119878lowast 119864lowast 119860lowast 119862lowast119872lowast) and we prove the local stability of
endemic equilibrium in the following
Theorem2 For1198770gt 1 the endemic equilibrium119863lowast of system
(3) is locally asymptotical stable if the following conditionshold
12057411198852120575119900119878lowast2
gt 12057411205732120581
1198661gt 1198662
(13)
otherwise the system is unstable
Proof Here we prove the that the system (3) about theequilibrium point119863lowast is locally stable and for this we obtainthe Jacobian matrix 119869
lowast(120577) of the system (3) in the following
119869lowast(120577) = (
minus(120575 + 120575119900+ 119901 + 120573 (119860
lowast+ 120581119862lowast)) 120575
119900minus120573119878lowastminus 120575119900minus120575120587120578 minus 120573120581119878
lowastminus 120575119900
minus120575119900
120573 (119860lowast+ 120581119862lowast) minus (120575 + 120574
1) 120573119878
lowast120573120581119878lowast+ 120575120587120578 120583
1
0 1205741
minus1198761
0 1205832
0 0 1199021205742
minus (120575 + 1205743) 0
0 0 0 0 minus (120575 + 1205831+ 1205832)
) (14)
By elementary row operation and after simplification we getthe following Jacobian matrix
119869lowast(120577) = (
minus1198851
minus120575119900
minus120573119878lowastminus 120575119900
1198853
minus120575119900
0 minus1198851(120575 + 120574
1) minus 11988521205751199001198851120573119878lowastminus 1198852(120573119878lowast+ 120575119900) 119885
11198854+ 11988521198853
11988511205831minus 1198852120575119900
0 0 1198855
1205741(11988511198854+ 11988521198853) 119885
6
0 0 0 1198857
0
0 0 0 0 minus (1205831+ 1205832+ 120575)
) (15)
where
1198851= 120575 + 120575
119900+ 119901 + 120573 (119860
lowast+ 120581119862lowast)
1198852= 120573 (119860
lowast+ 120581119862lowast)
1198853= minus (120575120587120578 + 120573120581119878
lowast+ 120575119900)
1198854= (120573120581119878
lowast+ 120575120587120578)
1198855= minus119876
1(minus1198851(120575 + 120574
1) minus 1198852120575119900)
+ 1205741(1198854120573119878lowastminus 1198852(120573119878lowast+ 120575119900))
1198856= minus1205832(minus1198851(120575 + 120574
1) minus 1198852120575119900) + 1205741(11988511205831minus 1198852120575119900)
1198857= minus1198855(120575 + 120574
3) minus 11988551205741(11988511198854+ 11988521198853)
(16)
The eigenvalue 1205821= minus(120583
1+ 1205832+ 120575) lt 0 120582
2= minus1198851 and using
the value of 1198851and further 119862lowast we get 120582
2= minus(120575 + 120575
119900+ 119901 +
120573((120575 + 1205743) + 120581119902120574
2)119860lowast) lt 0 120582
3= minus(119885
1(120575 + 120574
1) + 1198852120575119900) lt 0
as 1198851gt 0 and 119885
2gt 0 120582
4= 1198855 1205824lt 0 if and only if 119885
5lt 0
After the simplifications we get
120573lowast119860lowast(12057411198852120575119900minus 12057411205732120581119878lowast2
)
+ 1205741120573120575120587120578120573
lowastlowastlowast119872lowast
minus [(1205741120573120575120587120578120573
lowastlowast+ 120573lowast1198761) 1198851(120575 + 120574
1)
+11987611198852120575119900120573lowast] gt 0
(17)
where 120573lowast = 1205741120573((120575 + 120574
3) + 120581119902120574
2) 120573lowastlowast = 119876
1(120575 + 120574
1)(120575 +
1205743)(1minus119877
0)+120581120573119878
1199001198761 and 120574
1120573120575120587120578120573
lowastlowastlowast119872lowastgt [(1205741120573120575120587120578120573
lowastlowast+
120573lowast1198761)1198851(120575+1205741)+11987611198852120575119900120573lowast] say119866
1= 1205741120573120575120587120578120573
lowastlowastlowast119872lowast and
6 BioMed Research International
1198662= [(1205741120573120575120587120578120573
lowastlowast+120573lowast1198761)1198851(120575+1205741)+11987611198852120575119900120573lowast]120573lowastlowastlowast
= (120575+
1205743)[(120575 + 120574
1) + 12057411205831] So 120582
4has negative real part if (120574
11198852120575119900gt
12057411205732120581119878lowast2) For 120582
5= 1198857 we obtained negative real parts
and by using the 1198855which is positive under the conditions
described in 1205824 and 119885
1gt 0 119885
2gt 0 we get the negative
real partsThus all the eigenvalues have negative real parts soby the Routh-Hurwitz criteria the endemic equilibrium point119863lowast is locally asymptotically stable when 119877
0gt 1 The proof is
completed
5 Global Stability of DFE
In this section we present the global stability of disease-freeequilibrium DFE of the system (3) For different biologicalmodel the Lyapunov function was used by [20 21] for theglobal stability For our model we define and constructLyapunov function in the following for the global stability ofDFE Further the global stability of endemic equilibrium weuse the Lyapunov function and find its global asymptoticalstability
Theorem 3 For 1198770le 1 the disease-free equilibrium of the
system (3) is stable globally asymptotically if 119878 = 119878119900 and
unstable for 1198770gt 1
Proof Here we define the Lyapunov function for the globalstability of disease-free equilibrium given by
119881 (119905) = [1198891(119878 minus 119878
119900) + 1198892119864 + 1198893119860 + 119889
4119862 + 119889
5119872] (18)
Differentiating the previous function with respect to 119905 andusing the system (3)
1198811015840(119905) = 119889
11198781015840+ 11988921198641015840+ 11988931198601015840+ 11988941198621015840+ 11988951198721015840
1198811015840(119905)
= 1198891[120575120587 (1 minus 120578119862 (119905)) minus 120575119878 (119905) minus 120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905)
+ 120575119900(1minus(119878 (119905) + 119864 (119905) +119860 (119905) + 119862 (119905) +119872 (119905)))minus119901119878 (119905)]
+ 1198892[120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905) minus 120575119864 (119905) + 120575120587120578119862 (119905)
minus1205741119864 (119905) + 120583
1119872(119905)]
+ 1198893[1205741119864 (119905) minus 120575119860 (119905) minus 119902120574
2119860 (119905) minus (1 minus 119902) 120574
1119860 (119905)
+1205832119872(119905)]
+ 1198894[1199021205742119860 (119905) minus 120575119862 (119905) minus 120574
3119862 (119905)]
+ 1198895[minus (1205831+ 1205832)119872 (119905) minus 120575119872 (119905)]
(19)
where 119889119894 119894 = 1 2 5 are some positive constants to be
chosen later
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Time (year)
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 2 The plot shows the HBV transmission model of hepatitisB with 120583
1= 090 and 120583
2= 090
Time (year)0 1 2 3 4 5 6 7 8 9 10
0
10
20
30
40
50
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 3 The plot shows the HBV transmission model of hepatitisB with 120583
1= 080 and 120583
2= 090
After the arrangement we obtain
1198811015840(119905) = [119889
2minus 1198891] (120573 (119860 + 120581119862)) 119878 + [119889
2minus 1198891] 120575120587120578119862
+ [11988931205741minus 1198892(120575 + 120574
1)] 119864 + [119889
21205831minus 1198895120575]119872
+ [11988941199021205742minus 11988931198761] 119860 + 119889
1120575120587 minus 119889
1120575119878 + 119889
1120575119900
minus 1198891120575119900(119878 + 119864 + 119860 + 119862 +119872) minus 119889
1119901119878
minus 1198894(120575 + 120574
3) 119862
(20)
BioMed Research International 7
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Time (year)
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 4 The plot shows the HBV transmission model of hepatitisB with 120583
1= 070 and 120583
2= 080
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Time (year)
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 5 The plot shows the HBV transmission model of hepatitisB with 120583
1= 050 and 120583
2= 060
Choosing the constants 1198891= 1198892= 1205741 1198893= (120575 + 120574
1) 1198894=
1198761(120575 + 120574
1)1199021205742 and 119889
5= 12057411205831120575
After the simplification we get
1198811015840(119905) = minus (119878 minus 119878
119900) minus 1205741120575119900119864 minus 1205741120575119900119860
minus (1205741120575119900+(120575 + 120574
3) (120575 + 120574
1) 1198761
1199021205742
)119862
minus (1205741120575119900+ (120575119900+ 1205741) 1205832)119872
(21)
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Time (year)
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 6 The plot shows the HBV transmission model of hepatitisB with 120583
1= 020 and 120583
2= 030
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Time (year)
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 7 The plot shows the HBV transmission model of hepatitisB with 120583
1= 020 and 120583
2= 040
where 119878119900 = (120575120587 + 120575119900)(120575 + 120575
119900+ 119901) 1198811015840(119905) = 0 if and only if
119878 = 119878119900 and 119864 = 119860 = 119862 = 119872 = 0 Also 1198811015840(119905) is negative for
(119878 gt 119878119900) So by [22] the DFE is globally asymptotically stable
in Γ The proof is completed
51 Global Stability of Endemic Equilibrium In this subsec-tion we show the global asymptotical stability of the system(3) To do this we state and prove the following theorem forthe global stability of endemic equilibrium
8 BioMed Research International
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Time (year)
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 8 The plot shows the HBV transmission model of hepatitisB with 120583
1= 010 and 120583
2= 020
Theorem 4 For 1198770gt 1 system (3) is globally asymptotically
stable if 119878 = 119878lowast and 120575119900gt 1205831 and unstable for 119877
0le 1
Proof To prove that system (3) is globally asymptoticallystable we define the Lyapunov in the following
119871 (119905) =(1205831+ 1205832+ 1205833) 1205741
1205741+ 120575
(119878 minus 119878lowast) +
(1205831+ 1205832+ 1205833) 1205741
1205741+ 120575
119864
+ (1205831+ 1205832+ 1205833) 119860 +
(1205831+ 1205832+ 1205833) 1198761
1199021205742
119862 + 1205832119872
(22)
Taking the derivative with respect to time 119905 using the system(3)
1198711015840(119905) =
(1205831+ 1205832+ 1205833) 1205741
1205741+ 120575
times [120575120587 (1 minus 120578119862 (119905)) minus 120575119878 (119905)minus120573 (119860 (119905)+120581119862 (119905)) 119878 (119905)
+ 120575119900(1 minus (119878 (119905) + 119864 (119905) + 119860 (119905) + 119862 (119905) + 119872 (119905)))
minus119901119878 (119905)]
+(1205831+ 1205832+ 1205833) 1205741
1205741+ 120575
[120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905) minus 120575119864 (119905)
+120575120587120578119862 (119905) minus 1205741119864 (119905) + 120583
1119872(119905)]
+ (1205831+ 1205832+ 1205833) [1205741119864 (119905) minus 120575119860 (119905) minus 119902120574
2119860 (119905)
minus (1 minus 119902) 1205741119860 (119905) + 120583
2119872(119905)]
+(1205831+ 1205832+ 1205833) 1198761
1199021205742
[1199021205742119860 (119905) minus 120575119862 (119905) minus 120574
3119862 (119905)]
+ 1205832[minus (1205831+ 1205832+ 120575)119872]
(23)Simplifying we obtained
1198711015840(119905) = minus
(1205831+ 1205832+ 1205833) 1205741
(1205741+ 120575) (120575 + 120575
119900+ 119901)
(119878 minus 119878lowast)
minus(1205831+ 1205832+ 1205833) 1205741120575119900
(1205741+ 120575)
[119864 + 119860 + 119862]
minus(1205831+ 1205832+ 1205833) 1205741
(1205741+ 120575)
[120575119900minus 1205831]119872
(24)
The endemic equilibrium of the system (3) is globally asymp-totically stable for 119877
0gt 1 if 119878 = 119878
lowast and 120575119900gt 1205831
So the endemic equilibrium of the system (3) is globallyasymptotically stable The proof is completed
6 Numerical Simulations
In this section we present the numerical simulation of theproposed model (3) by using the Runge-Kutta order fourscheme For different values of the parameters the numericalresults are presented in Figures 2 3 4 5 6 7 and 8 Thevariation of migration parameters 120583
1and 120583
2 with different
values is presented In the numerical solution of the modelthe parameters and their values are presented in Table 1In our simulation the susceptible individuals are shown bydashed line the exposed individuals by dotted dashed theacute individuals by bold line the carrier by dashed lineand the migrated individuals by red bold line Figures 2to 8 represent the compartmental population of hepatitis Bindividuals with migration effect The values presented inTable 1 are fixed except for120583
1and1205832 In Figure 2 by the values
for 1205831= 09 and 120583
2= 09 we see that the population of
exposed and acute individuals is increasing In Figure 3 weset 1205831= 08 and 120583
2= 09 and the population of exposed
and acute individuals decreased Decreasing the values of 1205831
and 1205832 we obtain different results see Figures 2 to 8 When
we decrease the population of immigrants who have the HBVvirus we see the decrease in the population of exposed andacute individuals The parameters presented in Table 1 wereused by different authors for example the natural deathrate equally birth rate by [4] the rate at which the latentindividuals become infectious by [3] the rate at which theindividuals move to the carrier class by [3] the rate at whichthe individuals move to the carrier class [3] the transmissioncoefficient 120573 by [24] 120575
119900the loss of immunity rate by [13] the
value of 120578 by [13] the value of 120581 by [10] and120587 and 119902 from [23]We assume the values for the parameters 120574
3 119901 1205831 and 120583
2in
our simulation
7 Conclusion
A compartmental model of HBV transmission virus hasbeen presented A mathematical model has been obtained
BioMed Research International 9
Table 1 Parameter values used in numerical simulations
Notation Parameter description Range Source120575 Natural death rate equally birth rate 00143 [4]120587 The failure immunization 0-1 [23]1205741
The rate at which the latent individuals become infectious 6 [3]1205742
The rate at which the individuals move to the carrier class 4 [3]1205743
The rate of flow from carrier to the vaccinated class 034 Assumed120573 The transmission coefficient 08 [24]119902 The proportion of individuals become carrier 0005 [23]120575119900
Represent the loss of immunity 006ndash003 [13]119901 Represent the vaccination of susceptible 03 Assumed1205831
The rate of flow fromMigrated class to exposed class 023 Assumed1205832
The rate of flow fromMigrated class to acute class 056 Assumed120578 Unimmunized children born to carrier mothers 07 [13]120581 The infectiousness of carriers related to acute infection 0-1 [10]
by adding (1) the migrated class (2) HBV transmission ratebetween migrated class and exposed class (3) transmissionrate between migrated class and acute class and (4) deathrate of individuals in the migrated class By adding thesenew features we have obtained a compartmental model ofHBVwithmigration effect First we obtained the basic repro-duction number for the proposed model The disease-freeequilibrium is locally as well as globally asymptotically stablefor 1198770le 1 We obtained the local and global asymptotical
stability for the endemic equilibrium For the reproductionnumber 119877
0gt 1 the endemic equilibrium is locally as
well as globally asymptotically stable Furthermore we havesolved the compartment model numerically and the resultsare presented in Figures 2 to 8 By changing the values of 120583
1
and 1205832 different results have been obtained It is concluded
that when the value of 1205831and 120583
2decreases the population
of (exposed acute and carrier) individuals also decreaseThe proportion of infected individuals decreases when theproportion ofmigrated individuals (who have theHBVvirus)decreases So the number of infected individuals is directlyproportional to the number of migrated individuals
Last yet not the least the authors of this work have agreedto devise in a course of time a more advanced model onrestraining HBV transmission through migration
Acknowledgment
Theauthors would like to thank the Kanury V S Rao for theircareful reading of the original manuscripts and their manyvaluable comments and suggestions that greatly improve thepresentation of this work
References
[1] F Ahmed and G R Foster ldquoGlobal hepatitis migration and itsimpact onWestern healthcarerdquoGut vol 59 no 8 pp 1009ndash10112010
[2] ldquoChinese center for disease control and prevention (CCDC)rdquo 2010httpwwwchinacdccnn272442n272530n3479265n347930337095html
[3] 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
[4] 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
[5] R M Anderson and R M May Infectious Disease of HumansDynamics and Control Oxford University Press Oxford UK1991
[6] S Zhao Z Xu and Y Lu ldquoA mathematical model of hepatitis Bvirus transmission and its application for vaccination strategyin Chinardquo International Journal of Epidemiology vol 29 no 4pp 744ndash752 2000
[7] K Wang W Wang and S Song ldquoDynamics of an HBV modelwith diffusion and delayrdquo Journal ofTheoretical Biology vol 253no 1 pp 36ndash44 2008
[8] R Xu andZMa ldquoAnHBVmodelwith diffusion and time delayrdquoJournal of Theoretical Biology vol 257 no 3 pp 499ndash509 2009
[9] 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
[10] 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
[11] K P Maier Hepatitis-Hepatitisfolgen Georg Thieme VerlagStuttgart Germany 2000
[12] G L Mandell R G Douglas and J E Bennett Principles andPractice of Infectious Diseases A Wiley Medical PublicationJohn Wiley and Sons New York NY USA 1979
[13] CW Shepard E P Simard L Finelli A E Fiore and B P BellldquoHepatitis B virus infection epidemiology and vaccinationrdquoEpidemiologic Reviews vol 28 no 1 pp 112ndash125 2006
[14] X Weng and Y Zhang Infectious Diseases Fudan UniversityPress Shanghai China 2003
[15] J Wu and Y Luo Infectious Diseases Central South UniversityPress Changsha China 2004
[16] J BWyngaarden L H Smith and J C BennettCecil Text Bookof Medicine WB Saunders Philadelphia Pa USA 19th edition1992
10 BioMed Research International
[17] B JMcMahonW LM Alward andD BHall ldquoAcute hepatitisB virus infection relation of age to the clinical expressionof disease and subsequent development of the carrier staterdquoJournal of Infectious Diseases vol 151 no 4 pp 599ndash603 1985
[18] P Van Den Driessche and J Watmough ldquoReproduction num-bers and sub-threshold endemic equilibria for compartmentalmodels of disease transmissionrdquoMathematical Biosciences vol180 pp 29ndash48 2002
