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Research ArticleOptimal (Control of) Intervention Strategies for MalariaEpidemic in Karonga District Malawi

Peter M Mwamtobe12 Shirley Abelman1 J Michel Tchuenche3 and Ansley Kasambara2

1 School of Computational and Applied Mathematics University of Witwatersrand Private Bag 3 WitsJohannesburg 2050 South Africa

2Department of Mathematics and Statistics University of Malawi The Malawi Polytechnic Private Bag 303 ChichiriBlantyre 3 Malawi

3 3253 Flowers Road South Atlanta GA 30341 USA

Correspondence should be addressed to Peter M Mwamtobe pmwamtobegmailcom and J Michel Tchuenchejmtchuenchegmailcom

Received 18 February 2014 Revised 22 April 2014 Accepted 29 April 2014 Published 19 June 2014

Academic Editor Mariano Torrisi

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

Malaria is a public health problem for more than 2 billion people globally About 219 million cases of malaria occur worldwideand 660000 people die mostly (91) in the African Region despite decades of efforts to control the disease Although the diseaseis preventable it is life-threatening and parasitically transmitted by the bite of the female Anopheles mosquito A deterministicmathematical model with intervention strategies is developed in order to investigate the effectiveness and optimal control strategiesof indoor residual spraying (IRS) insecticide treated nets (ITNs) and treatment on the transmission dynamics ofmalaria inKarongaDistrict Malawi The effective reproduction number is analytically computed and the existence and stability conditions of theequilibria are explored The model does not exhibit backward bifurcation Pontryaginrsquos Maximum Principle which uses both theLagrangian and Hamiltonian principles with respect to a time dependent constant is used to derive the necessary conditions forthe optimal control of the disease Numerical simulations indicate that the prevention strategies lead to the reduction of both themosquito population and infected human individuals Effective treatment consolidates the prevention strategiesThus malaria canbe eradicated in Karonga District by concurrently applying vector control via ITNs and IRS complemented with timely treatmentof infected people

1 Introduction

Malaria is a vector-borne infectious disease found mainlyin tropical regions (Sub-Saharan Africa Central and SouthAmerica the Indian subcontinent Southeast Asia and thePacific islands) [1] It is a life-threatening disease transmittedthrough the bites of infected mosquitoes [2] There are fourdifferent types of Plasmodium parasites Plasmodium falci-parum (the only parasite which causes malignant malaria)Plasmodium vivax (causes benign malaria with less severesymptoms the vector can remain in the liver for up to threeyears and can lead to a relapse) Plasmodium malariae (alsocauses benignmalaria and is relatively rare) and Plasmodiumovale (causes benign malaria and can remain in the blood

and liver for many years without causing symptoms) Thisstudy focuses mainly on malignant malaria Severe malariacan affect the patientrsquos brain and central nervous system andcan be fatal [2 3] FurthermoreMedicinenet [4] has reportedanother relatively new species Plasmodium knowlesi whichhas been causing malaria in Malaysia and areas of SoutheastAsia It is also a dangerous species that is typically foundonly in long-tailed and pigtail macaque monkeys Like Pfalciparum P knowlesi may be deadly to anyone infected[5]

Beyond the human toll malaria wreaks significant eco-nomic havoc in endemic regions decreasing gross domesticproduct (GDP) by as much as 13 in countries with highlevels of transmissionmdashthe disease accounts for up to 40

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 594256 20 pageshttpdxdoiorg1011552014594256

2 Abstract and Applied Analysis

of public health expenditures 30ndash50 of in-patient hospitaladmissions and up to 60 of out-patient health clinic visits[2] The Government of Malawi has put in place severalcontrol strategies through the National Malaria ControlProgramme to reduce and possibly (ultimately) eliminatemalaria The main strategic areas that have been identifiedfor scaling-up of malaria control activities include malariacase management intermittent preventive treatment (IPT) ofpregnantwomenusing sulfadoxine-pyrimethamine (SP) andmalaria prevention with special emphasis on the use of ITNsas well as IRS [8]

Various compartmental models for the spread of malariahave been proposed [10ndash12] Extensive review of mathemat-ical models of malaria dynamics using 119878119864119868119877119878-type modelsand their variants can be found in Chiyaka et al [13 14]For information on the unexpected stability of malaria elimi-nation and eradication and additional literature on malariadynamics see Smith et al [15] who also showed that ifmosquito birth rate is increased by 25 then the minimaleffective spraying period is reduced by half and if doubledthe period is reduced by three-quarters Our goal is to assessthe role of optimal control of preventive measures namelyITNs and IRS as well as treatment of the transmission dynam-ics of malaria in Karonga District Malawi without actuallytargeting a certain high-risk group Despite a plethora ofstudies on the dynamics of malaria and its control (seeChiyaka et al [14] and Smith et al [15]) to the best of ourknowledge the proposed model with exposed immigrants isseemingly new the use of optimal values of a combinationof three intervention strategies namely indoor residualspraying (IRS) insecticide treated nets (ITNs) and treatmentto investigate the effectiveness and optimal control strategiesin the transmission dynamics of malaria in a locality in aresource constrained setting