[19] O Diekmann and J A P HeesterbeekMathematical Epideniol-ogy of Infectious Diseases 2000
[20] A Mwasa and J M Tchuenche ldquoMathematical analysis of acholera model with public health interventionsrdquo BioSystemsvol 105 no 3 pp 190ndash200 2011
[21] J Li and Y Yang ldquoSIR-SVS epidemic models with continuousand impulsive vaccination strategiesrdquo Journal of TheoreticalBiology vol 280 no 1 pp 108ndash116 2011
[22] J P LaSalle ldquoStability of nonautonomous systemsrdquo NonlinearAnalysis vol 1 no 1 pp 83ndash90 1976
[23] World Health Organization ldquoHepatitis B WHOCDSCSRLYO20022Hepatitis Brdquo httpwwwwhointcsrdiseasehepa-titiswhocdscsrlyo20022en
[24] W J Edmunds G FMedley andD J Nokes ldquoThe transmissiondynamics and control of hepatitis B virus in the GambiardquoStatistics in Medicine vol 15 pp 2215ndash2233 2002
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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BioMed Research International 3
Subject to the initial conditions
119878 (0) ge 0 119864 (0) ge 0 119860 (0) ge 0
119862 (0) ge 0 119881 (0) ge 0 119872 (0) ge 0
(2)
The proportion of failure immunization is shown by 120587120575 represent both the death and birth rate At the 120574
1rate the
exposed individuals become infectious andmove to theAcuteclass 120574
2is the rate at which the individuals move to the
carrier class 1205743is the flow of carrier to vaccinated class 120573
shows the transmission coefficient 120581 represents the carrierinfectiousness to acute infection 119902 is the proportion of acuteindividuals that become carrier 120575
119900represent the loss of
immunity rate and the individual become the susceptibleagain 119901 represents the vaccination of susceptible individuals1205831represents the rate of flow from migrated class to exposed
class and 1205742is the rate of transmission from migrated class
to acute class 120578 is the unimmunized children born to carriermothers 120575(1 minus 120587) measures the successful immunization ofnewborn babies and the term 120575120587(1 minus 120578119862(119905)) shows that thenewborns are unimmunized and become susceptible again
We assume that the total population size is equal to 1and just for simplifications 119878(119905) is the susceptible 119864(119905) theexposed 119860(119905) the acute 119862(119905) the carrier 119881(119905) the immunityand119872(119905) is themigrated class representing the state variablesin our proposed population model The sum of the totalpopulation is 119878(119905) + 119864(119905) + 119860(119905) + 119862(119905) + 119881(119905) + 119872(119905) = 1holds We just add (1) and we can easily get We ignore thefifth equation in system (1) so the new models become
119889119878 (119905)
119889119905= 120575120587 (1 minus 120578119862 (119905)) minus 120575119878 (119905) minus 120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905)
+ 120575119900(1minus(119878(119905)+119864(119905)+119860 (119905)+119862(119905)+119872(119905)))minus119901119878(119905)
119889119864 (119905)
119889119905= 120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905) minus 120575119864 (119905) + 120575120587120578119862 (119905)
minus 1205741119864 (119905) + 120583
1119872(119905)
119889119860 (119905)
119889119905= 1205741119864 (119905) minus 120575119860 (119905) minus 119902120574
2119860 (119905) minus (1 minus 119902) 120574
1119860 (119905)
+ 1205832119872(119905)
119889119862 (119905)
119889119905= 1199021205742119860 (119905) minus 120575119862 (119905) minus 120574
3119862 (119905)
119889119872 (119905)
119889119905= minus (120583
1+ 1205832)119872 (119905) minus 120575119872 (119905)
(3)
Let
Γ = (119878 119864 119860 119862119872) isin R5
+ | 119878 (119905) le
120575120587 + 120575119900
120575 + 120575119900+ 119901
(119878 + 119864 + 119860 + 119862 +119872 le120575120587 + 120575
119900
120575 + 120575119900
)
(4)
Here Γ is a positively invariant set All the solutions lie insideΓ which is our main focus
3 Basic ReproductionNumberThreshold Quantity
The basic reproduction number or the threshold quantityR0for the proposed model gives an average number of
secondary infection when an infection is introduced in apurely susceptible population We use the idea developed by[18] and also for detail see [19] We have
F = [
[
0 120573119878119900120573120581119878119900
0 0 0
0 0 0
]
]
119881 = [
[
120575 + 1205741
0 minus120575120587120578
minus1205741
120575 + 1199021205742+ (1 minus 119902) 120574
10
0 minus1199021205742
120575 + 1205743
]
]
119881minus1=
[
(120575+1199021205742+(1minus119902)1205741)(120575+1205743) 1199021205742120575120587120578 120575120587120578(120575+1199021205742+(1minus119902)1205741)
1205741(120575+1205743) (120575+1205741)(120575+1205743) 120575120587120578120574
1
11990212057411205742
1199021205742(120575+1205741) (120575+1205741)(120575+1199021205742+(1minus119902)1205741)
]
(120575 + 1205741) (120575 + 119902120574
2+ (1 minus 119902) 120574
1) (120575 + 120574
3) minus 120575120587120578119902120574
11205742
(5)
where 1198761= (120575 + 119902120574
2+ (1 minus 119902)120574
1) and
119865119881minus1=
[[[[[
[
0120573119878119900120575120587120578119902120574
2
(120575 + 1205741) (120575 + 119902120574
2+ (1 minus 119902) 120574
1) (120575 + 120574
3) minus 120575120587120578119902119903
11199032
120581120573119878119900(120575 + 120574
1(1 minus 119902) + 119902120574
2)
(120575 + 1205741) (120575 + 119902120574
2+ (1 minus 119902) 120574
1) (120575 + 120574
3) minus 120575120587120578119902119903
11199032
0 0 0
0 0 0
]]]]]
]
(6)
4 BioMed Research International
So the reproduction number given by 120588(119865119881minus1) is
R119900=120581120573119878119900(120575 + 120574
1(1 minus 119902) + 119902120574
2) + 120575120587120578119902119903
11199032
(120575 + 1205741) (120575 + 119902120574
2+ (1 minus 119902) 120574
1) (120575 + 120574
3) (7)
Here 119878119900 shows the disease-free equilibrium (DFE) and119863119900=
(119878119900 0 0 0 0) giving 119878119900 = (120575120587+120575
119900)(120575+120575
119900+119901) The endemic
equilibrium point 119879lowast = (119878lowast 119864lowast 119860lowast 119862lowast119872lowast) for system
(3) whose endemic equilibrium is given in the followingsubsection
Endemic Equilibria To find the endemic equilibria of thesystem (3) by setting 119878 = 119878
lowast 119864 = 119864lowast 119860 = 119860
lowast 119862 = 119862lowast
and 119872 = 119872lowast equating left side of the system (3) equal to
zero we obtained
119878lowast=1198761(120575 + 120574
1) (120575 + 120574
3) (1 minus 119877
0) + 120581120573119878
1199001198761
1205741120573 ((120575 + 120574
3) + 120581119902120574
2)
minus(120575 + 120574
3) [(120575 + 120574
1) + 12058311205741]119872lowast
1205741120573 ((120575 + 120574
3) + 120581119902120574
2) 119860lowast
119864lowast=1198761119860lowastminus 1205832119872lowast
1205741
119862lowast=
1199021205742
(120575 + 1205743)119860lowast
(8)
4 Local Stability Analysis
In this section we find the local stability of disease-free andendemic equilibria First we show the local asymptoticalstability of DFE equilibrium and then we find the localasymptotical stability of endemic equilibrium Now we showthe local stability of DFE about the point 119863
119900= (119878119900 0 0 0 0)
in the following theorem
Theorem 1 For 1198770le 1 the disease-free equilibrium of the
system (3) about an equilibrium point 119863119900= (119878119900 0 0 0 0) is
locally asymptotically stable if ((1198761(120575 + 120575
119900+ 119901)(120575 + 120574
1) gt
1205731205741(120575120587 + 120575
119900)) otherwise the disease-free equilibrium of the
system (3) is unstable for 1198770gt 1
Proof To show the local stability of the system (3) about thepoint119863
119900 we set the left-hand side of the system (3) equating
to zero and we obtain the following Jacobian matrix 119869119900(120577)
119869119900(120577) = (
minus120575 minus 120575119900minus 119901 120575
119900minus120573119878119900minus 120575119900minus120575120587120578 minus 120573120581119878
119900minus 120575119900
0
0 minus (120575 + 1205741) 120573119878
119900120573120581119878119900+ 120575120587120578 120583
1
0 1205741
minus1198761
0 1205832
0 0 1199021205742
minus (120575 + 1205743) 0
0 0 0 0 minus (1205831+ 1205832+ 120575)
) (9)
By the elementary row operation we get the followingmatrix
119869119900(120577) = (
minus(120575 + 120575119900+ 119901) minus120575
119900minus120573119878119900minus 120575119900
minus120575120587120578 minus 120573120581119878119900
0
0 minus (120575 + 1205741) 120573119878
119900120573120581119878119900+ 120575120587120578 120583
1
0 0 minus1198761(120575 + 120574
1) + 12057411205731198781199001205741(120573120581119878119900+ 120575120587120578) (120575 + 120574
1) 1205832+ 12058311205741
0 0 0 1198791
0
0 0 0 0 minus (1205831+ 1205832+ 120575)
) (10)
where 1198761= (120575 + 119902120574
2+ (1 minus 119902)120574
1) and 119879
1= minus(minus119876
1(120575 + 120574
1) +
1205741120573119878119900) minus 119902120574
11205742(120573120581119878119900+ 120575120587120578) The characteristic equation to
the previous Jacobian matrix is given by
(120582+120575 + 120575119900+119901) (120582 + 120583
1+1205832+ 120575) (120582
3+ 11988611205822+1198862120582+1198863) = 0
(11)
The first two eigenvalues minus(120575+120575119900+119901) and minus(120583
1+1205832+120575) have
negative real parts For the rest of the eigenvalues we get
1198861= (120575 + 120575
119900+ 119901) (119876
1(120575 + 120574
3) + 1 + 119876
1) (1 minus 119877
0)
+ 120573 (120575120587 + 120575119900) ((1205741(120575 + 120574
3) + 119902120574
11205742) minus (119876
1120581 + 1205741))
times (120575 + 120575119900+ 119901)minus1
1198862= (1198761(120575 + 120574
1) (120575 + 120575
119900+ 119901) minus 120573120574
1(120575120587 + 120575
119900))
times (120575 + 120575119900+ 119901) [(120575 + 120574
1) + (120575 + 120574
1)2
]
+ (120575 + 1205741) [1198761(120575 + 120574
1) (120575 + 120575
119900+ 119901) minus 120573120574
1(120575120587 + 120575
119900)]
+ 11990212057411205742(120575 + 120574
1) [120573120581 (120575120587 + 120575
119900) + 120575120587120578 (120575 + 120575 + 119901)]
times (120575 + 120575119900+ 119901)minus2
+ 11990212057411205742(120575 + 120574
1) (120573120581 (120575120587 + 120575
119900) + 120575120587120578 (120575 + 120575
119900+ 119901))
times (120575 + 120575119900+ 119901)minus1
BioMed Research International 5
1198863=
(120575 + 1205741)
(120575 + 120575119900+ 119901)
(1198761(120575 + 120575
119900+ 119901) (120575 + 120574
1) minus 120573120574
1(120575120587 + 120575
119900))
times [11990212057411205742(120573120581 (120575120587 + 120575
119900) + 120575120587120578)
+ (1198761(120575 + 120574
1) (120575 + 120575
119900+ 119901) minus 120573120574
1(120575120587 + 120575
119900))]
(12)
By the Routh-Hurwitz criteria 1198861gt 0 119886
3gt 0 and 119886
11198862gt 1198863
Here 1198861gt 0 when 119877
0le 1 and ((119876
1(120575 + 120575
119900+ 119901)(120575 + 120574
1) gt
1205731205741(120575120587 + 120575
119900)) Also 119886
2gt 0 and 119886
3gt 0 and then 119886
11198862gt
1198863 So according to the Routh-Hurwitz criteria the Jacobian
matrix has negative real parts if and only if 1198770le 1 Thus by
Routh-Hurwitz criteria the DFE of the system (3) is locallyasymptotically stable about the point119863
119900= (119878119900 0 0 0 0) The
proof is completed
The stability of the disease-free equilibrium for 1198770le 1
means that the disease dies out from the population Next weshow that the endemic equilibrium of the system (3) is locallyasymptotically stable for 119877
0gt 1 When the disease-free
equilibrium is locally asymptotically stable for 1198770le 1 then
the endemic equilibrium does not exist but we are interestedto know about the properties of an endemic equilibrium for1198770gt 1
41 Stability of Endemic Equilibrium (EE) In this subsectionwe find the local asymptotic stability of EE about 119863lowast =
(119878lowast 119864lowast 119860lowast 119862lowast119872lowast) and we prove the local stability of
endemic equilibrium in the following
Theorem2 For1198770gt 1 the endemic equilibrium119863lowast of system
(3) is locally asymptotical stable if the following conditionshold
12057411198852120575119900119878lowast2
gt 12057411205732120581
1198661gt 1198662
(13)
otherwise the system is unstable
Proof Here we prove the that the system (3) about theequilibrium point119863lowast is locally stable and for this we obtainthe Jacobian matrix 119869
lowast(120577) of the system (3) in the following
119869lowast(120577) = (
minus(120575 + 120575119900+ 119901 + 120573 (119860
lowast+ 120581119862lowast)) 120575
119900minus120573119878lowastminus 120575119900minus120575120587120578 minus 120573120581119878
lowastminus 120575119900
minus120575119900
120573 (119860lowast+ 120581119862lowast) minus (120575 + 120574
1) 120573119878
lowast120573120581119878lowast+ 120575120587120578 120583
1
0 1205741
minus1198761
0 1205832
0 0 1199021205742
minus (120575 + 1205743) 0
0 0 0 0 minus (120575 + 1205831+ 1205832)
) (14)
By elementary row operation and after simplification we getthe following Jacobian matrix
119869lowast(120577) = (
minus1198851
minus120575119900
minus120573119878lowastminus 120575119900
1198853
minus120575119900
0 minus1198851(120575 + 120574
1) minus 11988521205751199001198851120573119878lowastminus 1198852(120573119878lowast+ 120575119900) 119885
11198854+ 11988521198853
11988511205831minus 1198852120575119900
0 0 1198855
1205741(11988511198854+ 11988521198853) 119885
6
0 0 0 1198857
0
0 0 0 0 minus (1205831+ 1205832+ 120575)
) (15)
where
1198851= 120575 + 120575
119900+ 119901 + 120573 (119860
lowast+ 120581119862lowast)
1198852= 120573 (119860
lowast+ 120581119862lowast)
1198853= minus (120575120587120578 + 120573120581119878
lowast+ 120575119900)
1198854= (120573120581119878
lowast+ 120575120587120578)
1198855= minus119876
1(minus1198851(120575 + 120574
1) minus 1198852120575119900)
+ 1205741(1198854120573119878lowastminus 1198852(120573119878lowast+ 120575119900))
1198856= minus1205832(minus1198851(120575 + 120574
1) minus 1198852120575119900) + 1205741(11988511205831minus 1198852120575119900)
1198857= minus1198855(120575 + 120574
3) minus 11988551205741(11988511198854+ 11988521198853)
(16)
The eigenvalue 1205821= minus(120583
1+ 1205832+ 120575) lt 0 120582
2= minus1198851 and using
the value of 1198851and further 119862lowast we get 120582
2= minus(120575 + 120575
119900+ 119901 +
120573((120575 + 1205743) + 120581119902120574
2)119860lowast) lt 0 120582
3= minus(119885
1(120575 + 120574
1) + 1198852120575119900) lt 0
as 1198851gt 0 and 119885
2gt 0 120582
4= 1198855 1205824lt 0 if and only if 119885
5lt 0
After the simplifications we get
120573lowast119860lowast(12057411198852120575119900minus 12057411205732120581119878lowast2
)
+ 1205741120573120575120587120578120573
lowastlowastlowast119872lowast
minus [(1205741120573120575120587120578120573
lowastlowast+ 120573lowast1198761) 1198851(120575 + 120574
1)
+11987611198852120575119900120573lowast] gt 0
(17)
where 120573lowast = 1205741120573((120575 + 120574
3) + 120581119902120574
2) 120573lowastlowast = 119876
1(120575 + 120574
1)(120575 +
1205743)(1minus119877
0)+120581120573119878
1199001198761 and 120574
1120573120575120587120578120573
lowastlowastlowast119872lowastgt [(1205741120573120575120587120578120573
lowastlowast+
120573lowast1198761)1198851(120575+1205741)+11987611198852120575119900120573lowast] say119866
1= 1205741120573120575120587120578120573
lowastlowastlowast119872lowast and
6 BioMed Research International
1198662= [(1205741120573120575120587120578120573
lowastlowast+120573lowast1198761)1198851(120575+1205741)+11987611198852120575119900120573lowast]120573lowastlowastlowast
= (120575+
1205743)[(120575 + 120574
1) + 12057411205831] So 120582
4has negative real part if (120574
11198852120575119900gt
12057411205732120581119878lowast2) For 120582
5= 1198857 we obtained negative real parts
and by using the 1198855which is positive under the conditions
described in 1205824 and 119885
1gt 0 119885
2gt 0 we get the negative
real partsThus all the eigenvalues have negative real parts soby the Routh-Hurwitz criteria the endemic equilibrium point119863lowast is locally asymptotically stable when 119877
0gt 1 The proof is
completed
5 Global Stability of DFE
In this section we present the global stability of disease-freeequilibrium DFE of the system (3) For different biologicalmodel the Lyapunov function was used by [20 21] for theglobal stability For our model we define and constructLyapunov function in the following for the global stability ofDFE Further the global stability of endemic equilibrium weuse the Lyapunov function and find its global asymptoticalstability
Theorem 3 For 1198770le 1 the disease-free equilibrium of the
system (3) is stable globally asymptotically if 119878 = 119878119900 and
unstable for 1198770gt 1
Proof Here we define the Lyapunov function for the globalstability of disease-free equilibrium given by
119881 (119905) = [1198891(119878 minus 119878
119900) + 1198892119864 + 1198893119860 + 119889
4119862 + 119889
5119872] (18)
Differentiating the previous function with respect to 119905 andusing the system (3)
1198811015840(119905) = 119889
11198781015840+ 11988921198641015840+ 11988931198601015840+ 11988941198621015840+ 11988951198721015840
1198811015840(119905)
= 1198891[120575120587 (1 minus 120578119862 (119905)) minus 120575119878 (119905) minus 120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905)
+ 120575119900(1minus(119878 (119905) + 119864 (119905) +119860 (119905) + 119862 (119905) +119872 (119905)))minus119901119878 (119905)]
+ 1198892[120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905) minus 120575119864 (119905) + 120575120587120578119862 (119905)
minus1205741119864 (119905) + 120583
1119872(119905)]
+ 1198893[1205741119864 (119905) minus 120575119860 (119905) minus 119902120574
2119860 (119905) minus (1 minus 119902) 120574
1119860 (119905)
+1205832119872(119905)]
+ 1198894[1199021205742119860 (119905) minus 120575119862 (119905) minus 120574
3119862 (119905)]
+ 1198895[minus (1205831+ 1205832)119872 (119905) minus 120575119872 (119905)]
(19)
where 119889119894 119894 = 1 2 5 are some positive constants to be