2 Model Formulation and Analysis

We formulate an optimal control model for malaria with thepopulation under study being subdivided into compartmentsaccording to individualsrsquo disease status We consider thetotal population sizes denoted by 119873ℎ(119905) and 119873V(119905) for thehuman hosts and Anopheles female mosquitoes respectivelyWe employ the 119878119864119868119877119878 framework to describe a disease withtemporary immunity on recovery from infection The 119878119864119868119877119878model indicates that the passage of individuals is from thesusceptible class 119878ℎ to the exposed class 119864ℎ then to theinfectious class 119868ℎ and finally to the recovery class 119877ℎ 119878ℎ(119905)represents the number of individuals not yet infectedwith themalaria parasite at time 119905 The latent or exposed class 119864ℎ(119905)represents individuals who are infected but not yet infectiousIndividuals in the 119868ℎ(119905) class are infected with malariaand are capable of transmitting the disease to susceptiblemosquitoes 119877ℎ(119905) represents the class of individuals whohave temporarily recovered from the disease The susceptiblehuman population is increased by recruitment (birth) ata constant rate Λ ℎ while others are generated throughmigration by (1 minus 1205811)120579119873ℎ where 1205811 is the proportion ofinfected immigrants into the exposed class and 120579 is the rate

at which people migrate into Karonga District All the re-cruited individuals are assumed to be naive when joining thecommunity

There is some finite probability 120573Vℎ that the parasites(in the form of sporozoites) will be passed onto the humanswhen an infectious female Anopheles mosquito bites a sus-ceptible human The parasite then moves to the liver whereit develops into its next life stage merozoites Using theapproach adopted in [6] susceptible individuals acquiremalaria through contact with infectious mosquitoes at therate 120599 The infected person moves to the exposed class atthe rate (1 minus 1199061)120582ℎ119868V119878ℎ The preventive variable 1199061(119905) isin

[0 1] represents the use of ITNs as a means of minimizingor eliminating mosquito-human contacts After a certainperiod of time the parasite (in the form of merozoites)enters the blood stream usually signaling the clinical onsetof malaria Then the exposed individuals become infectiousand progress to the infected state at a constant rate 120572ℎ Theindividuals who have experienced infectionmay recover withtemporary immunity at a constant rate 120588 and move to therecovery class while some infectious humans after recoverywithout immunity become immediately susceptible again atthe rate (1minus120588) Infectious individuals recover due to treatmentat a rate 1205781199062 with 1199062(119905) isin [0 1] representing the controleffort on treatment and 120601 being the proportion of indi-viduals who recover spontaneously Recovered individualslose immunity at a rate 120595 The natural and disease induceddeath rates are 120583ℎ and 120575ℎ respectively The disease induceddeath rate is very small in comparison with the recoveryrate

The mosquito population 119873V is divided into three com-partments susceptible 119878V(119905) exposed 119864V(119905) and infectious119868V(119905) FemaleAnophelesmosquitoes enter the susceptible classthrough birth at a rate Λ V The parasites in the form ofgametocytes enter the mosquito population with probability120573ℎV This happens when the mosquito bites an infectioushuman and the mosquito moves from the susceptible tothe exposed class Mosquitoes are assumed to suffer deathdue to natural causes at a rate 120583V The exposed mosquitoesprogress to the class of symptomatic mosquitoes 119868V at a rate120572V It is assumed that the disease does not induce deathto the mosquito population Finally the mortality rate ofthe mosquito population increases at a rate proportional to1205911199063(119905) where 120591 gt 0 is a rate constant when using IRS and1199063(119905) isin [0 1] is a constant rate of IRS Figure 1 represents aschematic flow diagram of the proposed model

The model state variables are represented in Table 1Table 2 represents prevention and control strategies practisedin the district Table 3 shows parameters of the model

The state variables in Table 1 the prevention and controlparameters in Table 2 and the model parameters in Table 3for the malaria model satisfy (1) It is assumed that allstate variables and parameters of the model which moni-tors human and mosquito populations are positive for all119905 ge 0 We will therefore analyse the model in a suitableregion

The assumptions lead to the following deterministicsystem of nonlinear ordinary differential equations which

Abstract and Applied Analysis 3

Sh

S

E

Rh

Ih

I

Eh

120583hEh

120572hEh

Λh

120595Rh120583hSh

120583hIh 120575hIh

120583hRh

Λ

120583I

120583E

120583S

120572E 120582S

120591u3I

120591u3E

120591u3S

(1 minus u1)120582hSh

1( (120601 + 120578u2) minus 120588)Ih

(120601 + 120578u2)120588Ih

1205811120579Nh

(1 minus k1)120579Nh

Figure 1 The malaria model with interventions flowchart

describe the evolutionary dynamics of a malaria model witha combination of interventions