chosen later
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Time (year)
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 2 The plot shows the HBV transmission model of hepatitisB with 120583
1= 090 and 120583
2= 090
Time (year)0 1 2 3 4 5 6 7 8 9 10
0
10
20
30
40
50
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 3 The plot shows the HBV transmission model of hepatitisB with 120583
1= 080 and 120583
2= 090
After the arrangement we obtain
1198811015840(119905) = [119889
2minus 1198891] (120573 (119860 + 120581119862)) 119878 + [119889
2minus 1198891] 120575120587120578119862
+ [11988931205741minus 1198892(120575 + 120574
1)] 119864 + [119889
21205831minus 1198895120575]119872
+ [11988941199021205742minus 11988931198761] 119860 + 119889
1120575120587 minus 119889
1120575119878 + 119889
1120575119900
minus 1198891120575119900(119878 + 119864 + 119860 + 119862 +119872) minus 119889
1119901119878
minus 1198894(120575 + 120574
3) 119862
(20)
BioMed Research International 7
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Time (year)
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 4 The plot shows the HBV transmission model of hepatitisB with 120583
1= 070 and 120583
2= 080
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Time (year)
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 5 The plot shows the HBV transmission model of hepatitisB with 120583
1= 050 and 120583
2= 060
Choosing the constants 1198891= 1198892= 1205741 1198893= (120575 + 120574
1) 1198894=
1198761(120575 + 120574
1)1199021205742 and 119889
5= 12057411205831120575
After the simplification we get
1198811015840(119905) = minus (119878 minus 119878
119900) minus 1205741120575119900119864 minus 1205741120575119900119860
minus (1205741120575119900+(120575 + 120574
3) (120575 + 120574
1) 1198761
1199021205742
)119862
minus (1205741120575119900+ (120575119900+ 1205741) 1205832)119872
(21)
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Time (year)
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 6 The plot shows the HBV transmission model of hepatitisB with 120583
1= 020 and 120583
2= 030
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Time (year)
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 7 The plot shows the HBV transmission model of hepatitisB with 120583
1= 020 and 120583
2= 040
where 119878119900 = (120575120587 + 120575119900)(120575 + 120575
119900+ 119901) 1198811015840(119905) = 0 if and only if
119878 = 119878119900 and 119864 = 119860 = 119862 = 119872 = 0 Also 1198811015840(119905) is negative for
(119878 gt 119878119900) So by [22] the DFE is globally asymptotically stable
in Γ The proof is completed
51 Global Stability of Endemic Equilibrium In this subsec-tion we show the global asymptotical stability of the system(3) To do this we state and prove the following theorem forthe global stability of endemic equilibrium
8 BioMed Research International
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Time (year)
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 8 The plot shows the HBV transmission model of hepatitisB with 120583
1= 010 and 120583
2= 020
Theorem 4 For 1198770gt 1 system (3) is globally asymptotically
stable if 119878 = 119878lowast and 120575119900gt 1205831 and unstable for 119877
0le 1
Proof To prove that system (3) is globally asymptoticallystable we define the Lyapunov in the following
119871 (119905) =(1205831+ 1205832+ 1205833) 1205741
1205741+ 120575
(119878 minus 119878lowast) +
(1205831+ 1205832+ 1205833) 1205741
1205741+ 120575
119864
+ (1205831+ 1205832+ 1205833) 119860 +
(1205831+ 1205832+ 1205833) 1198761
1199021205742
119862 + 1205832119872
(22)
Taking the derivative with respect to time 119905 using the system(3)
1198711015840(119905) =
(1205831+ 1205832+ 1205833) 1205741
1205741+ 120575
times [120575120587 (1 minus 120578119862 (119905)) minus 120575119878 (119905)minus120573 (119860 (119905)+120581119862 (119905)) 119878 (119905)
+ 120575119900(1 minus (119878 (119905) + 119864 (119905) + 119860 (119905) + 119862 (119905) + 119872 (119905)))
minus119901119878 (119905)]
+(1205831+ 1205832+ 1205833) 1205741
1205741+ 120575
[120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905) minus 120575119864 (119905)
+120575120587120578119862 (119905) minus 1205741119864 (119905) + 120583
1119872(119905)]
+ (1205831+ 1205832+ 1205833) [1205741119864 (119905) minus 120575119860 (119905) minus 119902120574
2119860 (119905)
minus (1 minus 119902) 1205741119860 (119905) + 120583
2119872(119905)]
+(1205831+ 1205832+ 1205833) 1198761
1199021205742
[1199021205742119860 (119905) minus 120575119862 (119905) minus 120574
3119862 (119905)]
+ 1205832[minus (1205831+ 1205832+ 120575)119872]
(23)Simplifying we obtained
1198711015840(119905) = minus
(1205831+ 1205832+ 1205833) 1205741
(1205741+ 120575) (120575 + 120575
119900+ 119901)
(119878 minus 119878lowast)
minus(1205831+ 1205832+ 1205833) 1205741120575119900
(1205741+ 120575)
[119864 + 119860 + 119862]
minus(1205831+ 1205832+ 1205833) 1205741
(1205741+ 120575)
[120575119900minus 1205831]119872
(24)
The endemic equilibrium of the system (3) is globally asymp-totically stable for 119877
0gt 1 if 119878 = 119878
lowast and 120575119900gt 1205831
So the endemic equilibrium of the system (3) is globallyasymptotically stable The proof is completed
6 Numerical Simulations
In this section we present the numerical simulation of theproposed model (3) by using the Runge-Kutta order fourscheme For different values of the parameters the numericalresults are presented in Figures 2 3 4 5 6 7 and 8 Thevariation of migration parameters 120583
1and 120583
2 with different
values is presented In the numerical solution of the modelthe parameters and their values are presented in Table 1In our simulation the susceptible individuals are shown bydashed line the exposed individuals by dotted dashed theacute individuals by bold line the carrier by dashed lineand the migrated individuals by red bold line Figures 2to 8 represent the compartmental population of hepatitis Bindividuals with migration effect The values presented inTable 1 are fixed except for120583
1and1205832 In Figure 2 by the values
for 1205831= 09 and 120583
2= 09 we see that the population of
exposed and acute individuals is increasing In Figure 3 weset 1205831= 08 and 120583
2= 09 and the population of exposed
and acute individuals decreased Decreasing the values of 1205831
and 1205832 we obtain different results see Figures 2 to 8 When
we decrease the population of immigrants who have the HBVvirus we see the decrease in the population of exposed andacute individuals The parameters presented in Table 1 wereused by different authors for example the natural deathrate equally birth rate by [4] the rate at which the latentindividuals become infectious by [3] the rate at which theindividuals move to the carrier class by [3] the rate at whichthe individuals move to the carrier class [3] the transmissioncoefficient 120573 by [24] 120575
119900the loss of immunity rate by [13] the
value of 120578 by [13] the value of 120581 by [10] and120587 and 119902 from [23]We assume the values for the parameters 120574
3 119901 1205831 and 120583
2in
our simulation
7 Conclusion
A compartmental model of HBV transmission virus hasbeen presented A mathematical model has been obtained
BioMed Research International 9
Table 1 Parameter values used in numerical simulations
Notation Parameter description Range Source120575 Natural death rate equally birth rate 00143 [4]120587 The failure immunization 0-1 [23]1205741
The rate at which the latent individuals become infectious 6 [3]1205742
The rate at which the individuals move to the carrier class 4 [3]1205743
The rate of flow from carrier to the vaccinated class 034 Assumed120573 The transmission coefficient 08 [24]119902 The proportion of individuals become carrier 0005 [23]120575119900
Represent the loss of immunity 006ndash003 [13]119901 Represent the vaccination of susceptible 03 Assumed1205831
The rate of flow fromMigrated class to exposed class 023 Assumed1205832
The rate of flow fromMigrated class to acute class 056 Assumed120578 Unimmunized children born to carrier mothers 07 [13]120581 The infectiousness of carriers related to acute infection 0-1 [10]
by adding (1) the migrated class (2) HBV transmission ratebetween migrated class and exposed class (3) transmissionrate between migrated class and acute class and (4) deathrate of individuals in the migrated class By adding thesenew features we have obtained a compartmental model ofHBVwithmigration effect First we obtained the basic repro-duction number for the proposed model The disease-freeequilibrium is locally as well as globally asymptotically stablefor 1198770le 1 We obtained the local and global asymptotical
stability for the endemic equilibrium For the reproductionnumber 119877
0gt 1 the endemic equilibrium is locally as
well as globally asymptotically stable Furthermore we havesolved the compartment model numerically and the resultsare presented in Figures 2 to 8 By changing the values of 120583
1
and 1205832 different results have been obtained It is concluded
that when the value of 1205831and 120583
2decreases the population
of (exposed acute and carrier) individuals also decreaseThe proportion of infected individuals decreases when theproportion ofmigrated individuals (who have theHBVvirus)decreases So the number of infected individuals is directlyproportional to the number of migrated individuals
Last yet not the least the authors of this work have agreedto devise in a course of time a more advanced model onrestraining HBV transmission through migration
Acknowledgment
Theauthors would like to thank the Kanury V S Rao for theircareful reading of the original manuscripts and their manyvaluable comments and suggestions that greatly improve thepresentation of this work
References
[1] F Ahmed and G R Foster ldquoGlobal hepatitis migration and itsimpact onWestern healthcarerdquoGut vol 59 no 8 pp 1009ndash10112010
[2] ldquoChinese center for disease control and prevention (CCDC)rdquo 2010httpwwwchinacdccnn272442n272530n3479265n347930337095html
[3] 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
[4] 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
[5] R M Anderson and R M May Infectious Disease of HumansDynamics and Control Oxford University Press Oxford UK1991
[6] S Zhao Z Xu and Y Lu ldquoA mathematical model of hepatitis Bvirus transmission and its application for vaccination strategyin Chinardquo International Journal of Epidemiology vol 29 no 4pp 744ndash752 2000
[7] K Wang W Wang and S Song ldquoDynamics of an HBV modelwith diffusion and delayrdquo Journal ofTheoretical Biology vol 253no 1 pp 36ndash44 2008
[8] R Xu andZMa ldquoAnHBVmodelwith diffusion and time delayrdquoJournal of Theoretical Biology vol 257 no 3 pp 499ndash509 2009
[9] 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
[10] 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
[11] K P Maier Hepatitis-Hepatitisfolgen Georg Thieme VerlagStuttgart Germany 2000
[12] G L Mandell R G Douglas and J E Bennett Principles andPractice of Infectious Diseases A Wiley Medical PublicationJohn Wiley and Sons New York NY USA 1979
[13] CW Shepard E P Simard L Finelli A E Fiore and B P BellldquoHepatitis B virus infection epidemiology and vaccinationrdquoEpidemiologic Reviews vol 28 no 1 pp 112ndash125 2006
[14] X Weng and Y Zhang Infectious Diseases Fudan UniversityPress Shanghai China 2003
[15] J Wu and Y Luo Infectious Diseases Central South UniversityPress Changsha China 2004
[16] J BWyngaarden L H Smith and J C BennettCecil Text Bookof Medicine WB Saunders Philadelphia Pa USA 19th edition1992
10 BioMed Research International
[17] B JMcMahonW LM Alward andD BHall ldquoAcute hepatitisB virus infection relation of age to the clinical expressionof disease and subsequent development of the carrier staterdquoJournal of Infectious Diseases vol 151 no 4 pp 599ndash603 1985
[18] P Van Den Driessche and J Watmough ldquoReproduction num-bers and sub-threshold endemic equilibria for compartmentalmodels of disease transmissionrdquoMathematical Biosciences vol180 pp 29ndash48 2002
[19] O Diekmann and J A P HeesterbeekMathematical Epideniol-ogy of Infectious Diseases 2000
[20] A Mwasa and J M Tchuenche ldquoMathematical analysis of acholera model with public health interventionsrdquo BioSystemsvol 105 no 3 pp 190ndash200 2011
[21] J Li and Y Yang ldquoSIR-SVS epidemic models with continuousand impulsive vaccination strategiesrdquo Journal of TheoreticalBiology vol 280 no 1 pp 108ndash116 2011
[22] J P LaSalle ldquoStability of nonautonomous systemsrdquo NonlinearAnalysis vol 1 no 1 pp 83ndash90 1976
[23] World Health Organization ldquoHepatitis B WHOCDSCSRLYO20022Hepatitis Brdquo httpwwwwhointcsrdiseasehepa-titiswhocdscsrlyo20022en
[24] W J Edmunds G FMedley andD J Nokes ldquoThe transmissiondynamics and control of hepatitis B virus in the GambiardquoStatistics in Medicine vol 15 pp 2215ndash2233 2002
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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International Journal of
Microbiology
4 BioMed Research International
So the reproduction number given by 120588(119865119881minus1) is
R119900=120581120573119878119900(120575 + 120574
1(1 minus 119902) + 119902120574
2) + 120575120587120578119902119903
11199032
(120575 + 1205741) (120575 + 119902120574
2+ (1 minus 119902) 120574
1) (120575 + 120574
3) (7)
Here 119878119900 shows the disease-free equilibrium (DFE) and119863119900=
(119878119900 0 0 0 0) giving 119878119900 = (120575120587+120575
119900)(120575+120575
119900+119901) The endemic
equilibrium point 119879lowast = (119878lowast 119864lowast 119860lowast 119862lowast119872lowast) for system
(3) whose endemic equilibrium is given in the followingsubsection
Endemic Equilibria To find the endemic equilibria of thesystem (3) by setting 119878 = 119878
lowast 119864 = 119864lowast 119860 = 119860
lowast 119862 = 119862lowast
and 119872 = 119872lowast equating left side of the system (3) equal to
zero we obtained
119878lowast=1198761(120575 + 120574
1) (120575 + 120574
3) (1 minus 119877
0) + 120581120573119878
1199001198761
1205741120573 ((120575 + 120574
3) + 120581119902120574
2)
minus(120575 + 120574
3) [(120575 + 120574
1) + 12058311205741]119872lowast
1205741120573 ((120575 + 120574
3) + 120581119902120574
2) 119860lowast
119864lowast=1198761119860lowastminus 1205832119872lowast
1205741
119862lowast=
1199021205742
(120575 + 1205743)119860lowast
(8)
4 Local Stability Analysis
In this section we find the local stability of disease-free andendemic equilibria First we show the local asymptoticalstability of DFE equilibrium and then we find the localasymptotical stability of endemic equilibrium Now we showthe local stability of DFE about the point 119863
119900= (119878119900 0 0 0 0)
in the following theorem
Theorem 1 For 1198770le 1 the disease-free equilibrium of the
system (3) about an equilibrium point 119863119900= (119878119900 0 0 0 0) is
locally asymptotically stable if ((1198761(120575 + 120575
119900+ 119901)(120575 + 120574
1) gt
1205731205741(120575120587 + 120575
119900)) otherwise the disease-free equilibrium of the
system (3) is unstable for 1198770gt 1
Proof To show the local stability of the system (3) about thepoint119863
119900 we set the left-hand side of the system (3) equating
to zero and we obtain the following Jacobian matrix 119869119900(120577)
119869119900(120577) = (
minus120575 minus 120575119900minus 119901 120575
119900minus120573119878119900minus 120575119900minus120575120587120578 minus 120573120581119878
119900minus 120575119900
0
0 minus (120575 + 1205741) 120573119878
119900120573120581119878119900+ 120575120587120578 120583
1
0 1205741
minus1198761
0 1205832
0 0 1199021205742
minus (120575 + 1205743) 0
0 0 0 0 minus (1205831+ 1205832+ 120575)
) (9)
By the elementary row operation we get the followingmatrix
119869119900(120577) = (
minus(120575 + 120575119900+ 119901) minus120575
119900minus120573119878119900minus 120575119900
minus120575120587120578 minus 120573120581119878119900
0
0 minus (120575 + 1205741) 120573119878
119900120573120581119878119900+ 120575120587120578 120583
1
0 0 minus1198761(120575 + 120574
1) + 12057411205731198781199001205741(120573120581119878119900+ 120575120587120578) (120575 + 120574
1) 1205832+ 12058311205741
0 0 0 1198791
0
0 0 0 0 minus (1205831+ 1205832+ 120575)
) (10)
where 1198761= (120575 + 119902120574
2+ (1 minus 119902)120574
1) and 119879
1= minus(minus119876
1(120575 + 120574
1) +
1205741120573119878119900) minus 119902120574
11205742(120573120581119878119900+ 120575120587120578) The characteristic equation to
the previous Jacobian matrix is given by
(120582+120575 + 120575119900+119901) (120582 + 120583
1+1205832+ 120575) (120582
3+ 11988611205822+1198862120582+1198863) = 0
(11)
The first two eigenvalues minus(120575+120575119900+119901) and minus(120583
1+1205832+120575) have
negative real parts For the rest of the eigenvalues we get
1198861= (120575 + 120575
119900+ 119901) (119876
1(120575 + 120574
3) + 1 + 119876
1) (1 minus 119877
0)
+ 120573 (120575120587 + 120575119900) ((1205741(120575 + 120574
3) + 119902120574
11205742) minus (119876
1120581 + 1205741))
times (120575 + 120575119900+ 119901)minus1
1198862= (1198761(120575 + 120574
1) (120575 + 120575
119900+ 119901) minus 120573120574
1(120575120587 + 120575
119900))
times (120575 + 120575119900+ 119901) [(120575 + 120574
1) + (120575 + 120574
1)2
]
+ (120575 + 1205741) [1198761(120575 + 120574
1) (120575 + 120575
119900+ 119901) minus 120573120574
1(120575120587 + 120575
119900)]
+ 11990212057411205742(120575 + 120574
1) [120573120581 (120575120587 + 120575
119900) + 120575120587120578 (120575 + 120575 + 119901)]
times (120575 + 120575119900+ 119901)minus2
+ 11990212057411205742(120575 + 120574
1) (120573120581 (120575120587 + 120575
119900) + 120575120587120578 (120575 + 120575
119900+ 119901))
times (120575 + 120575119900+ 119901)minus1
BioMed Research International 5
1198863=
(120575 + 1205741)
(120575 + 120575119900+ 119901)
(1198761(120575 + 120575
119900+ 119901) (120575 + 120574
1) minus 120573120574
1(120575120587 + 120575
119900))
times [11990212057411205742(120573120581 (120575120587 + 120575
119900) + 120575120587120578)
+ (1198761(120575 + 120574
1) (120575 + 120575
119900+ 119901) minus 120573120574
1(120575120587 + 120575
119900))]
(12)
By the Routh-Hurwitz criteria 1198861gt 0 119886
3gt 0 and 119886
11198862gt 1198863
Here 1198861gt 0 when 119877
0le 1 and ((119876
1(120575 + 120575
119900+ 119901)(120575 + 120574
1) gt
1205731205741(120575120587 + 120575
119900)) Also 119886
2gt 0 and 119886
3gt 0 and then 119886
11198862gt
1198863 So according to the Routh-Hurwitz criteria the Jacobian
matrix has negative real parts if and only if 1198770le 1 Thus by
Routh-Hurwitz criteria the DFE of the system (3) is locallyasymptotically stable about the point119863
119900= (119878119900 0 0 0 0) The
proof is completed
The stability of the disease-free equilibrium for 1198770le 1
means that the disease dies out from the population Next weshow that the endemic equilibrium of the system (3) is locallyasymptotically stable for 119877
0gt 1 When the disease-free
equilibrium is locally asymptotically stable for 1198770le 1 then
the endemic equilibrium does not exist but we are interestedto know about the properties of an endemic equilibrium for1198770gt 1
41 Stability of Endemic Equilibrium (EE) In this subsectionwe find the local asymptotic stability of EE about 119863lowast =
(119878lowast 119864lowast 119860lowast 119862lowast119872lowast) and we prove the local stability of
endemic equilibrium in the following
Theorem2 For1198770gt 1 the endemic equilibrium119863lowast of system
(3) is locally asymptotical stable if the following conditionshold
12057411198852120575119900119878lowast2
gt 12057411205732120581
1198661gt 1198662
(13)
otherwise the system is unstable
Proof Here we prove the that the system (3) about theequilibrium point119863lowast is locally stable and for this we obtainthe Jacobian matrix 119869
lowast(120577) of the system (3) in the following
119869lowast(120577) = (
minus(120575 + 120575119900+ 119901 + 120573 (119860
lowast+ 120581119862lowast)) 120575
119900minus120573119878lowastminus 120575119900minus120575120587120578 minus 120573120581119878
lowastminus 120575119900
minus120575119900
120573 (119860lowast+ 120581119862lowast) minus (120575 + 120574
1) 120573119878
lowast120573120581119878lowast+ 120575120587120578 120583
1
0 1205741
minus1198761
0 1205832
0 0 1199021205742
minus (120575 + 1205743) 0
0 0 0 0 minus (120575 + 1205831+ 1205832)
) (14)
By elementary row operation and after simplification we getthe following Jacobian matrix
119869lowast(120577) = (
minus1198851
minus120575119900
minus120573119878lowastminus 120575119900
1198853
minus120575119900
0 minus1198851(120575 + 120574
1) minus 11988521205751199001198851120573119878lowastminus 1198852(120573119878lowast+ 120575119900) 119885
11198854+ 11988521198853
11988511205831minus 1198852120575119900
0 0 1198855
1205741(11988511198854+ 11988521198853) 119885
6
0 0 0 1198857
0
0 0 0 0 minus (1205831+ 1205832+ 120575)
) (15)
where
1198851= 120575 + 120575
119900+ 119901 + 120573 (119860
lowast+ 120581119862lowast)
1198852= 120573 (119860
lowast+ 120581119862lowast)
1198853= minus (120575120587120578 + 120573120581119878
lowast+ 120575119900)
1198854= (120573120581119878
lowast+ 120575120587120578)
1198855= minus119876
1(minus1198851(120575 + 120574
1) minus 1198852120575119900)
+ 1205741(1198854120573119878lowastminus 1198852(120573119878lowast+ 120575119900))
1198856= minus1205832(minus1198851(120575 + 120574
1) minus 1198852120575119900) + 1205741(11988511205831minus 1198852120575119900)
1198857= minus1198855(120575 + 120574
3) minus 11988551205741(11988511198854+ 11988521198853)
(16)
The eigenvalue 1205821= minus(120583
1+ 1205832+ 120575) lt 0 120582
2= minus1198851 and using
the value of 1198851and further 119862lowast we get 120582
2= minus(120575 + 120575
119900+ 119901 +
120573((120575 + 1205743) + 120581119902120574
2)119860lowast) lt 0 120582
3= minus(119885
1(120575 + 120574
1) + 1198852120575119900) lt 0
as 1198851gt 0 and 119885
2gt 0 120582
4= 1198855 1205824lt 0 if and only if 119885
5lt 0
After the simplifications we get
120573lowast119860lowast(12057411198852120575119900minus 12057411205732120581119878lowast2
)
+ 1205741120573120575120587120578120573
lowastlowastlowast119872lowast
minus [(1205741120573120575120587120578120573
lowastlowast+ 120573lowast1198761) 1198851(120575 + 120574
1)
+11987611198852120575119900120573lowast] gt 0
(17)
where 120573lowast = 1205741120573((120575 + 120574
3) + 120581119902120574
2) 120573lowastlowast = 119876
1(120575 + 120574
1)(120575 +
1205743)(1minus119877
0)+120581120573119878
1199001198761 and 120574
1120573120575120587120578120573
lowastlowastlowast119872lowastgt [(1205741120573120575120587120578120573
lowastlowast+
120573lowast1198761)1198851(120575+1205741)+11987611198852120575119900120573lowast] say119866
1= 1205741120573120575120587120578120573
lowastlowastlowast119872lowast and
6 BioMed Research International
1198662= [(1205741120573120575120587120578120573
lowastlowast+120573lowast1198761)1198851(120575+1205741)+11987611198852120575119900120573lowast]120573lowastlowastlowast
= (120575+
1205743)[(120575 + 120574
1) + 12057411205831] So 120582
4has negative real part if (120574
11198852120575119900gt
12057411205732120581119878lowast2) For 120582
5= 1198857 we obtained negative real parts
and by using the 1198855which is positive under the conditions
described in 1205824 and 119885
1gt 0 119885
2gt 0 we get the negative
real partsThus all the eigenvalues have negative real parts soby the Routh-Hurwitz criteria the endemic equilibrium point119863lowast is locally asymptotically stable when 119877
0gt 1 The proof is
completed
5 Global Stability of DFE
In this section we present the global stability of disease-freeequilibrium DFE of the system (3) For different biologicalmodel the Lyapunov function was used by [20 21] for theglobal stability For our model we define and constructLyapunov function in the following for the global stability ofDFE Further the global stability of endemic equilibrium weuse the Lyapunov function and find its global asymptoticalstability
Theorem 3 For 1198770le 1 the disease-free equilibrium of the
system (3) is stable globally asymptotically if 119878 = 119878119900 and
unstable for 1198770gt 1
Proof Here we define the Lyapunov function for the globalstability of disease-free equilibrium given by
119881 (119905) = [1198891(119878 minus 119878
119900) + 1198892119864 + 1198893119860 + 119889
4119862 + 119889
5119872] (18)
Differentiating the previous function with respect to 119905 andusing the system (3)
1198811015840(119905) = 119889
11198781015840+ 11988921198641015840+ 11988931198601015840+ 11988941198621015840+ 11988951198721015840
1198811015840(119905)
= 1198891[120575120587 (1 minus 120578119862 (119905)) minus 120575119878 (119905) minus 120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905)
+ 120575119900(1minus(119878 (119905) + 119864 (119905) +119860 (119905) + 119862 (119905) +119872 (119905)))minus119901119878 (119905)]
+ 1198892[120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905) minus 120575119864 (119905) + 120575120587120578119862 (119905)
minus1205741119864 (119905) + 120583
1119872(119905)]
+ 1198893[1205741119864 (119905) minus 120575119860 (119905) minus 119902120574
2119860 (119905) minus (1 minus 119902) 120574
1119860 (119905)
+1205832119872(119905)]
+ 1198894[1199021205742119860 (119905) minus 120575119862 (119905) minus 120574
3119862 (119905)]
+ 1198895[minus (1205831+ 1205832)119872 (119905) minus 120575119872 (119905)]
(19)
where 119889119894 119894 = 1 2 5 are some positive constants to be
chosen later
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Time (year)
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 2 The plot shows the HBV transmission model of hepatitisB with 120583
1= 090 and 120583
2= 090
Time (year)0 1 2 3 4 5 6 7 8 9 10
0
10
20
30
40
50
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 3 The plot shows the HBV transmission model of hepatitisB with 120583
1= 080 and 120583
2= 090
After the arrangement we obtain
1198811015840(119905) = [119889
2minus 1198891] (120573 (119860 + 120581119862)) 119878 + [119889
2minus 1198891] 120575120587120578119862
+ [11988931205741minus 1198892(120575 + 120574
1)] 119864 + [119889
21205831minus 1198895120575]119872
+ [11988941199021205742minus 11988931198761] 119860 + 119889
1120575120587 minus 119889
1120575119878 + 119889
1120575119900
minus 1198891120575119900(119878 + 119864 + 119860 + 119862 +119872) minus 119889
1119901119878
minus 1198894(120575 + 120574
3) 119862
(20)
BioMed Research International 7
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Time (year)
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 4 The plot shows the HBV transmission model of hepatitisB with 120583
1= 070 and 120583
2= 080
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Time (year)
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 5 The plot shows the HBV transmission model of hepatitisB with 120583
1= 050 and 120583
2= 060
Choosing the constants 1198891= 1198892= 1205741 1198893= (120575 + 120574
1) 1198894=
1198761(120575 + 120574
1)1199021205742 and 119889
5= 12057411205831120575
After the simplification we get
1198811015840(119905) = minus (119878 minus 119878
119900) minus 1205741120575119900119864 minus 1205741120575119900119860
minus (1205741120575119900+(120575 + 120574
3) (120575 + 120574
1) 1198761
1199021205742
)119862
minus (1205741120575119900+ (120575119900+ 1205741) 1205832)119872
(21)
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Time (year)
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 6 The plot shows the HBV transmission model of hepatitisB with 120583
1= 020 and 120583
2= 030
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Time (year)
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 7 The plot shows the HBV transmission model of hepatitisB with 120583
1= 020 and 120583
2= 040
where 119878119900 = (120575120587 + 120575119900)(120575 + 120575
119900+ 119901) 1198811015840(119905) = 0 if and only if
119878 = 119878119900 and 119864 = 119860 = 119862 = 119872 = 0 Also 1198811015840(119905) is negative for
(119878 gt 119878119900) So by [22] the DFE is globally asymptotically stable
in Γ The proof is completed
51 Global Stability of Endemic Equilibrium In this subsec-tion we show the global asymptotical stability of the system(3) To do this we state and prove the following theorem forthe global stability of endemic equilibrium
8 BioMed Research International
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Time (year)
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 8 The plot shows the HBV transmission model of hepatitisB with 120583
1= 010 and 120583
2= 020
Theorem 4 For 1198770gt 1 system (3) is globally asymptotically
stable if 119878 = 119878lowast and 120575119900gt 1205831 and unstable for 119877
0le 1
Proof To prove that system (3) is globally asymptoticallystable we define the Lyapunov in the following
119871 (119905) =(1205831+ 1205832+ 1205833) 1205741
1205741+ 120575
(119878 minus 119878lowast) +
(1205831+ 1205832+ 1205833) 1205741
1205741+ 120575
119864
+ (1205831+ 1205832+ 1205833) 119860 +
(1205831+ 1205832+ 1205833) 1198761
1199021205742
119862 + 1205832119872
(22)
Taking the derivative with respect to time 119905 using the system(3)
1198711015840(119905) =
(1205831+ 1205832+ 1205833) 1205741
1205741+ 120575
times [120575120587 (1 minus 120578119862 (119905)) minus 120575119878 (119905)minus120573 (119860 (119905)+120581119862 (119905)) 119878 (119905)
+ 120575119900(1 minus (119878 (119905) + 119864 (119905) + 119860 (119905) + 119862 (119905) + 119872 (119905)))
minus119901119878 (119905)]
+(1205831+ 1205832+ 1205833) 1205741
1205741+ 120575
[120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905) minus 120575119864 (119905)
+120575120587120578119862 (119905) minus 1205741119864 (119905) + 120583
1119872(119905)]
+ (1205831+ 1205832+ 1205833) [1205741119864 (119905) minus 120575119860 (119905) minus 119902120574
2119860 (119905)
minus (1 minus 119902) 1205741119860 (119905) + 120583
2119872(119905)]
+(1205831+ 1205832+ 1205833) 1198761
1199021205742
[1199021205742119860 (119905) minus 120575119862 (119905) minus 120574
3119862 (119905)]
+ 1205832[minus (1205831+ 1205832+ 120575)119872]
(23)Simplifying we obtained
1198711015840(119905) = minus
(1205831+ 1205832+ 1205833) 1205741
(1205741+ 120575) (120575 + 120575
119900+ 119901)
(119878 minus 119878lowast)
minus(1205831+ 1205832+ 1205833) 1205741120575119900
(1205741+ 120575)
[119864 + 119860 + 119862]
minus(1205831+ 1205832+ 1205833) 1205741
(1205741+ 120575)
[120575119900minus 1205831]119872
(24)
The endemic equilibrium of the system (3) is globally asymp-totically stable for 119877
0gt 1 if 119878 = 119878
lowast and 120575119900gt 1205831
So the endemic equilibrium of the system (3) is globallyasymptotically stable The proof is completed
6 Numerical Simulations
In this section we present the numerical simulation of theproposed model (3) by using the Runge-Kutta order fourscheme For different values of the parameters the numericalresults are presented in Figures 2 3 4 5 6 7 and 8 Thevariation of migration parameters 120583
1and 120583
2 with different
values is presented In the numerical solution of the modelthe parameters and their values are presented in Table 1In our simulation the susceptible individuals are shown bydashed line the exposed individuals by dotted dashed theacute individuals by bold line the carrier by dashed lineand the migrated individuals by red bold line Figures 2to 8 represent the compartmental population of hepatitis Bindividuals with migration effect The values presented inTable 1 are fixed except for120583
1and1205832 In Figure 2 by the values
for 1205831= 09 and 120583
2= 09 we see that the population of
exposed and acute individuals is increasing In Figure 3 weset 1205831= 08 and 120583
2= 09 and the population of exposed
and acute individuals decreased Decreasing the values of 1205831
and 1205832 we obtain different results see Figures 2 to 8 When
we decrease the population of immigrants who have the HBVvirus we see the decrease in the population of exposed andacute individuals The parameters presented in Table 1 wereused by different authors for example the natural deathrate equally birth rate by [4] the rate at which the latentindividuals become infectious by [3] the rate at which theindividuals move to the carrier class by [3] the rate at whichthe individuals move to the carrier class [3] the transmissioncoefficient 120573 by [24] 120575
119900the loss of immunity rate by [13] the
value of 120578 by [13] the value of 120581 by [10] and120587 and 119902 from [23]We assume the values for the parameters 120574
3 119901 1205831 and 120583
2in
our simulation
7 Conclusion
A compartmental model of HBV transmission virus hasbeen presented A mathematical model has been obtained
BioMed Research International 9
Table 1 Parameter values used in numerical simulations
Notation Parameter description Range Source120575 Natural death rate equally birth rate 00143 [4]120587 The failure immunization 0-1 [23]1205741
The rate at which the latent individuals become infectious 6 [3]1205742
The rate at which the individuals move to the carrier class 4 [3]1205743
The rate of flow from carrier to the vaccinated class 034 Assumed120573 The transmission coefficient 08 [24]119902 The proportion of individuals become carrier 0005 [23]120575119900
Represent the loss of immunity 006ndash003 [13]119901 Represent the vaccination of susceptible 03 Assumed1205831
The rate of flow fromMigrated class to exposed class 023 Assumed1205832
The rate of flow fromMigrated class to acute class 056 Assumed120578 Unimmunized children born to carrier mothers 07 [13]120581 The infectiousness of carriers related to acute infection 0-1 [10]
by adding (1) the migrated class (2) HBV transmission ratebetween migrated class and exposed class (3) transmissionrate between migrated class and acute class and (4) deathrate of individuals in the migrated class By adding thesenew features we have obtained a compartmental model ofHBVwithmigration effect First we obtained the basic repro-duction number for the proposed model The disease-freeequilibrium is locally as well as globally asymptotically stablefor 1198770le 1 We obtained the local and global asymptotical
stability for the endemic equilibrium For the reproductionnumber 119877
0gt 1 the endemic equilibrium is locally as
well as globally asymptotically stable Furthermore we havesolved the compartment model numerically and the resultsare presented in Figures 2 to 8 By changing the values of 120583
1
and 1205832 different results have been obtained It is concluded
that when the value of 1205831and 120583
2decreases the population
of (exposed acute and carrier) individuals also decreaseThe proportion of infected individuals decreases when theproportion ofmigrated individuals (who have theHBVvirus)decreases So the number of infected individuals is directlyproportional to the number of migrated individuals
Last yet not the least the authors of this work have agreedto devise in a course of time a more advanced model onrestraining HBV transmission through migration
Acknowledgment
Theauthors would like to thank the Kanury V S Rao for theircareful reading of the original manuscripts and their manyvaluable comments and suggestions that greatly improve thepresentation of this work
References
[1] F Ahmed and G R Foster ldquoGlobal hepatitis migration and itsimpact onWestern healthcarerdquoGut vol 59 no 8 pp 1009ndash10112010
[2] ldquoChinese center for disease control and prevention (CCDC)rdquo 2010httpwwwchinacdccnn272442n272530n3479265n347930337095html
[3] 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
[4] 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
[5] R M Anderson and R M May Infectious Disease of HumansDynamics and Control Oxford University Press Oxford UK1991
[6] S Zhao Z Xu and Y Lu ldquoA mathematical model of hepatitis Bvirus transmission and its application for vaccination strategyin Chinardquo International Journal of Epidemiology vol 29 no 4pp 744ndash752 2000
[7] K Wang W Wang and S Song ldquoDynamics of an HBV modelwith diffusion and delayrdquo Journal ofTheoretical Biology vol 253no 1 pp 36ndash44 2008
[8] R Xu andZMa ldquoAnHBVmodelwith diffusion and time delayrdquoJournal of Theoretical Biology vol 257 no 3 pp 499ndash509 2009
[9] 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