119889119878ℎ

119889119905= Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119868ℎ

minus (1 minus 1199061) 120582ℎ119878ℎ minus 120583ℎ119878ℎ + 120595119877ℎ

119889119864ℎ

119889119905= (1 minus 1199061) 120582ℎ119878ℎ + 1205811120579119873ℎ minus 120572ℎ119864ℎ minus 120583ℎ119864ℎ

119889119868ℎ

119889119905= 120572ℎ119864ℎ minus (120601 + 1205781199062) (1 minus 120588) 119868ℎ minus (120601 + 1205781199062) 120588119868ℎ

minus (120583ℎ + 120575ℎ) 119868ℎ

119889119877ℎ

119889119905= (120601 + 1205781199062) 120588119868ℎ minus (120583ℎ + 120595)119877ℎ

119889119878V

119889119905= Λ V minus 120582V119878V minus (120583V + 1205911199063) 119878V

119889119864V

119889119905= 120582V119878V minus 120572V119864V minus (120583V + 1205911199063) 119864V

119889119868V

119889119905= 120572V119864V minus (120583V + 1205911199063) 119868V

(1)

where 120582ℎ = 120573Vℎ120599119868V119873ℎ 120582119881 = 120573ℎV120599119868ℎ119873ℎThe term 120573Vℎ120599119878ℎ119868V119873ℎ denotes the rate at which the

human hosts 119878ℎ become infected by infectious mosquitoes 119868Vand 120573ℎV120599119878V119868ℎ119873ℎ refers to the rate at which the susceptiblemosquitoes 119878V are infected by the infectious human hosts 119868ℎ

Adding the first six equations of model (1) and assumingthat there is no disease induced death that is 120575ℎ = 0 gives119889119873ℎ119889119905 = Λ ℎ minus 120583ℎ119873ℎ so that 119873ℎ(119905) rarr Λ ℎ120583ℎ as 119905 rarr infin

[16] Thus Λ ℎ120583ℎ is an upper bound of 119873ℎ(119905) provided that119873ℎ(0) le Λ120583ℎ Further if 119873ℎ(0) gt Λ ℎ120583ℎ then 119873ℎ(119905)

will decrease to this level Λ ℎ120583ℎ Similar calculation for thevector equations shows that119873V rarr Λ V120583V as 119905 rarr infin

Table 1 State variables of the malaria model

Symbol Description119878ℎ(119905) Number of susceptible individuals at time t119864ℎ(119905) Number of exposed individuals at time t119868ℎ(119905) Number of infectious humans at time 119905119877ℎ(119905) Number of recovered humans at time 119905119878V(119905) Number of susceptible mosquitoes at time 119905119864V(119905) Number of infected mosquitoes at time 119905119868V(119905) Number of infectious mosquitoes at time 119905119873ℎ(119905) Total number of individuals at time 119905119873V(119905) Total mosquito population at time 119905

Table 2 Prevention and control variables in the model

Symbol Description

1199061(119905)Preventive measure using insecticide treated bed nets(ITNs)

1199062(119905)The control effort on treatment of infectiousindividuals

1199063(119905)Preventing measure using indoor residual spraying(IRS)

120591 Rate constant due to use of indoor residual spraying120578 Rate constant due to use of treatment effort

Table 3 Parameters variables of the malaria model

Symbol DescriptionΛ ℎ Recruitment rate of individuals by birth1205811

Proportion of exposed immigrants into exposed class120579 Proportion of people migrating into Karonga District120583ℎ Per capita natural death rate of humans120583V Per capita natural death rate of mosquitoes120575ℎ

Per capita disease induced death rate for humans

120572ℎ

Progression rate of humans from exposed state to theinfectious state

120573VℎProbability that a bite results in transmission ofinfection to human

120573ℎV

Probability that a bite results in transmission of theparasite from an infectious human to the susceptiblemosquitoes

120599 Biting rate of mosquito

120582ℎForce of infection for susceptible humans to exposedindividuals

120582VForce of infection for susceptible mosquitoes toexposed mosquito class

120588 Per capita rate of recovery with temporary immunity120601 Proportion of spontaneous individual recovery120595 Per capita rate of loss immunity

120572VProgression of exposed mosquitoes into infectedmosquitoes


The feasible region

Φ = (119878ℎ 119864ℎ 119868ℎ 119877ℎ 119878V 119864V 119868V) isin R7

+119873ℎ le

Λ ℎ

120583ℎ

= 119873lowast

ℎ

119873V leΛ V

120583V= 119873lowast

V

(2)

is positive-invariant and attracting

Lemma 1 The region Φ isin R7+is positively invariant for the

model system (1) with initial conditions in R7+

Proof Let 119905 = sup119905 gt 0 119878ℎ gt 0 119864ℎ gt 0 119868ℎ gt 0 119877ℎ gt 0 119878V gt0 119864V gt 0 119868V gt 0 isin [0 119905] give 119905 gt 0 The first equation ofmodel (1) gives