[10] 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
[11] K P Maier Hepatitis-Hepatitisfolgen Georg Thieme VerlagStuttgart Germany 2000
[12] G L Mandell R G Douglas and J E Bennett Principles andPractice of Infectious Diseases A Wiley Medical PublicationJohn Wiley and Sons New York NY USA 1979
[13] CW Shepard E P Simard L Finelli A E Fiore and B P BellldquoHepatitis B virus infection epidemiology and vaccinationrdquoEpidemiologic Reviews vol 28 no 1 pp 112ndash125 2006
[14] X Weng and Y Zhang Infectious Diseases Fudan UniversityPress Shanghai China 2003
[15] J Wu and Y Luo Infectious Diseases Central South UniversityPress Changsha China 2004
[16] J BWyngaarden L H Smith and J C BennettCecil Text Bookof Medicine WB Saunders Philadelphia Pa USA 19th edition1992
10 BioMed Research International
[17] B JMcMahonW LM Alward andD BHall ldquoAcute hepatitisB virus infection relation of age to the clinical expressionof disease and subsequent development of the carrier staterdquoJournal of Infectious Diseases vol 151 no 4 pp 599ndash603 1985
[18] P Van Den Driessche and J Watmough ldquoReproduction num-bers and sub-threshold endemic equilibria for compartmentalmodels of disease transmissionrdquoMathematical Biosciences vol180 pp 29ndash48 2002
[19] O Diekmann and J A P HeesterbeekMathematical Epideniol-ogy of Infectious Diseases 2000
[20] A Mwasa and J M Tchuenche ldquoMathematical analysis of acholera model with public health interventionsrdquo BioSystemsvol 105 no 3 pp 190ndash200 2011
[21] J Li and Y Yang ldquoSIR-SVS epidemic models with continuousand impulsive vaccination strategiesrdquo Journal of TheoreticalBiology vol 280 no 1 pp 108ndash116 2011
[22] J P LaSalle ldquoStability of nonautonomous systemsrdquo NonlinearAnalysis vol 1 no 1 pp 83ndash90 1976
[23] World Health Organization ldquoHepatitis B WHOCDSCSRLYO20022Hepatitis Brdquo httpwwwwhointcsrdiseasehepa-titiswhocdscsrlyo20022en
[24] W J Edmunds G FMedley andD J Nokes ldquoThe transmissiondynamics and control of hepatitis B virus in the GambiardquoStatistics in Medicine vol 15 pp 2215ndash2233 2002
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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International Journal of
Microbiology
BioMed Research International 5
1198863=
(120575 + 1205741)
(120575 + 120575119900+ 119901)
(1198761(120575 + 120575
119900+ 119901) (120575 + 120574
1) minus 120573120574
1(120575120587 + 120575
119900))
times [11990212057411205742(120573120581 (120575120587 + 120575
119900) + 120575120587120578)
+ (1198761(120575 + 120574
1) (120575 + 120575
119900+ 119901) minus 120573120574
1(120575120587 + 120575
119900))]
(12)
By the Routh-Hurwitz criteria 1198861gt 0 119886
3gt 0 and 119886
11198862gt 1198863
Here 1198861gt 0 when 119877
0le 1 and ((119876
1(120575 + 120575
119900+ 119901)(120575 + 120574
1) gt
1205731205741(120575120587 + 120575
119900)) Also 119886
2gt 0 and 119886
3gt 0 and then 119886
11198862gt
1198863 So according to the Routh-Hurwitz criteria the Jacobian
matrix has negative real parts if and only if 1198770le 1 Thus by
Routh-Hurwitz criteria the DFE of the system (3) is locallyasymptotically stable about the point119863
119900= (119878119900 0 0 0 0) The
proof is completed
The stability of the disease-free equilibrium for 1198770le 1
means that the disease dies out from the population Next weshow that the endemic equilibrium of the system (3) is locallyasymptotically stable for 119877
0gt 1 When the disease-free
equilibrium is locally asymptotically stable for 1198770le 1 then
the endemic equilibrium does not exist but we are interestedto know about the properties of an endemic equilibrium for1198770gt 1
41 Stability of Endemic Equilibrium (EE) In this subsectionwe find the local asymptotic stability of EE about 119863lowast =
(119878lowast 119864lowast 119860lowast 119862lowast119872lowast) and we prove the local stability of
endemic equilibrium in the following
Theorem2 For1198770gt 1 the endemic equilibrium119863lowast of system
(3) is locally asymptotical stable if the following conditionshold
12057411198852120575119900119878lowast2
gt 12057411205732120581
1198661gt 1198662
(13)
otherwise the system is unstable
Proof Here we prove the that the system (3) about theequilibrium point119863lowast is locally stable and for this we obtainthe Jacobian matrix 119869
lowast(120577) of the system (3) in the following
119869lowast(120577) = (
minus(120575 + 120575119900+ 119901 + 120573 (119860
lowast+ 120581119862lowast)) 120575
119900minus120573119878lowastminus 120575119900minus120575120587120578 minus 120573120581119878
lowastminus 120575119900
minus120575119900
120573 (119860lowast+ 120581119862lowast) minus (120575 + 120574
1) 120573119878
lowast120573120581119878lowast+ 120575120587120578 120583
1
0 1205741
minus1198761
0 1205832
0 0 1199021205742
minus (120575 + 1205743) 0
0 0 0 0 minus (120575 + 1205831+ 1205832)
) (14)
By elementary row operation and after simplification we getthe following Jacobian matrix
119869lowast(120577) = (
minus1198851
minus120575119900
minus120573119878lowastminus 120575119900
1198853
minus120575119900
0 minus1198851(120575 + 120574
1) minus 11988521205751199001198851120573119878lowastminus 1198852(120573119878lowast+ 120575119900) 119885
11198854+ 11988521198853
11988511205831minus 1198852120575119900
0 0 1198855
1205741(11988511198854+ 11988521198853) 119885
6
0 0 0 1198857
0
0 0 0 0 minus (1205831+ 1205832+ 120575)
) (15)
where
1198851= 120575 + 120575
119900+ 119901 + 120573 (119860
lowast+ 120581119862lowast)
1198852= 120573 (119860
lowast+ 120581119862lowast)
1198853= minus (120575120587120578 + 120573120581119878
lowast+ 120575119900)
1198854= (120573120581119878
lowast+ 120575120587120578)
1198855= minus119876
1(minus1198851(120575 + 120574
1) minus 1198852120575119900)
+ 1205741(1198854120573119878lowastminus 1198852(120573119878lowast+ 120575119900))
1198856= minus1205832(minus1198851(120575 + 120574
1) minus 1198852120575119900) + 1205741(11988511205831minus 1198852120575119900)
1198857= minus1198855(120575 + 120574
3) minus 11988551205741(11988511198854+ 11988521198853)
(16)
The eigenvalue 1205821= minus(120583
1+ 1205832+ 120575) lt 0 120582
2= minus1198851 and using
the value of 1198851and further 119862lowast we get 120582
2= minus(120575 + 120575
119900+ 119901 +
120573((120575 + 1205743) + 120581119902120574
2)119860lowast) lt 0 120582
3= minus(119885
1(120575 + 120574
1) + 1198852120575119900) lt 0
as 1198851gt 0 and 119885
2gt 0 120582
4= 1198855 1205824lt 0 if and only if 119885
5lt 0
After the simplifications we get
120573lowast119860lowast(12057411198852120575119900minus 12057411205732120581119878lowast2
)
+ 1205741120573120575120587120578120573
lowastlowastlowast119872lowast
minus [(1205741120573120575120587120578120573
lowastlowast+ 120573lowast1198761) 1198851(120575 + 120574
1)
+11987611198852120575119900120573lowast] gt 0
(17)
where 120573lowast = 1205741120573((120575 + 120574
3) + 120581119902120574
2) 120573lowastlowast = 119876
1(120575 + 120574
1)(120575 +
1205743)(1minus119877
0)+120581120573119878
1199001198761 and 120574
1120573120575120587120578120573
lowastlowastlowast119872lowastgt [(1205741120573120575120587120578120573
lowastlowast+
120573lowast1198761)1198851(120575+1205741)+11987611198852120575119900120573lowast] say119866
1= 1205741120573120575120587120578120573
lowastlowastlowast119872lowast and
6 BioMed Research International
1198662= [(1205741120573120575120587120578120573
lowastlowast+120573lowast1198761)1198851(120575+1205741)+11987611198852120575119900120573lowast]120573lowastlowastlowast
= (120575+
1205743)[(120575 + 120574
1) + 12057411205831] So 120582
4has negative real part if (120574
11198852120575119900gt
12057411205732120581119878lowast2) For 120582
5= 1198857 we obtained negative real parts
and by using the 1198855which is positive under the conditions
described in 1205824 and 119885
1gt 0 119885
2gt 0 we get the negative
real partsThus all the eigenvalues have negative real parts soby the Routh-Hurwitz criteria the endemic equilibrium point119863lowast is locally asymptotically stable when 119877
0gt 1 The proof is
completed
5 Global Stability of DFE
In this section we present the global stability of disease-freeequilibrium DFE of the system (3) For different biologicalmodel the Lyapunov function was used by [20 21] for theglobal stability For our model we define and constructLyapunov function in the following for the global stability ofDFE Further the global stability of endemic equilibrium weuse the Lyapunov function and find its global asymptoticalstability
Theorem 3 For 1198770le 1 the disease-free equilibrium of the
system (3) is stable globally asymptotically if 119878 = 119878119900 and
unstable for 1198770gt 1
Proof Here we define the Lyapunov function for the globalstability of disease-free equilibrium given by
119881 (119905) = [1198891(119878 minus 119878
119900) + 1198892119864 + 1198893119860 + 119889
4119862 + 119889
5119872] (18)
Differentiating the previous function with respect to 119905 andusing the system (3)
1198811015840(119905) = 119889
11198781015840+ 11988921198641015840+ 11988931198601015840+ 11988941198621015840+ 11988951198721015840
1198811015840(119905)
= 1198891[120575120587 (1 minus 120578119862 (119905)) minus 120575119878 (119905) minus 120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905)
+ 120575119900(1minus(119878 (119905) + 119864 (119905) +119860 (119905) + 119862 (119905) +119872 (119905)))minus119901119878 (119905)]
+ 1198892[120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905) minus 120575119864 (119905) + 120575120587120578119862 (119905)
minus1205741119864 (119905) + 120583
1119872(119905)]
+ 1198893[1205741119864 (119905) minus 120575119860 (119905) minus 119902120574
2119860 (119905) minus (1 minus 119902) 120574
1119860 (119905)
+1205832119872(119905)]
+ 1198894[1199021205742119860 (119905) minus 120575119862 (119905) minus 120574
3119862 (119905)]
+ 1198895[minus (1205831+ 1205832)119872 (119905) minus 120575119872 (119905)]
(19)
where 119889119894 119894 = 1 2 5 are some positive constants to be
chosen later
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Time (year)
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 2 The plot shows the HBV transmission model of hepatitisB with 120583
1= 090 and 120583
2= 090
Time (year)0 1 2 3 4 5 6 7 8 9 10
0
10
20
30
40
50
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 3 The plot shows the HBV transmission model of hepatitisB with 120583
1= 080 and 120583
2= 090
After the arrangement we obtain
1198811015840(119905) = [119889
2minus 1198891] (120573 (119860 + 120581119862)) 119878 + [119889
2minus 1198891] 120575120587120578119862
+ [11988931205741minus 1198892(120575 + 120574
1)] 119864 + [119889
21205831minus 1198895120575]119872
+ [11988941199021205742minus 11988931198761] 119860 + 119889
1120575120587 minus 119889
1120575119878 + 119889
1120575119900
minus 1198891120575119900(119878 + 119864 + 119860 + 119862 +119872) minus 119889
1119901119878
minus 1198894(120575 + 120574
3) 119862
(20)
BioMed Research International 7
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Time (year)
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 4 The plot shows the HBV transmission model of hepatitisB with 120583
1= 070 and 120583
2= 080
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Time (year)
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 5 The plot shows the HBV transmission model of hepatitisB with 120583
1= 050 and 120583
2= 060
Choosing the constants 1198891= 1198892= 1205741 1198893= (120575 + 120574
1) 1198894=
1198761(120575 + 120574
1)1199021205742 and 119889
5= 12057411205831120575
After the simplification we get
1198811015840(119905) = minus (119878 minus 119878
119900) minus 1205741120575119900119864 minus 1205741120575119900119860
minus (1205741120575119900+(120575 + 120574
3) (120575 + 120574
1) 1198761
1199021205742
)119862
minus (1205741120575119900+ (120575119900+ 1205741) 1205832)119872
(21)
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Time (year)
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 6 The plot shows the HBV transmission model of hepatitisB with 120583
1= 020 and 120583
2= 030
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Time (year)
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 7 The plot shows the HBV transmission model of hepatitisB with 120583
1= 020 and 120583
2= 040
where 119878119900 = (120575120587 + 120575119900)(120575 + 120575
119900+ 119901) 1198811015840(119905) = 0 if and only if
119878 = 119878119900 and 119864 = 119860 = 119862 = 119872 = 0 Also 1198811015840(119905) is negative for
(119878 gt 119878119900) So by [22] the DFE is globally asymptotically stable
in Γ The proof is completed
51 Global Stability of Endemic Equilibrium In this subsec-tion we show the global asymptotical stability of the system(3) To do this we state and prove the following theorem forthe global stability of endemic equilibrium
8 BioMed Research International
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Time (year)
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 8 The plot shows the HBV transmission model of hepatitisB with 120583
1= 010 and 120583
2= 020
Theorem 4 For 1198770gt 1 system (3) is globally asymptotically
stable if 119878 = 119878lowast and 120575119900gt 1205831 and unstable for 119877
0le 1
Proof To prove that system (3) is globally asymptoticallystable we define the Lyapunov in the following
119871 (119905) =(1205831+ 1205832+ 1205833) 1205741
1205741+ 120575
(119878 minus 119878lowast) +
(1205831+ 1205832+ 1205833) 1205741
1205741+ 120575
119864
+ (1205831+ 1205832+ 1205833) 119860 +
(1205831+ 1205832+ 1205833) 1198761
1199021205742
119862 + 1205832119872
(22)
Taking the derivative with respect to time 119905 using the system(3)
1198711015840(119905) =
(1205831+ 1205832+ 1205833) 1205741
1205741+ 120575
times [120575120587 (1 minus 120578119862 (119905)) minus 120575119878 (119905)minus120573 (119860 (119905)+120581119862 (119905)) 119878 (119905)
+ 120575119900(1 minus (119878 (119905) + 119864 (119905) + 119860 (119905) + 119862 (119905) + 119872 (119905)))
minus119901119878 (119905)]
+(1205831+ 1205832+ 1205833) 1205741
1205741+ 120575
[120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905) minus 120575119864 (119905)
+120575120587120578119862 (119905) minus 1205741119864 (119905) + 120583
1119872(119905)]
+ (1205831+ 1205832+ 1205833) [1205741119864 (119905) minus 120575119860 (119905) minus 119902120574
2119860 (119905)
minus (1 minus 119902) 1205741119860 (119905) + 120583
2119872(119905)]
+(1205831+ 1205832+ 1205833) 1198761
1199021205742
[1199021205742119860 (119905) minus 120575119862 (119905) minus 120574
3119862 (119905)]
+ 1205832[minus (1205831+ 1205832+ 120575)119872]
(23)Simplifying we obtained
1198711015840(119905) = minus
(1205831+ 1205832+ 1205833) 1205741
(1205741+ 120575) (120575 + 120575
119900+ 119901)
(119878 minus 119878lowast)
minus(1205831+ 1205832+ 1205833) 1205741120575119900
(1205741+ 120575)
[119864 + 119860 + 119862]
minus(1205831+ 1205832+ 1205833) 1205741
(1205741+ 120575)
[120575119900minus 1205831]119872
(24)
The endemic equilibrium of the system (3) is globally asymp-totically stable for 119877
0gt 1 if 119878 = 119878
lowast and 120575119900gt 1205831
So the endemic equilibrium of the system (3) is globallyasymptotically stable The proof is completed
6 Numerical Simulations
In this section we present the numerical simulation of theproposed model (3) by using the Runge-Kutta order fourscheme For different values of the parameters the numericalresults are presented in Figures 2 3 4 5 6 7 and 8 Thevariation of migration parameters 120583
1and 120583
2 with different
values is presented In the numerical solution of the modelthe parameters and their values are presented in Table 1In our simulation the susceptible individuals are shown bydashed line the exposed individuals by dotted dashed theacute individuals by bold line the carrier by dashed lineand the migrated individuals by red bold line Figures 2to 8 represent the compartmental population of hepatitis Bindividuals with migration effect The values presented inTable 1 are fixed except for120583
1and1205832 In Figure 2 by the values
for 1205831= 09 and 120583
2= 09 we see that the population of
exposed and acute individuals is increasing In Figure 3 weset 1205831= 08 and 120583
2= 09 and the population of exposed
and acute individuals decreased Decreasing the values of 1205831
and 1205832 we obtain different results see Figures 2 to 8 When
we decrease the population of immigrants who have the HBVvirus we see the decrease in the population of exposed andacute individuals The parameters presented in Table 1 wereused by different authors for example the natural deathrate equally birth rate by [4] the rate at which the latentindividuals become infectious by [3] the rate at which theindividuals move to the carrier class by [3] the rate at whichthe individuals move to the carrier class [3] the transmissioncoefficient 120573 by [24] 120575
119900the loss of immunity rate by [13] the
value of 120578 by [13] the value of 120581 by [10] and120587 and 119902 from [23]We assume the values for the parameters 120574
3 119901 1205831 and 120583
2in
our simulation
7 Conclusion
A compartmental model of HBV transmission virus hasbeen presented A mathematical model has been obtained
BioMed Research International 9
Table 1 Parameter values used in numerical simulations
Notation Parameter description Range Source120575 Natural death rate equally birth rate 00143 [4]120587 The failure immunization 0-1 [23]1205741
The rate at which the latent individuals become infectious 6 [3]1205742
The rate at which the individuals move to the carrier class 4 [3]1205743
The rate of flow from carrier to the vaccinated class 034 Assumed120573 The transmission coefficient 08 [24]119902 The proportion of individuals become carrier 0005 [23]120575119900
Represent the loss of immunity 006ndash003 [13]119901 Represent the vaccination of susceptible 03 Assumed1205831
The rate of flow fromMigrated class to exposed class 023 Assumed1205832
The rate of flow fromMigrated class to acute class 056 Assumed120578 Unimmunized children born to carrier mothers 07 [13]120581 The infectiousness of carriers related to acute infection 0-1 [10]
by adding (1) the migrated class (2) HBV transmission ratebetween migrated class and exposed class (3) transmissionrate between migrated class and acute class and (4) deathrate of individuals in the migrated class By adding thesenew features we have obtained a compartmental model ofHBVwithmigration effect First we obtained the basic repro-duction number for the proposed model The disease-freeequilibrium is locally as well as globally asymptotically stablefor 1198770le 1 We obtained the local and global asymptotical
stability for the endemic equilibrium For the reproductionnumber 119877
0gt 1 the endemic equilibrium is locally as
well as globally asymptotically stable Furthermore we havesolved the compartment model numerically and the resultsare presented in Figures 2 to 8 By changing the values of 120583
1
and 1205832 different results have been obtained It is concluded
that when the value of 1205831and 120583
2decreases the population
of (exposed acute and carrier) individuals also decreaseThe proportion of infected individuals decreases when theproportion ofmigrated individuals (who have theHBVvirus)decreases So the number of infected individuals is directlyproportional to the number of migrated individuals
Last yet not the least the authors of this work have agreedto devise in a course of time a more advanced model onrestraining HBV transmission through migration
Acknowledgment
Theauthors would like to thank the Kanury V S Rao for theircareful reading of the original manuscripts and their manyvaluable comments and suggestions that greatly improve thepresentation of this work
References
[1] F Ahmed and G R Foster ldquoGlobal hepatitis migration and itsimpact onWestern healthcarerdquoGut vol 59 no 8 pp 1009ndash10112010
[2] ldquoChinese center for disease control and prevention (CCDC)rdquo 2010httpwwwchinacdccnn272442n272530n3479265n347930337095html
[3] 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
[4] 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
[5] R M Anderson and R M May Infectious Disease of HumansDynamics and Control Oxford University Press Oxford UK1991
[6] S Zhao Z Xu and Y Lu ldquoA mathematical model of hepatitis Bvirus transmission and its application for vaccination strategyin Chinardquo International Journal of Epidemiology vol 29 no 4pp 744ndash752 2000
[7] K Wang W Wang and S Song ldquoDynamics of an HBV modelwith diffusion and delayrdquo Journal ofTheoretical Biology vol 253no 1 pp 36ndash44 2008
[8] R Xu andZMa ldquoAnHBVmodelwith diffusion and time delayrdquoJournal of Theoretical Biology vol 257 no 3 pp 499ndash509 2009
[9] 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
[10] 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
[11] K P Maier Hepatitis-Hepatitisfolgen Georg Thieme VerlagStuttgart Germany 2000
[12] G L Mandell R G Douglas and J E Bennett Principles andPractice of Infectious Diseases A Wiley Medical PublicationJohn Wiley and Sons New York NY USA 1979
[13] CW Shepard E P Simard L Finelli A E Fiore and B P BellldquoHepatitis B virus infection epidemiology and vaccinationrdquoEpidemiologic Reviews vol 28 no 1 pp 112ndash125 2006
[14] X Weng and Y Zhang Infectious Diseases Fudan UniversityPress Shanghai China 2003
[15] J Wu and Y Luo Infectious Diseases Central South UniversityPress Changsha China 2004
[16] J BWyngaarden L H Smith and J C BennettCecil Text Bookof Medicine WB Saunders Philadelphia Pa USA 19th edition1992
10 BioMed Research International
[17] B JMcMahonW LM Alward andD BHall ldquoAcute hepatitisB virus infection relation of age to the clinical expressionof disease and subsequent development of the carrier staterdquoJournal of Infectious Diseases vol 151 no 4 pp 599ndash603 1985
[18] P Van Den Driessche and J Watmough ldquoReproduction num-bers and sub-threshold endemic equilibria for compartmentalmodels of disease transmissionrdquoMathematical Biosciences vol180 pp 29ndash48 2002
[19] O Diekmann and J A P HeesterbeekMathematical Epideniol-ogy of Infectious Diseases 2000
[20] A Mwasa and J M Tchuenche ldquoMathematical analysis of acholera model with public health interventionsrdquo BioSystemsvol 105 no 3 pp 190ndash200 2011
[21] J Li and Y Yang ldquoSIR-SVS epidemic models with continuousand impulsive vaccination strategiesrdquo Journal of TheoreticalBiology vol 280 no 1 pp 108ndash116 2011
[22] J P LaSalle ldquoStability of nonautonomous systemsrdquo NonlinearAnalysis vol 1 no 1 pp 83ndash90 1976
[23] World Health Organization ldquoHepatitis B WHOCDSCSRLYO20022Hepatitis Brdquo httpwwwwhointcsrdiseasehepa-titiswhocdscsrlyo20022en
[24] W J Edmunds G FMedley andD J Nokes ldquoThe transmissiondynamics and control of hepatitis B virus in the GambiardquoStatistics in Medicine vol 15 pp 2215ndash2233 2002
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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International Journal of
Microbiology
6 BioMed Research International
1198662= [(1205741120573120575120587120578120573
lowastlowast+120573lowast1198761)1198851(120575+1205741)+11987611198852120575119900120573lowast]120573lowastlowastlowast
= (120575+
1205743)[(120575 + 120574
1) + 12057411205831] So 120582
4has negative real part if (120574
11198852120575119900gt
12057411205732120581119878lowast2) For 120582
5= 1198857 we obtained negative real parts
and by using the 1198855which is positive under the conditions
described in 1205824 and 119885
1gt 0 119885
2gt 0 we get the negative
real partsThus all the eigenvalues have negative real parts soby the Routh-Hurwitz criteria the endemic equilibrium point119863lowast is locally asymptotically stable when 119877
0gt 1 The proof is
completed
5 Global Stability of DFE
In this section we present the global stability of disease-freeequilibrium DFE of the system (3) For different biologicalmodel the Lyapunov function was used by [20 21] for theglobal stability For our model we define and constructLyapunov function in the following for the global stability ofDFE Further the global stability of endemic equilibrium weuse the Lyapunov function and find its global asymptoticalstability
Theorem 3 For 1198770le 1 the disease-free equilibrium of the
system (3) is stable globally asymptotically if 119878 = 119878119900 and
unstable for 1198770gt 1
Proof Here we define the Lyapunov function for the globalstability of disease-free equilibrium given by
119881 (119905) = [1198891(119878 minus 119878
119900) + 1198892119864 + 1198893119860 + 119889
4119862 + 119889
5119872] (18)
Differentiating the previous function with respect to 119905 andusing the system (3)
1198811015840(119905) = 119889
11198781015840+ 11988921198641015840+ 11988931198601015840+ 11988941198621015840+ 11988951198721015840
1198811015840(119905)
= 1198891[120575120587 (1 minus 120578119862 (119905)) minus 120575119878 (119905) minus 120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905)
+ 120575119900(1minus(119878 (119905) + 119864 (119905) +119860 (119905) + 119862 (119905) +119872 (119905)))minus119901119878 (119905)]
+ 1198892[120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905) minus 120575119864 (119905) + 120575120587120578119862 (119905)
minus1205741119864 (119905) + 120583
1119872(119905)]
+ 1198893[1205741119864 (119905) minus 120575119860 (119905) minus 119902120574
2119860 (119905) minus (1 minus 119902) 120574
1119860 (119905)
+1205832119872(119905)]
+ 1198894[1199021205742119860 (119905) minus 120575119862 (119905) minus 120574
3119862 (119905)]
+ 1198895[minus (1205831+ 1205832)119872 (119905) minus 120575119872 (119905)]
(19)
where 119889119894 119894 = 1 2 5 are some positive constants to be
chosen later
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Time (year)
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 2 The plot shows the HBV transmission model of hepatitisB with 120583
1= 090 and 120583
2= 090
Time (year)0 1 2 3 4 5 6 7 8 9 10
0
10
20
30
40
50
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 3 The plot shows the HBV transmission model of hepatitisB with 120583
1= 080 and 120583
2= 090
After the arrangement we obtain
1198811015840(119905) = [119889
2minus 1198891] (120573 (119860 + 120581119862)) 119878 + [119889
2minus 1198891] 120575120587120578119862
+ [11988931205741minus 1198892(120575 + 120574
1)] 119864 + [119889
21205831minus 1198895120575]119872
+ [11988941199021205742minus 11988931198761] 119860 + 119889
1120575120587 minus 119889
1120575119878 + 119889
1120575119900
minus 1198891120575119900(119878 + 119864 + 119860 + 119862 +119872) minus 119889
1119901119878
minus 1198894(120575 + 120574
3) 119862
(20)
BioMed Research International 7
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Time (year)
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 4 The plot shows the HBV transmission model of hepatitisB with 120583
1= 070 and 120583
2= 080
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Time (year)
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 5 The plot shows the HBV transmission model of hepatitisB with 120583
1= 050 and 120583
2= 060
Choosing the constants 1198891= 1198892= 1205741 1198893= (120575 + 120574
1) 1198894=
1198761(120575 + 120574
1)1199021205742 and 119889
5= 12057411205831120575
After the simplification we get
1198811015840(119905) = minus (119878 minus 119878
119900) minus 1205741120575119900119864 minus 1205741120575119900119860
minus (1205741120575119900+(120575 + 120574
3) (120575 + 120574
1) 1198761
1199021205742
)119862
minus (1205741120575119900+ (120575119900+ 1205741) 1205832)119872
(21)
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Time (year)
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 6 The plot shows the HBV transmission model of hepatitisB with 120583
1= 020 and 120583
2= 030
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Time (year)
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 7 The plot shows the HBV transmission model of hepatitisB with 120583
1= 020 and 120583
2= 040
where 119878119900 = (120575120587 + 120575119900)(120575 + 120575
119900+ 119901) 1198811015840(119905) = 0 if and only if
119878 = 119878119900 and 119864 = 119860 = 119862 = 119872 = 0 Also 1198811015840(119905) is negative for
(119878 gt 119878119900) So by [22] the DFE is globally asymptotically stable
in Γ The proof is completed
51 Global Stability of Endemic Equilibrium In this subsec-tion we show the global asymptotical stability of the system(3) To do this we state and prove the following theorem forthe global stability of endemic equilibrium
8 BioMed Research International
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Time (year)
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 8 The plot shows the HBV transmission model of hepatitisB with 120583
1= 010 and 120583
2= 020
Theorem 4 For 1198770gt 1 system (3) is globally asymptotically
stable if 119878 = 119878lowast and 120575119900gt 1205831 and unstable for 119877
0le 1
Proof To prove that system (3) is globally asymptoticallystable we define the Lyapunov in the following
119871 (119905) =(1205831+ 1205832+ 1205833) 1205741
1205741+ 120575
(119878 minus 119878lowast) +
(1205831+ 1205832+ 1205833) 1205741
1205741+ 120575
119864
+ (1205831+ 1205832+ 1205833) 119860 +
(1205831+ 1205832+ 1205833) 1198761
1199021205742
119862 + 1205832119872
(22)
Taking the derivative with respect to time 119905 using the system(3)
1198711015840(119905) =
(1205831+ 1205832+ 1205833) 1205741
1205741+ 120575
times [120575120587 (1 minus 120578119862 (119905)) minus 120575119878 (119905)minus120573 (119860 (119905)+120581119862 (119905)) 119878 (119905)
+ 120575119900(1 minus (119878 (119905) + 119864 (119905) + 119860 (119905) + 119862 (119905) + 119872 (119905)))
minus119901119878 (119905)]
+(1205831+ 1205832+ 1205833) 1205741
1205741+ 120575
[120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905) minus 120575119864 (119905)
+120575120587120578119862 (119905) minus 1205741119864 (119905) + 120583
1119872(119905)]
+ (1205831+ 1205832+ 1205833) [1205741119864 (119905) minus 120575119860 (119905) minus 119902120574
2119860 (119905)
minus (1 minus 119902) 1205741119860 (119905) + 120583
2119872(119905)]
+(1205831+ 1205832+ 1205833) 1198761
1199021205742
[1199021205742119860 (119905) minus 120575119862 (119905) minus 120574
3119862 (119905)]
+ 1205832[minus (1205831+ 1205832+ 120575)119872]
(23)Simplifying we obtained
1198711015840(119905) = minus
(1205831+ 1205832+ 1205833) 1205741
(1205741+ 120575) (120575 + 120575
119900+ 119901)
(119878 minus 119878lowast)
minus(1205831+ 1205832+ 1205833) 1205741120575119900
(1205741+ 120575)
[119864 + 119860 + 119862]
minus(1205831+ 1205832+ 1205833) 1205741
(1205741+ 120575)
[120575119900minus 1205831]119872
(24)
The endemic equilibrium of the system (3) is globally asymp-totically stable for 119877
0gt 1 if 119878 = 119878
lowast and 120575119900gt 1205831
So the endemic equilibrium of the system (3) is globallyasymptotically stable The proof is completed
6 Numerical Simulations
In this section we present the numerical simulation of theproposed model (3) by using the Runge-Kutta order fourscheme For different values of the parameters the numericalresults are presented in Figures 2 3 4 5 6 7 and 8 Thevariation of migration parameters 120583
1and 120583
2 with different
values is presented In the numerical solution of the modelthe parameters and their values are presented in Table 1In our simulation the susceptible individuals are shown bydashed line the exposed individuals by dotted dashed theacute individuals by bold line the carrier by dashed lineand the migrated individuals by red bold line Figures 2to 8 represent the compartmental population of hepatitis Bindividuals with migration effect The values presented inTable 1 are fixed except for120583
1and1205832 In Figure 2 by the values
for 1205831= 09 and 120583
2= 09 we see that the population of
exposed and acute individuals is increasing In Figure 3 weset 1205831= 08 and 120583
2= 09 and the population of exposed
and acute individuals decreased Decreasing the values of 1205831
and 1205832 we obtain different results see Figures 2 to 8 When
we decrease the population of immigrants who have the HBVvirus we see the decrease in the population of exposed andacute individuals The parameters presented in Table 1 wereused by different authors for example the natural deathrate equally birth rate by [4] the rate at which the latentindividuals become infectious by [3] the rate at which theindividuals move to the carrier class by [3] the rate at whichthe individuals move to the carrier class [3] the transmissioncoefficient 120573 by [24] 120575
119900the loss of immunity rate by [13] the
value of 120578 by [13] the value of 120581 by [10] and120587 and 119902 from [23]We assume the values for the parameters 120574
3 119901 1205831 and 120583
2in
our simulation
7 Conclusion
A compartmental model of HBV transmission virus hasbeen presented A mathematical model has been obtained
BioMed Research International 9
Table 1 Parameter values used in numerical simulations
Notation Parameter description Range Source120575 Natural death rate equally birth rate 00143 [4]120587 The failure immunization 0-1 [23]1205741
The rate at which the latent individuals become infectious 6 [3]1205742
The rate at which the individuals move to the carrier class 4 [3]1205743
The rate of flow from carrier to the vaccinated class 034 Assumed120573 The transmission coefficient 08 [24]119902 The proportion of individuals become carrier 0005 [23]120575119900
Represent the loss of immunity 006ndash003 [13]119901 Represent the vaccination of susceptible 03 Assumed1205831
The rate of flow fromMigrated class to exposed class 023 Assumed1205832
The rate of flow fromMigrated class to acute class 056 Assumed120578 Unimmunized children born to carrier mothers 07 [13]120581 The infectiousness of carriers related to acute infection 0-1 [10]
by adding (1) the migrated class (2) HBV transmission ratebetween migrated class and exposed class (3) transmissionrate between migrated class and acute class and (4) deathrate of individuals in the migrated class By adding thesenew features we have obtained a compartmental model ofHBVwithmigration effect First we obtained the basic repro-duction number for the proposed model The disease-freeequilibrium is locally as well as globally asymptotically stablefor 1198770le 1 We obtained the local and global asymptotical
stability for the endemic equilibrium For the reproductionnumber 119877
0gt 1 the endemic equilibrium is locally as
well as globally asymptotically stable Furthermore we havesolved the compartment model numerically and the resultsare presented in Figures 2 to 8 By changing the values of 120583
1
and 1205832 different results have been obtained It is concluded
that when the value of 1205831and 120583
2decreases the population
of (exposed acute and carrier) individuals also decreaseThe proportion of infected individuals decreases when theproportion ofmigrated individuals (who have theHBVvirus)decreases So the number of infected individuals is directlyproportional to the number of migrated individuals
Last yet not the least the authors of this work have agreedto devise in a course of time a more advanced model onrestraining HBV transmission through migration
Acknowledgment
Theauthors would like to thank the Kanury V S Rao for theircareful reading of the original manuscripts and their manyvaluable comments and suggestions that greatly improve thepresentation of this work
References
[1] F Ahmed and G R Foster ldquoGlobal hepatitis migration and itsimpact onWestern healthcarerdquoGut vol 59 no 8 pp 1009ndash10112010
[2] ldquoChinese center for disease control and prevention (CCDC)rdquo 2010httpwwwchinacdccnn272442n272530n3479265n347930337095html
[3] 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
[4] 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
[5] R M Anderson and R M May Infectious Disease of HumansDynamics and Control Oxford University Press Oxford UK1991
[6] S Zhao Z Xu and Y Lu ldquoA mathematical model of hepatitis Bvirus transmission and its application for vaccination strategyin Chinardquo International Journal of Epidemiology vol 29 no 4pp 744ndash752 2000
[7] K Wang W Wang and S Song ldquoDynamics of an HBV modelwith diffusion and delayrdquo Journal ofTheoretical Biology vol 253no 1 pp 36ndash44 2008
[8] R Xu andZMa ldquoAnHBVmodelwith diffusion and time delayrdquoJournal of Theoretical Biology vol 257 no 3 pp 499ndash509 2009