119889119878ℎ

119889119905= Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119868ℎ


ge Λ ℎ minus ((1 minus 1199061) 120582ℎ + 120583ℎ) 119878ℎ

(3)

which can be rewritten as

119889

119889119905[119878ℎ (119905) 119890

int119905

0(1minus1199061)120582ℎ(119904)119889119904+120583

ℎ119905]

ge Λ ℎ119890int119905

0(1minus1199061)120582ℎ(119904)119889119904+120583

ℎ119905

(4)

Therefore

119878ℎ (119905) 119890int119905

0(1minus119906

1)120582ℎ(119904)119889119904+120583

ℎ119905minus 119878ℎ (0)

ge int

119905

0

Λ ℎ119890int119906

0(1minus1199061)120582ℎ(119908)119889119908+119906

ℎ119906119889119906

(5)

so that

119878ℎ (119905) ge 119878ℎ (0) 119890minus(int119905

0(1minus1199061)120582ℎ(119904)119889119904+120583

ℎ119905)

+ 119890minus(int119905

0(1minus1199061)120582ℎ(119904)119889119904+120583

ℎ119905)

times int

119905

0

Λ ℎ119890int119906

0(1minus1199061)120582ℎ(119908)119889119908+120583

ℎ119906119889119906 gt 0

(6)

Similarly it can be shown that 119864ℎ gt 0 119868ℎ gt 0 119877ℎ gt 0 119878V gt 0119864V gt 0 and 119868V gt 0 for all 119905 gt 0This completes the proof

21 Existence and Stability of Equilibrium Points We analysesystem (1) to obtain the equilibrium points of the system andtheir stability Let (119878lowast

ℎ 119864lowast

ℎ 119868lowast

ℎ 119877lowast

ℎ 119878lowast

V 119864lowast

V 119868lowast

V ) be the equilib-rium points of system (1) At an equilibrium point we have

1198781015840

ℎ(119905) = 119864

1015840

ℎ(119905) = 119868

1015840

ℎ(119905) = 119877

1015840

ℎ(119905) = 119878

1015840

V (119905) = 1198641015840

V (119905) = 1198681015840

V (119905) = 0

(7)

211 Disease-Free Equilibrium 1198640 In the absence of malariathat is 119864lowast

ℎ= 119868lowast

ℎ= 119877lowast

ℎ= 119864lowast

V = 119868lowast

V = 0 model system (1) hasan equilibrium point called the disease-free equilibrium 1198640and is given by

1198640 = (Λ ℎ

120583ℎ

0 0 0Λ ℎ

120583V 0 0) (8)

To establish the linear stability of 1198640 we employ van denDriessche and Watmoughrsquos next generation matrix approach[17] A reproduction number obtained this way determinesthe local stability of the disease-free equilibrium point for119877119890 lt 1 and instability for119877119890 gt 1 Following van denDriesscheand Watmough [17] the associated next generation matrices119865 and 119881 of system (1) can be determined from F119894 and V119894respectively where

F119894 =

[[[[[[[[[

[

(1 minus 1199061) 120573Vℎ120599119868V119878ℎ

119873ℎ

+ 1205811120579119873ℎ

0

120573ℎV120599119868ℎ119878V

119873ℎ

0

]]]]]]]]]

]

V119894 =[[[[

[

(120572ℎ + 120583ℎ) 119864ℎ

(120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ minus 120572ℎ119864ℎ

(120572V + 120583V + 1205911199063) 119864V(120583V + 1205911199063) 119868V minus 120572V119864V

]]]]

]

(9)

The Jacobian matrix ofF119894 andV119894 is given by

119865 =[[[

[

0 0 0 1199031

0 0 0 0

0 119895 0 0

0 0 0 0

]]]

]

119881 =[[[

[

119898 0 0 0

minus120572ℎ 119899 0 0

0 0 119886 0

0 0 minus120572V b

]]]

]

(10)

where

1199031 = (1 minus 1199061) 120573Vℎ120599 119895 =120573ℎV120599Λ V120583ℎ

Λ ℎ120583V

119898 = 120572ℎ + 120583ℎ 119899 = 120601 + 1205781199062 + 120583ℎ + 120575ℎ

119886 = 120572V + 120583V + 1205911199063 119887 = 120583V + 1205911199063

(11)

Algebraic manipulation of the matrices leads to the effectivereproduction number

R119890 = (120573ℎV120573Vℎ1205992120572ℎ120572V (1 minus 1199061) Λ V120583ℎ

times ( (120572ℎ + 120583ℎ) (120601 + 1205781199062 + 120583ℎ + 120575ℎ)

times(120572V + 120583V + 1205911199063)(120583V + 1205911199063)120583VΛ ℎ)minus1