[9] 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
[10] 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
[11] K P Maier Hepatitis-Hepatitisfolgen Georg Thieme VerlagStuttgart Germany 2000
[12] G L Mandell R G Douglas and J E Bennett Principles andPractice of Infectious Diseases A Wiley Medical PublicationJohn Wiley and Sons New York NY USA 1979
[13] CW Shepard E P Simard L Finelli A E Fiore and B P BellldquoHepatitis B virus infection epidemiology and vaccinationrdquoEpidemiologic Reviews vol 28 no 1 pp 112ndash125 2006
[14] X Weng and Y Zhang Infectious Diseases Fudan UniversityPress Shanghai China 2003
[15] J Wu and Y Luo Infectious Diseases Central South UniversityPress Changsha China 2004
[16] J BWyngaarden L H Smith and J C BennettCecil Text Bookof Medicine WB Saunders Philadelphia Pa USA 19th edition1992
10 BioMed Research International
[17] B JMcMahonW LM Alward andD BHall ldquoAcute hepatitisB virus infection relation of age to the clinical expressionof disease and subsequent development of the carrier staterdquoJournal of Infectious Diseases vol 151 no 4 pp 599ndash603 1985
[18] P Van Den Driessche and J Watmough ldquoReproduction num-bers and sub-threshold endemic equilibria for compartmentalmodels of disease transmissionrdquoMathematical Biosciences vol180 pp 29ndash48 2002
[19] O Diekmann and J A P HeesterbeekMathematical Epideniol-ogy of Infectious Diseases 2000
[20] A Mwasa and J M Tchuenche ldquoMathematical analysis of acholera model with public health interventionsrdquo BioSystemsvol 105 no 3 pp 190ndash200 2011
[21] J Li and Y Yang ldquoSIR-SVS epidemic models with continuousand impulsive vaccination strategiesrdquo Journal of TheoreticalBiology vol 280 no 1 pp 108ndash116 2011
[22] J P LaSalle ldquoStability of nonautonomous systemsrdquo NonlinearAnalysis vol 1 no 1 pp 83ndash90 1976
[23] World Health Organization ldquoHepatitis B WHOCDSCSRLYO20022Hepatitis Brdquo httpwwwwhointcsrdiseasehepa-titiswhocdscsrlyo20022en
[24] W J Edmunds G FMedley andD J Nokes ldquoThe transmissiondynamics and control of hepatitis B virus in the GambiardquoStatistics in Medicine vol 15 pp 2215ndash2233 2002
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Anatomy Research International
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International Journal of
Volume 2014
Zoology
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Molecular Biology International
GenomicsInternational Journal of
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioinformaticsAdvances in
Marine BiologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Signal TransductionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
Evolutionary BiologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Biochemistry Research International
ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Genetics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Virolog y
Hindawi Publishing Corporationhttpwwwhindawicom
Nucleic AcidsJournal of
Volume 2014
Stem CellsInternational
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Enzyme Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Microbiology
BioMed Research International 7
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Time (year)
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 4 The plot shows the HBV transmission model of hepatitisB with 120583
1= 070 and 120583
2= 080
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Time (year)
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 5 The plot shows the HBV transmission model of hepatitisB with 120583
1= 050 and 120583
2= 060
Choosing the constants 1198891= 1198892= 1205741 1198893= (120575 + 120574
1) 1198894=
1198761(120575 + 120574
1)1199021205742 and 119889
5= 12057411205831120575
After the simplification we get
1198811015840(119905) = minus (119878 minus 119878
119900) minus 1205741120575119900119864 minus 1205741120575119900119860
minus (1205741120575119900+(120575 + 120574
3) (120575 + 120574
1) 1198761
1199021205742
)119862
minus (1205741120575119900+ (120575119900+ 1205741) 1205832)119872
(21)
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Time (year)
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 6 The plot shows the HBV transmission model of hepatitisB with 120583
1= 020 and 120583
2= 030
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Time (year)
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 7 The plot shows the HBV transmission model of hepatitisB with 120583
1= 020 and 120583
2= 040
where 119878119900 = (120575120587 + 120575119900)(120575 + 120575
119900+ 119901) 1198811015840(119905) = 0 if and only if
119878 = 119878119900 and 119864 = 119860 = 119862 = 119872 = 0 Also 1198811015840(119905) is negative for
(119878 gt 119878119900) So by [22] the DFE is globally asymptotically stable
in Γ The proof is completed
51 Global Stability of Endemic Equilibrium In this subsec-tion we show the global asymptotical stability of the system(3) To do this we state and prove the following theorem forthe global stability of endemic equilibrium
8 BioMed Research International
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Time (year)
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 8 The plot shows the HBV transmission model of hepatitisB with 120583
1= 010 and 120583
2= 020
Theorem 4 For 1198770gt 1 system (3) is globally asymptotically
stable if 119878 = 119878lowast and 120575119900gt 1205831 and unstable for 119877
0le 1
Proof To prove that system (3) is globally asymptoticallystable we define the Lyapunov in the following
119871 (119905) =(1205831+ 1205832+ 1205833) 1205741
1205741+ 120575
(119878 minus 119878lowast) +
(1205831+ 1205832+ 1205833) 1205741
1205741+ 120575
119864
+ (1205831+ 1205832+ 1205833) 119860 +
(1205831+ 1205832+ 1205833) 1198761
1199021205742
119862 + 1205832119872
(22)
Taking the derivative with respect to time 119905 using the system(3)
1198711015840(119905) =
(1205831+ 1205832+ 1205833) 1205741
1205741+ 120575
times [120575120587 (1 minus 120578119862 (119905)) minus 120575119878 (119905)minus120573 (119860 (119905)+120581119862 (119905)) 119878 (119905)
+ 120575119900(1 minus (119878 (119905) + 119864 (119905) + 119860 (119905) + 119862 (119905) + 119872 (119905)))
minus119901119878 (119905)]
+(1205831+ 1205832+ 1205833) 1205741
1205741+ 120575
[120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905) minus 120575119864 (119905)
+120575120587120578119862 (119905) minus 1205741119864 (119905) + 120583
1119872(119905)]
+ (1205831+ 1205832+ 1205833) [1205741119864 (119905) minus 120575119860 (119905) minus 119902120574
2119860 (119905)
minus (1 minus 119902) 1205741119860 (119905) + 120583
2119872(119905)]
+(1205831+ 1205832+ 1205833) 1198761
1199021205742
[1199021205742119860 (119905) minus 120575119862 (119905) minus 120574
3119862 (119905)]
+ 1205832[minus (1205831+ 1205832+ 120575)119872]
(23)Simplifying we obtained
1198711015840(119905) = minus
(1205831+ 1205832+ 1205833) 1205741
(1205741+ 120575) (120575 + 120575
119900+ 119901)
(119878 minus 119878lowast)
minus(1205831+ 1205832+ 1205833) 1205741120575119900
(1205741+ 120575)
[119864 + 119860 + 119862]
minus(1205831+ 1205832+ 1205833) 1205741
(1205741+ 120575)
[120575119900minus 1205831]119872
(24)
The endemic equilibrium of the system (3) is globally asymp-totically stable for 119877
0gt 1 if 119878 = 119878
lowast and 120575119900gt 1205831
So the endemic equilibrium of the system (3) is globallyasymptotically stable The proof is completed
6 Numerical Simulations
In this section we present the numerical simulation of theproposed model (3) by using the Runge-Kutta order fourscheme For different values of the parameters the numericalresults are presented in Figures 2 3 4 5 6 7 and 8 Thevariation of migration parameters 120583
1and 120583
2 with different
values is presented In the numerical solution of the modelthe parameters and their values are presented in Table 1In our simulation the susceptible individuals are shown bydashed line the exposed individuals by dotted dashed theacute individuals by bold line the carrier by dashed lineand the migrated individuals by red bold line Figures 2to 8 represent the compartmental population of hepatitis Bindividuals with migration effect The values presented inTable 1 are fixed except for120583
1and1205832 In Figure 2 by the values
for 1205831= 09 and 120583
2= 09 we see that the population of
exposed and acute individuals is increasing In Figure 3 weset 1205831= 08 and 120583
2= 09 and the population of exposed
and acute individuals decreased Decreasing the values of 1205831
and 1205832 we obtain different results see Figures 2 to 8 When
we decrease the population of immigrants who have the HBVvirus we see the decrease in the population of exposed andacute individuals The parameters presented in Table 1 wereused by different authors for example the natural deathrate equally birth rate by [4] the rate at which the latentindividuals become infectious by [3] the rate at which theindividuals move to the carrier class by [3] the rate at whichthe individuals move to the carrier class [3] the transmissioncoefficient 120573 by [24] 120575
119900the loss of immunity rate by [13] the
value of 120578 by [13] the value of 120581 by [10] and120587 and 119902 from [23]We assume the values for the parameters 120574
3 119901 1205831 and 120583
2in
our simulation
7 Conclusion
A compartmental model of HBV transmission virus hasbeen presented A mathematical model has been obtained
BioMed Research International 9
Table 1 Parameter values used in numerical simulations
Notation Parameter description Range Source120575 Natural death rate equally birth rate 00143 [4]120587 The failure immunization 0-1 [23]1205741
The rate at which the latent individuals become infectious 6 [3]1205742
The rate at which the individuals move to the carrier class 4 [3]1205743
The rate of flow from carrier to the vaccinated class 034 Assumed120573 The transmission coefficient 08 [24]119902 The proportion of individuals become carrier 0005 [23]120575119900
Represent the loss of immunity 006ndash003 [13]119901 Represent the vaccination of susceptible 03 Assumed1205831
The rate of flow fromMigrated class to exposed class 023 Assumed1205832
The rate of flow fromMigrated class to acute class 056 Assumed120578 Unimmunized children born to carrier mothers 07 [13]120581 The infectiousness of carriers related to acute infection 0-1 [10]
by adding (1) the migrated class (2) HBV transmission ratebetween migrated class and exposed class (3) transmissionrate between migrated class and acute class and (4) deathrate of individuals in the migrated class By adding thesenew features we have obtained a compartmental model ofHBVwithmigration effect First we obtained the basic repro-duction number for the proposed model The disease-freeequilibrium is locally as well as globally asymptotically stablefor 1198770le 1 We obtained the local and global asymptotical
stability for the endemic equilibrium For the reproductionnumber 119877
0gt 1 the endemic equilibrium is locally as
well as globally asymptotically stable Furthermore we havesolved the compartment model numerically and the resultsare presented in Figures 2 to 8 By changing the values of 120583
1
and 1205832 different results have been obtained It is concluded
that when the value of 1205831and 120583
2decreases the population
of (exposed acute and carrier) individuals also decreaseThe proportion of infected individuals decreases when theproportion ofmigrated individuals (who have theHBVvirus)decreases So the number of infected individuals is directlyproportional to the number of migrated individuals
Last yet not the least the authors of this work have agreedto devise in a course of time a more advanced model onrestraining HBV transmission through migration
Acknowledgment
Theauthors would like to thank the Kanury V S Rao for theircareful reading of the original manuscripts and their manyvaluable comments and suggestions that greatly improve thepresentation of this work
References
[1] F Ahmed and G R Foster ldquoGlobal hepatitis migration and itsimpact onWestern healthcarerdquoGut vol 59 no 8 pp 1009ndash10112010
[2] ldquoChinese center for disease control and prevention (CCDC)rdquo 2010httpwwwchinacdccnn272442n272530n3479265n347930337095html
[3] 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
[4] 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
[5] R M Anderson and R M May Infectious Disease of HumansDynamics and Control Oxford University Press Oxford UK1991
[6] S Zhao Z Xu and Y Lu ldquoA mathematical model of hepatitis Bvirus transmission and its application for vaccination strategyin Chinardquo International Journal of Epidemiology vol 29 no 4pp 744ndash752 2000
[7] K Wang W Wang and S Song ldquoDynamics of an HBV modelwith diffusion and delayrdquo Journal ofTheoretical Biology vol 253no 1 pp 36ndash44 2008
[8] R Xu andZMa ldquoAnHBVmodelwith diffusion and time delayrdquoJournal of Theoretical Biology vol 257 no 3 pp 499ndash509 2009
[9] 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
[10] 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
[11] K P Maier Hepatitis-Hepatitisfolgen Georg Thieme VerlagStuttgart Germany 2000
[12] G L Mandell R G Douglas and J E Bennett Principles andPractice of Infectious Diseases A Wiley Medical PublicationJohn Wiley and Sons New York NY USA 1979
[13] CW Shepard E P Simard L Finelli A E Fiore and B P BellldquoHepatitis B virus infection epidemiology and vaccinationrdquoEpidemiologic Reviews vol 28 no 1 pp 112ndash125 2006
[14] X Weng and Y Zhang Infectious Diseases Fudan UniversityPress Shanghai China 2003
[15] J Wu and Y Luo Infectious Diseases Central South UniversityPress Changsha China 2004
[16] J BWyngaarden L H Smith and J C BennettCecil Text Bookof Medicine WB Saunders Philadelphia Pa USA 19th edition1992
10 BioMed Research International
[17] B JMcMahonW LM Alward andD BHall ldquoAcute hepatitisB virus infection relation of age to the clinical expressionof disease and subsequent development of the carrier staterdquoJournal of Infectious Diseases vol 151 no 4 pp 599ndash603 1985
[18] P Van Den Driessche and J Watmough ldquoReproduction num-bers and sub-threshold endemic equilibria for compartmentalmodels of disease transmissionrdquoMathematical Biosciences vol180 pp 29ndash48 2002
[19] O Diekmann and J A P HeesterbeekMathematical Epideniol-ogy of Infectious Diseases 2000
[20] A Mwasa and J M Tchuenche ldquoMathematical analysis of acholera model with public health interventionsrdquo BioSystemsvol 105 no 3 pp 190ndash200 2011
[21] J Li and Y Yang ldquoSIR-SVS epidemic models with continuousand impulsive vaccination strategiesrdquo Journal of TheoreticalBiology vol 280 no 1 pp 108ndash116 2011
[22] J P LaSalle ldquoStability of nonautonomous systemsrdquo NonlinearAnalysis vol 1 no 1 pp 83ndash90 1976
[23] World Health Organization ldquoHepatitis B WHOCDSCSRLYO20022Hepatitis Brdquo httpwwwwhointcsrdiseasehepa-titiswhocdscsrlyo20022en
[24] W J Edmunds G FMedley andD J Nokes ldquoThe transmissiondynamics and control of hepatitis B virus in the GambiardquoStatistics in Medicine vol 15 pp 2215ndash2233 2002
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Anatomy Research International
PeptidesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
International Journal of
Volume 2014
Zoology
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Molecular Biology International
GenomicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioinformaticsAdvances in
Marine BiologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Signal TransductionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
Evolutionary BiologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Biochemistry Research International
ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Genetics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Virolog y
Hindawi Publishing Corporationhttpwwwhindawicom
Nucleic AcidsJournal of
Volume 2014
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Enzyme Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Microbiology
8 BioMed Research International
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Time (year)
Com
part
men
tal p
opul
atio
ns
Population behavior of individuals
SEA
CM
Figure 8 The plot shows the HBV transmission model of hepatitisB with 120583
1= 010 and 120583
2= 020
Theorem 4 For 1198770gt 1 system (3) is globally asymptotically
stable if 119878 = 119878lowast and 120575119900gt 1205831 and unstable for 119877
0le 1
Proof To prove that system (3) is globally asymptoticallystable we define the Lyapunov in the following
119871 (119905) =(1205831+ 1205832+ 1205833) 1205741
1205741+ 120575
(119878 minus 119878lowast) +
(1205831+ 1205832+ 1205833) 1205741
1205741+ 120575
119864
+ (1205831+ 1205832+ 1205833) 119860 +
(1205831+ 1205832+ 1205833) 1198761
1199021205742
119862 + 1205832119872
(22)
Taking the derivative with respect to time 119905 using the system(3)
1198711015840(119905) =
(1205831+ 1205832+ 1205833) 1205741
1205741+ 120575
times [120575120587 (1 minus 120578119862 (119905)) minus 120575119878 (119905)minus120573 (119860 (119905)+120581119862 (119905)) 119878 (119905)
+ 120575119900(1 minus (119878 (119905) + 119864 (119905) + 119860 (119905) + 119862 (119905) + 119872 (119905)))
minus119901119878 (119905)]
+(1205831+ 1205832+ 1205833) 1205741
1205741+ 120575
[120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905) minus 120575119864 (119905)
+120575120587120578119862 (119905) minus 1205741119864 (119905) + 120583
1119872(119905)]
+ (1205831+ 1205832+ 1205833) [1205741119864 (119905) minus 120575119860 (119905) minus 119902120574
2119860 (119905)
minus (1 minus 119902) 1205741119860 (119905) + 120583
2119872(119905)]
+(1205831+ 1205832+ 1205833) 1198761
1199021205742
[1199021205742119860 (119905) minus 120575119862 (119905) minus 120574
3119862 (119905)]
+ 1205832[minus (1205831+ 1205832+ 120575)119872]
(23)Simplifying we obtained
1198711015840(119905) = minus
(1205831+ 1205832+ 1205833) 1205741
(1205741+ 120575) (120575 + 120575
119900+ 119901)
(119878 minus 119878lowast)
minus(1205831+ 1205832+ 1205833) 1205741120575119900
(1205741+ 120575)
[119864 + 119860 + 119862]
minus(1205831+ 1205832+ 1205833) 1205741
(1205741+ 120575)
[120575119900minus 1205831]119872
(24)
The endemic equilibrium of the system (3) is globally asymp-totically stable for 119877