)

12

(12)

where

R119890V =120573ℎV120572V120599Λ V

120583V (120572V + 120583V + 1205911199063) (120583V + 1205911199063)(13)


is the contribution of themosquito populationwhen it infectsthe humans and

R119890ℎ =120573Vℎ120599120583ℎ120572ℎ (1 minus 1199061)

Λ ℎ (120572ℎ + 120583ℎ) (120601 + 1205781199062 + 120583ℎ + 120575ℎ)(14)

is the human contribution when they infect the mosquitoesThe expression for the effective reproduction number

R119890 has a biological meaning that is readily interpreted fromterms under the square root sign Consider the followingterms

(i) 120573Vℎ120599120572V(120572V+120583V+1205911199063)(120583V+1205911199063) represents the numberof secondary human infections caused by one infectedmosquito vector

(ii) 120573ℎV120599120572ℎ(120572ℎ + 120583ℎ)(120601 + 1205781199062 + 120583ℎ + 120575ℎ) represents thenumber of secondary mosquito infections caused byone infected human host

The square root represents the geometric mean of theaverage number of secondary host infections produced byone vector and the average number of secondary vectorinfections produced by one host This effective reproductionnumber serves as an invasion threshold both for predictingoutbreaks and evaluating control strategies that would reducethe spread of the disease The threshold quantity R119890 mea-sures the average number of secondary cases generated by asingle infected individual in a susceptible human population[18] where a fraction of the susceptible human population isunder prevention and the infected class is under treatmentIn the absence of any protective measure the effectivereproduction numberR119890 with treatment is

R119890119905 = radic120573ℎV120573Vℎ120599

2120572ℎ120572VΛ V120583ℎ

(120572ℎ + 120583ℎ) (120601 + 1205781199062 + 120583ℎ + 120575ℎ) (120572V + 120583V) 1205832VΛ ℎ

(15)

Also if ITNs are the only intervention strategy then

R119890119873 = radic120573ℎV120573Vℎ120599

2120572ℎ120572VΛ V120583ℎ (1 minus 1199061)

(120572ℎ + 120583ℎ) (120601 + 120583ℎ + 120575ℎ) (120572V + 120583V) 1205832VΛ ℎ

(16)

Similarly if IRS is the only means of protection then

R119890119904 = (120573ℎV120573Vℎ1205992120572ℎ120572V120583ℎΛ V

times ((120572ℎ + 120583ℎ) (120601 + 120583ℎ + 120575ℎ) (120572V + 120583V + 1205911199063)

times (120583V + 1205911199063) 120583VΛ ℎ)minus1)12

(17)

The value of the basic reproduction number can be obtainedfrom the value of the effective reproduction number whenthere are no control measures (1199061 = 1199062 = 1199063 = 0)

R0 = radic120573ℎV120573119906ℎ120599

2120572ℎ120572VΛ V120583ℎ

(120572ℎ + 120583ℎ) (120601 + 120583ℎ + 120575ℎ) (120572V + 120583V) 1205832VΛ ℎ

(18)

In general it is easy to prove that

R119890 le R0 (19)

for 0 le 1199061 1199063 le 1 due to reduction of likelihood of infectionby protection This implies that ITNs and IRS have a positiveimpact on the malaria dynamics as they contribute to thereduction of secondary infections Therefore from van denDriessche and Watmough [17] (Theorem 2) the followingresult holds

Lemma 2 The disease-free equilibrium 1198640 of the malariamodel with intervention strategies (1) given by (8) is locallyasymptotically stable ifR119890 lt 1 and unstable ifR119890 gt 1

212 Endemic Equilibrium Point 1198641 When malaria ispresent the model system (1) has a steady state 1198641 calledthe endemic equilibrium In order to establish the stability of1198641 we express system (1) in dimensionless variables namely119878ℎ119873ℎ = 119904 119864ℎ119873ℎ = 119890 119868ℎ119873ℎ = 119894 119877ℎ119873ℎ = 119903 119878V119873V = 119909119864V119873V = 119910 and 119868V119873V = 119911 where 119889119873ℎ119889119905 = Λ ℎminus120583ℎ119873ℎminus120575ℎ119868ℎ119889119873V119889119905 = Λ V minus 120583V119873V Then differentiating with respect totime 119905 respectively results in the following system

119889119904

119889119905=

1

119873ℎ

[119889119878ℎ

119889119905minus 119904

119889119873ℎ

119889119905]

=1

119873ℎ

[Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119894119873ℎ

minus (1 minus 1199061) 120573Vℎ120599119894119904119873ℎ minus 120583ℎ119904119873ℎ + 120595119903119873ℎ]

(20)

Let1198981 = (1minus1205811)120579 1198991 = (120601+1205781199062)(1minus120588) and 119902 = (1minus1199061)120573Vℎ120599Then

119889119904

119889119905=

1

119873ℎ

[Λ ℎ + 1198981119873ℎ + 1198991119894119873ℎ minus 119902119894119904119873ℎ minus119906ℎ119904119873ℎ + 120595119903119873ℎ]

minus119904

119873ℎ

[Λ ℎ minus 120575ℎ119894119873ℎ minus (120583ℎ minus 120579)119873ℎ]

=Λ ℎ

119873ℎ

minus [Λ ℎ

119873ℎ

+ 120579 minus 120575ℎ119894] 119904 + (1 minus 1205811) 120579

+ (120601 + 1205781199062) (1 minus 120588) 119894

minus (1 minus 1199061) 120573Vℎ120599119911119904 + 120595119903

119889119890

119889119905=

1

119873ℎ

[119889119864

119889119905minus 119890

119889119873ℎ

119889119905]

=1

119873ℎ

[(1 minus 1199061) 120573Vℎ120599119911119904119873ℎ + 1205811120579119873ℎ minus120572ℎ119890119873ℎ minus 120583ℎ119890119873ℎ]

minus119890

119873ℎ


= (1 minus 1199061) 120573Vℎ120599119911119904 + 1205811120579 minus [Λ ℎ

119873ℎ

minus 120575ℎ119894 + 120572ℎ + 120579] 119890


119889119894

119889119905=

1

119873ℎ

[119889119868

119889119905minus 119894119889119873ℎ

119889119905]

=1

119873ℎ

[120572ℎ119890119873ℎ minus (120601 + 1205781199062) (1 minus 120588) 119894119873ℎ

minus (120601 + 1205781199062) 120588119894119873ℎ minus (120583ℎ + 120575ℎ) 119894119873ℎ]

minus119894

119873ℎ


= 120572ℎ119890 minus [Λ ℎ

119873ℎ

+ 120601 + 1205781199062 + 120575ℎ + 120579 minus 120575ℎ119894] 119894

119889119903

119889119905=

1

119873ℎ

[119889119877

119889119905minus 119903

119889119873ℎ

119889119905]

=1

119873ℎ

[(120601 + 1205781199062) 120588119894119873ℎ minus (120583ℎ + 120595) 119903119873ℎ]

minus119903

119873ℎ


= (120601 + 1205781199062) 120588119894 minus [Λ ℎ

119873ℎ

+ 120595 + 120579 minus 120575ℎ119894] 119903

119889119909

119889119905=

1

119873V[119889119878V

119889119905minus 119909

119889119873V

119889119905]

=1

119873V[Λ V minus 120573ℎV120599119894119909119873V minus (120583V + 1205911199063) 119909119873V]

minus119909

119873V[Λ V minus 120583V119873V]

= [Λ V

119873Vminus 119909

Λ V

119873V] minus 120573ℎV120599119894119909 minus 1205911199063119909

119889119910

119889119905=

1

119873V[119889119864V

119889119905minus 119910

119889119873V

119889119905]

=1

119873V[120573ℎV120599119894119909119873V minus 120572V119910119873V minus (120583V + 1205911199063) 119910119873V]

minus119910

119873V[Λ V minus 120583V119873V]

= 120573ℎV120599119894119909 minus [Λ V

119873V+ 120572V + 1205911199063]119910

119889119911

119889119905=

1

119873V[119889119868V

119889119905minus 119885

119889119873V

119889119905]

=1

119873V[120572V119910119873V minus (120583V + 1205911199063) 119911119873V]

minus119911

119873V[Λ V minus 120583V119873V]

= 120572V119910 minus [Λ V

119873V+ 1205911199063] 119911

119889119873ℎ

119889119905= [

Λ ℎ

119873ℎ

minus 120583ℎ minus 120575ℎ119894]119873ℎ

119889119873V

119889119905= [

Λ V

119873Vminus 120583V]119873V

(21)

The system can now be reduced to a nine-dimensional systemby eliminating 119904 and 119909 since 119904 = 1minus 119890minus 119894minus 119903 and 119909 = 1minus119910minus119911The following feasible region

Φ1 = 119890 119894 119903119873ℎ 119910 119911119873V isin R7

+| 119890 ge 0 119894 ge 0 119903 ge 0

119890 + 119894 + 119903 le 1 119873ℎ leΛ ℎ

120583ℎ

119910 ge 0 119911 ge 0 119910 + 119911 le 1

119873V leΛ V

120583V

(22)

can be shown to be positively invariant R7+denotes the

nonnegative cone ofR7 including its lower dimensional facesThus we have the following system of equations