0gt 1 if 119878 = 119878
lowast and 120575119900gt 1205831
So the endemic equilibrium of the system (3) is globallyasymptotically stable The proof is completed
6 Numerical Simulations
In this section we present the numerical simulation of theproposed model (3) by using the Runge-Kutta order fourscheme For different values of the parameters the numericalresults are presented in Figures 2 3 4 5 6 7 and 8 Thevariation of migration parameters 120583
1and 120583
2 with different
values is presented In the numerical solution of the modelthe parameters and their values are presented in Table 1In our simulation the susceptible individuals are shown bydashed line the exposed individuals by dotted dashed theacute individuals by bold line the carrier by dashed lineand the migrated individuals by red bold line Figures 2to 8 represent the compartmental population of hepatitis Bindividuals with migration effect The values presented inTable 1 are fixed except for120583
1and1205832 In Figure 2 by the values
for 1205831= 09 and 120583
2= 09 we see that the population of
exposed and acute individuals is increasing In Figure 3 weset 1205831= 08 and 120583
2= 09 and the population of exposed
and acute individuals decreased Decreasing the values of 1205831
and 1205832 we obtain different results see Figures 2 to 8 When
we decrease the population of immigrants who have the HBVvirus we see the decrease in the population of exposed andacute individuals The parameters presented in Table 1 wereused by different authors for example the natural deathrate equally birth rate by [4] the rate at which the latentindividuals become infectious by [3] the rate at which theindividuals move to the carrier class by [3] the rate at whichthe individuals move to the carrier class [3] the transmissioncoefficient 120573 by [24] 120575
119900the loss of immunity rate by [13] the
value of 120578 by [13] the value of 120581 by [10] and120587 and 119902 from [23]We assume the values for the parameters 120574
3 119901 1205831 and 120583
2in
our simulation
7 Conclusion
A compartmental model of HBV transmission virus hasbeen presented A mathematical model has been obtained
BioMed Research International 9
Table 1 Parameter values used in numerical simulations
Notation Parameter description Range Source120575 Natural death rate equally birth rate 00143 [4]120587 The failure immunization 0-1 [23]1205741
The rate at which the latent individuals become infectious 6 [3]1205742
The rate at which the individuals move to the carrier class 4 [3]1205743
The rate of flow from carrier to the vaccinated class 034 Assumed120573 The transmission coefficient 08 [24]119902 The proportion of individuals become carrier 0005 [23]120575119900
Represent the loss of immunity 006ndash003 [13]119901 Represent the vaccination of susceptible 03 Assumed1205831
The rate of flow fromMigrated class to exposed class 023 Assumed1205832
The rate of flow fromMigrated class to acute class 056 Assumed120578 Unimmunized children born to carrier mothers 07 [13]120581 The infectiousness of carriers related to acute infection 0-1 [10]
by adding (1) the migrated class (2) HBV transmission ratebetween migrated class and exposed class (3) transmissionrate between migrated class and acute class and (4) deathrate of individuals in the migrated class By adding thesenew features we have obtained a compartmental model ofHBVwithmigration effect First we obtained the basic repro-duction number for the proposed model The disease-freeequilibrium is locally as well as globally asymptotically stablefor 1198770le 1 We obtained the local and global asymptotical
stability for the endemic equilibrium For the reproductionnumber 119877
0gt 1 the endemic equilibrium is locally as
well as globally asymptotically stable Furthermore we havesolved the compartment model numerically and the resultsare presented in Figures 2 to 8 By changing the values of 120583
1
and 1205832 different results have been obtained It is concluded
that when the value of 1205831and 120583
2decreases the population
of (exposed acute and carrier) individuals also decreaseThe proportion of infected individuals decreases when theproportion ofmigrated individuals (who have theHBVvirus)decreases So the number of infected individuals is directlyproportional to the number of migrated individuals
Last yet not the least the authors of this work have agreedto devise in a course of time a more advanced model onrestraining HBV transmission through migration
Acknowledgment
Theauthors would like to thank the Kanury V S Rao for theircareful reading of the original manuscripts and their manyvaluable comments and suggestions that greatly improve thepresentation of this work
References
[1] F Ahmed and G R Foster ldquoGlobal hepatitis migration and itsimpact onWestern healthcarerdquoGut vol 59 no 8 pp 1009ndash10112010
[2] ldquoChinese center for disease control and prevention (CCDC)rdquo 2010httpwwwchinacdccnn272442n272530n3479265n347930337095html
[3] 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
[4] 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
[5] R M Anderson and R M May Infectious Disease of HumansDynamics and Control Oxford University Press Oxford UK1991
[6] S Zhao Z Xu and Y Lu ldquoA mathematical model of hepatitis Bvirus transmission and its application for vaccination strategyin Chinardquo International Journal of Epidemiology vol 29 no 4pp 744ndash752 2000
[7] K Wang W Wang and S Song ldquoDynamics of an HBV modelwith diffusion and delayrdquo Journal ofTheoretical Biology vol 253no 1 pp 36ndash44 2008
[8] R Xu andZMa ldquoAnHBVmodelwith diffusion and time delayrdquoJournal of Theoretical Biology vol 257 no 3 pp 499ndash509 2009
[9] 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
[10] 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
[11] K P Maier Hepatitis-Hepatitisfolgen Georg Thieme VerlagStuttgart Germany 2000
[12] G L Mandell R G Douglas and J E Bennett Principles andPractice of Infectious Diseases A Wiley Medical PublicationJohn Wiley and Sons New York NY USA 1979
[13] CW Shepard E P Simard L Finelli A E Fiore and B P BellldquoHepatitis B virus infection epidemiology and vaccinationrdquoEpidemiologic Reviews vol 28 no 1 pp 112ndash125 2006
[14] X Weng and Y Zhang Infectious Diseases Fudan UniversityPress Shanghai China 2003
[15] J Wu and Y Luo Infectious Diseases Central South UniversityPress Changsha China 2004
[16] J BWyngaarden L H Smith and J C BennettCecil Text Bookof Medicine WB Saunders Philadelphia Pa USA 19th edition1992
10 BioMed Research International
[17] B JMcMahonW LM Alward andD BHall ldquoAcute hepatitisB virus infection relation of age to the clinical expressionof disease and subsequent development of the carrier staterdquoJournal of Infectious Diseases vol 151 no 4 pp 599ndash603 1985
[18] P Van Den Driessche and J Watmough ldquoReproduction num-bers and sub-threshold endemic equilibria for compartmentalmodels of disease transmissionrdquoMathematical Biosciences vol180 pp 29ndash48 2002
[19] O Diekmann and J A P HeesterbeekMathematical Epideniol-ogy of Infectious Diseases 2000
[20] A Mwasa and J M Tchuenche ldquoMathematical analysis of acholera model with public health interventionsrdquo BioSystemsvol 105 no 3 pp 190ndash200 2011
[21] J Li and Y Yang ldquoSIR-SVS epidemic models with continuousand impulsive vaccination strategiesrdquo Journal of TheoreticalBiology vol 280 no 1 pp 108ndash116 2011
[22] J P LaSalle ldquoStability of nonautonomous systemsrdquo NonlinearAnalysis vol 1 no 1 pp 83ndash90 1976
[23] World Health Organization ldquoHepatitis B WHOCDSCSRLYO20022Hepatitis Brdquo httpwwwwhointcsrdiseasehepa-titiswhocdscsrlyo20022en
[24] W J Edmunds G FMedley andD J Nokes ldquoThe transmissiondynamics and control of hepatitis B virus in the GambiardquoStatistics in Medicine vol 15 pp 2215ndash2233 2002
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Anatomy Research International
PeptidesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
International Journal of
Volume 2014
Zoology
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Molecular Biology International
GenomicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioinformaticsAdvances in
Marine BiologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Signal TransductionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
Evolutionary BiologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Biochemistry Research International
ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Genetics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Virolog y
Hindawi Publishing Corporationhttpwwwhindawicom
Nucleic AcidsJournal of
Volume 2014
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Enzyme Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Microbiology
BioMed Research International 9
Table 1 Parameter values used in numerical simulations
Notation Parameter description Range Source120575 Natural death rate equally birth rate 00143 [4]120587 The failure immunization 0-1 [23]1205741
The rate at which the latent individuals become infectious 6 [3]1205742
The rate at which the individuals move to the carrier class 4 [3]1205743
The rate of flow from carrier to the vaccinated class 034 Assumed120573 The transmission coefficient 08 [24]119902 The proportion of individuals become carrier 0005 [23]120575119900
Represent the loss of immunity 006ndash003 [13]119901 Represent the vaccination of susceptible 03 Assumed1205831
The rate of flow fromMigrated class to exposed class 023 Assumed1205832
The rate of flow fromMigrated class to acute class 056 Assumed120578 Unimmunized children born to carrier mothers 07 [13]120581 The infectiousness of carriers related to acute infection 0-1 [10]
by adding (1) the migrated class (2) HBV transmission ratebetween migrated class and exposed class (3) transmissionrate between migrated class and acute class and (4) deathrate of individuals in the migrated class By adding thesenew features we have obtained a compartmental model ofHBVwithmigration effect First we obtained the basic repro-duction number for the proposed model The disease-freeequilibrium is locally as well as globally asymptotically stablefor 1198770le 1 We obtained the local and global asymptotical
stability for the endemic equilibrium For the reproductionnumber 119877
0gt 1 the endemic equilibrium is locally as
well as globally asymptotically stable Furthermore we havesolved the compartment model numerically and the resultsare presented in Figures 2 to 8 By changing the values of 120583
1
and 1205832 different results have been obtained It is concluded
that when the value of 1205831and 120583
2decreases the population
of (exposed acute and carrier) individuals also decreaseThe proportion of infected individuals decreases when theproportion ofmigrated individuals (who have theHBVvirus)decreases So the number of infected individuals is directlyproportional to the number of migrated individuals
Last yet not the least the authors of this work have agreedto devise in a course of time a more advanced model onrestraining HBV transmission through migration
Acknowledgment
Theauthors would like to thank the Kanury V S Rao for theircareful reading of the original manuscripts and their manyvaluable comments and suggestions that greatly improve thepresentation of this work
References
[1] F Ahmed and G R Foster ldquoGlobal hepatitis migration and itsimpact onWestern healthcarerdquoGut vol 59 no 8 pp 1009ndash10112010
[2] ldquoChinese center for disease control and prevention (CCDC)rdquo 2010httpwwwchinacdccnn272442n272530n3479265n347930337095html
[3] 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
[4] 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
[5] R M Anderson and R M May Infectious Disease of HumansDynamics and Control Oxford University Press Oxford UK1991
[6] S Zhao Z Xu and Y Lu ldquoA mathematical model of hepatitis Bvirus transmission and its application for vaccination strategyin Chinardquo International Journal of Epidemiology vol 29 no 4pp 744ndash752 2000
[7] K Wang W Wang and S Song ldquoDynamics of an HBV modelwith diffusion and delayrdquo Journal ofTheoretical Biology vol 253no 1 pp 36ndash44 2008
[8] R Xu andZMa ldquoAnHBVmodelwith diffusion and time delayrdquoJournal of Theoretical Biology vol 257 no 3 pp 499ndash509 2009
[9] 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
[10] 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
[11] K P Maier Hepatitis-Hepatitisfolgen Georg Thieme VerlagStuttgart Germany 2000
[12] G L Mandell R G Douglas and J E Bennett Principles andPractice of Infectious Diseases A Wiley Medical PublicationJohn Wiley and Sons New York NY USA 1979
[13] CW Shepard E P Simard L Finelli A E Fiore and B P BellldquoHepatitis B virus infection epidemiology and vaccinationrdquoEpidemiologic Reviews vol 28 no 1 pp 112ndash125 2006
[14] X Weng and Y Zhang Infectious Diseases Fudan UniversityPress Shanghai China 2003
[15] J Wu and Y Luo Infectious Diseases Central South UniversityPress Changsha China 2004
[16] J BWyngaarden L H Smith and J C BennettCecil Text Bookof Medicine WB Saunders Philadelphia Pa USA 19th edition1992
10 BioMed Research International
[17] B JMcMahonW LM Alward andD BHall ldquoAcute hepatitisB virus infection relation of age to the clinical expressionof disease and subsequent development of the carrier staterdquoJournal of Infectious Diseases vol 151 no 4 pp 599ndash603 1985
[18] P Van Den Driessche and J Watmough ldquoReproduction num-bers and sub-threshold endemic equilibria for compartmentalmodels of disease transmissionrdquoMathematical Biosciences vol180 pp 29ndash48 2002
[19] O Diekmann and J A P HeesterbeekMathematical Epideniol-ogy of Infectious Diseases 2000
[20] A Mwasa and J M Tchuenche ldquoMathematical analysis of acholera model with public health interventionsrdquo BioSystemsvol 105 no 3 pp 190ndash200 2011
[21] J Li and Y Yang ldquoSIR-SVS epidemic models with continuousand impulsive vaccination strategiesrdquo Journal of TheoreticalBiology vol 280 no 1 pp 108ndash116 2011
[22] J P LaSalle ldquoStability of nonautonomous systemsrdquo NonlinearAnalysis vol 1 no 1 pp 83ndash90 1976
[23] World Health Organization ldquoHepatitis B WHOCDSCSRLYO20022Hepatitis Brdquo httpwwwwhointcsrdiseasehepa-titiswhocdscsrlyo20022en
[24] W J Edmunds G FMedley andD J Nokes ldquoThe transmissiondynamics and control of hepatitis B virus in the GambiardquoStatistics in Medicine vol 15 pp 2215ndash2233 2002
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Anatomy Research International
PeptidesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
International Journal of
Volume 2014
Zoology
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Molecular Biology International
GenomicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioinformaticsAdvances in
Marine BiologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Signal TransductionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
Evolutionary BiologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Biochemistry Research International
ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Genetics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Virolog y
Hindawi Publishing Corporationhttpwwwhindawicom
Nucleic AcidsJournal of
Volume 2014
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Enzyme Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Microbiology
10 BioMed Research International
[17] B JMcMahonW LM Alward andD BHall ldquoAcute hepatitisB virus infection relation of age to the clinical expressionof disease and subsequent development of the carrier staterdquoJournal of Infectious Diseases vol 151 no 4 pp 599ndash603 1985
[18] P Van Den Driessche and J Watmough ldquoReproduction num-bers and sub-threshold endemic equilibria for compartmentalmodels of disease transmissionrdquoMathematical Biosciences vol180 pp 29ndash48 2002
[19] O Diekmann and J A P HeesterbeekMathematical Epideniol-ogy of Infectious Diseases 2000
[20] A Mwasa and J M Tchuenche ldquoMathematical analysis of acholera model with public health interventionsrdquo BioSystemsvol 105 no 3 pp 190ndash200 2011
[21] J Li and Y Yang ldquoSIR-SVS epidemic models with continuousand impulsive vaccination strategiesrdquo Journal of TheoreticalBiology vol 280 no 1 pp 108ndash116 2011
[22] J P LaSalle ldquoStability of nonautonomous systemsrdquo NonlinearAnalysis vol 1 no 1 pp 83ndash90 1976
[23] World Health Organization ldquoHepatitis B WHOCDSCSRLYO20022Hepatitis Brdquo httpwwwwhointcsrdiseasehepa-titiswhocdscsrlyo20022en
[24] W J Edmunds G FMedley andD J Nokes ldquoThe transmissiondynamics and control of hepatitis B virus in the GambiardquoStatistics in Medicine vol 15 pp 2215ndash2233 2002
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Anatomy Research International
PeptidesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
International Journal of
Volume 2014
Zoology
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Molecular Biology International
GenomicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioinformaticsAdvances in
Marine BiologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Signal TransductionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
Evolutionary BiologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Biochemistry Research International
ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Genetics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Virolog y
Hindawi Publishing Corporationhttpwwwhindawicom
Nucleic AcidsJournal of
Volume 2014
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Enzyme Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Microbiology
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Anatomy Research International
PeptidesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
International Journal of
Volume 2014
Zoology
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Molecular Biology International
GenomicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioinformaticsAdvances in
Marine BiologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Signal TransductionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
Evolutionary BiologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Biochemistry Research International
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom
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