119889119890

119889119905= (1 minus 1199061) (1 minus 119890 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579

minus [Λ ℎ

119873ℎ

minus 120575ℎ119894 + 120572ℎ + 120579] 119890

119889119894

119889119905= 120572ℎ119890 minus [

Λ ℎ

119873ℎ

+ 120601 + 1205781199062 + 120575ℎ + 120579 minus 120575ℎ119894] 119894

119889119903

119889119905= (120601 + 1205781199062) 120588119894 minus [

Λ ℎ

119873ℎ

+ 120595 + 120579 + minus120575ℎ119894] 119903

119889119873ℎ

119889119905= Λ ℎ minus 120575ℎ119894119873ℎ minus 120583ℎ119873ℎ

119889119910

119889119905= (1 minus 119910 minus 119911) 120573ℎV120599119894 minus [

Λ V

119873V+ 120572V + 1205911199063] 119910

119889119911

119889119905= 120572V minus [

Λ V

119873V+ 1205911199063] 119911

119889119873V

119889119905= Λ V minus 120583V119873V

(23)

We seek to establish whether a unique endemic equilibriumexists This is done by making more realistic assumptionsnamely that the protective control measures may not betotally effective

Existence and Uniqueness of Endemic Equilibrium 1198641 Tocompute the steady states of the system (23) we set thederivative with respect to time in (23) equal to zero and


after simplification the following algebraic equations areobtained

(1 minus 1199061) (1 minus 119890 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579 = [Λ ℎ

119873ℎ

minus 120575ℎ119894 + 120572ℎ + 120579] 119890

120572ℎ119890 = [Λ ℎ

119873ℎ

+ 120601 + 1205781199062 + 120575ℎ + 120579 minus 120575ℎ119894] 119894

(120601 + 1205781199062) 120588119894 = [Λ ℎ

119873ℎ

+ 120595 + 120579 minus 120575ℎ119894] 119903

Λ ℎ

119873ℎ

= 120575ℎ119894 + 120583ℎ

(1 minus 119910 minus 119911) 120573ℎV120599119894 = [Λ V

119873V+ 120572V + 1205911199063]119910

120572V119910 = [Λ V

119873V+ 1205781199063] 119911

Λ V

119873V= 120583V

(24)

To calculate the dimensionless proportions in terms of 119894consider the first equation in the system (24)

(1 minus 119894 minus 119903) 119901119911 minus 119901119911119890 + 1205811120579 = [120583ℎ + 120572ℎ] 119890

(120583ℎ + 120572ℎ) 119890 + (1 minus 1199061) 120573Vℎ120599119911119890

= (1 minus 1199061) (1 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579

(120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ119911) 119890


119890 =(1 minus 1199061) (1 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579

120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911

(25)

where 119901 = (1 minus 1199061)120573Vℎ120599From the second equation we have

120572ℎ119890 = (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894

120572ℎ119890 = (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894

119890 =(120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894

120572ℎ

(26)

Equating these two equations for 119890 we obtain

(1 minus 1199061) (1 minus 119894 minus 119903) 120573Vℎ120599119911 + 1205811120579120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911

=(120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894

120572ℎ

(1 minus 1199061) 120573Vℎ120599120572ℎ119911119894

+ [120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911 (120583ℎ + 120601 + 1205781199062 + 120575ℎ)] 119894

= (1 minus 1199061) (1 minus 119903) 120573Vℎ120599120572ℎ119911 + 1205811120579120572ℎ

+ 120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911

(27)

Therefore 119894 with 119903 and 119911 becomes

119894 = ((1 minus 1199061) (1 minus 119903) 120573Vℎ120599120572ℎ119911 + 1205811120579120572ℎ

+120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911)

times ((1 minus 1199061) 120573Vℎ120599120572ℎ119911119894

+ [120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911 (120583ℎ + 120601 + 1205781199062 + 120575ℎ)] )minus1

(28)

Now solve for 119903 from the third equation

(120601 + 1205781199062) 120588119894 = (120583ℎ + 120595) 119903

119903 =(120601 + 1205781199062) 119894

120583ℎ + 120595

(29)

Let 1198861 = (120601+1205781199062)(120583ℎ+120595) 1198871 = 1minus1199061 119888 = 120573Vℎ120599120572ℎ 119889 = 1205811120579120572ℎ119892 = 120583ℎ + 120572ℎ ℎ = 120573Vℎ120599 and 119896 = 120583ℎ + 120601 + 1205781199062 + 120575ℎ Thensubstituting 119903 in the 119894 equation we obtain

119894 =1198871 (1 minus 1198861119894) 119888119911 + 119889 + 119892 + 119887ℎ119911

1198871119888119911 + 119892 + 1198871ℎ119896119911

(1198871119888119911 + 119892 + 1198871ℎ119896119911 + 11988711198881198861119911

1198871119888119911 + 119892 + 1198871ℎ119896119911) 119894 =

1198871119888119911 + 119889 + 119892 + 1198871ℎ119911

1198871119888119911 + 119892 + 1198871ℎ119896119911

119894 =1198871119888119911 + 119889 + 119892 + 1198871ℎ119911

1198871119888119911 + 119892 + 1198871ℎ119896119911 + 11988711198881198861119911

(30)

Using equation five of the system (24) we obtain

(120583V + 120572V + 1205911199063) 119910 + 120573ℎV119894119910 = 120573ℎV120599 (1 minus 119911) 119894

119910 =(1 minus 119911) 120573ℎV120599119894

120583V + 120572V + 1205911199063 + 120573ℎV120599119894

(31)

Solving for 119911 gives

120572V119910 = (120583V + 1205911199063) 119911

119911 =120572V119910

120583V + 1205911199063

(32)

Let 1198982 = 120572V120573ℎV120599 1198992 = 120583V + 1205911199063 1199011 = 120583V + 120572V + 1205911199063 and1199021 = 120573ℎV120599 Substituting 119910 in the 119911 equation gives

119911 =(1 minus 119911) 120572V120573ℎV120599119894

(120583V + 1205911199063) (120583V + 120572V + 1205911199063 + 120573ℎV120599119894)

=1198982119894

1198992 (1199011 + 1199021119894)minus

1198982119894119911

1198992 (1199011 + 1199021119894)

=1198982119894

11989921199011 + 11989921199021119894 + 1198982119894

(33)

Substituting 119911 in the 119894 equation gives

119894 =119889 + 119892 + (1198871119888 + 1198871ℎ) (1198982119894 (11989921199011 + 11989921199021119894 + 1198982119894))

119892 + (1198871119888 + 1198871ℎ119896 + 11988711198881198861) (1198982119894 (11989921199011 + 11989921199021119894 + 1198982119894))

(34)


The existence of the endemic equilibrium in Φ can bedetermined when 119894 = 0 After some algebraic manipulationswe obtain

1198601198942+ 119861119894 + 119862 = 0 (35)

where119860 = 11989211989921199021 + 1198921198982 + 11988711198881198982 + 1198871ℎ1198961198982 + 119887111988811988611198982

= (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 120572V120573ℎV120599

+ (1 minus 1199061) 120573Vℎ120573ℎV120572ℎ120572V1205992

+ (1 minus 1199061) (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 120573Vℎ120573ℎV120572V1205992

+ (1 minus 1199061) (120601 + 1205781199062

120583ℎ + 120595)120573Vℎ120573ℎV120572ℎ120572V120599

2

gt 0

119862 = minus11988911989921199011 minus 11989921199011

= minus (120583V + 1205911199063) (120583V + 120572V + 1205911199063) [(1 minus 1199061)]

lt 0

119861 = 11988911989921199021 + 1198891198982 + 11989211989921199021 + 1198921198982 + 1198871ℎ1198982 + 11989211989921199011 minus 11988711198881198982

= (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599

+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 120572V120573ℎV120599

+ (1 minus 1199061) 120573Vℎ120573ℎV120572V1205992

+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) (120583V + 120572V + 1205911199063)

minus 120573ℎV120573Vℎ1205992120572ℎ120572V (1 minus 1199061)

= (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599

+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599

+ (120583ℎ + 120572ℎ) 120572V120573ℎV120599 + (1 minus 1199061) 120573Vℎ120573ℎV120572V1205992

+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) (120583V + 120572V + 1205911199063) (1 minusR2

119890)

(36)

ForR119890 gt 1 the existence of endemic equilibria is determinedby the presence of positive real solutions of the quadraticexpression (35) Consider

119861 = (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599

+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 1205812120579120572V120573ℎV120599

+ (1 minus 1199061) 120573Vℎ120573ℎV120572V1205992+ (120583ℎ + 120572ℎ) (120583V + 1205911199063)

times (120583V + 120572V + 1205911199063) (1 minusR2

119890)

lt 0

(37)

Since 119862 lt 0 and 119860 gt 0 then 119862119860 lt 0 Thus there existsexactly one positive endemic equilibrium for 119894 isin (0 1] when-ever R119890 gt 1 This gives the threshold for the endemic

persistence Therefore we have proved the existence anduniqueness of the endemic equilibrium1198641 for the system (1)This result is summarized in the following theorem

Theorem 3 If R119890 gt 1 then the model system (23) has aunique endemic equilibrium 1198641

The result in Theorem 3 indicates the impossibility ofbackward bifurcation in the malaria model since it has noendemic equilibrium when R119890 lt 1 Thus the model (1)has a globally asymptotically stable disease-free equilibriumwheneverR119890 le 1

Next we need to investigate the global stability property ofthe endemic equilibrium for the case when the disease doesnot induce death

213 Global Stability of the Endemic Equilibrium for 120575ℎ = 0The model system (1) when 120575ℎ = 0 has a unique endemicequilibrium By letting

Φ2 = (119878ℎ 119864ℎ 119868ℎ 119877ℎ 119878V 119864V 119868V) isin R7+ 119864ℎ = 119868ℎ

= 119864V = 119868V = 0 with R1198900 = R119890 | 120575ℎ = 0

(38)

then the following can be claimed

Theorem 4 The endemic equilibrium point of the malariamodel (1)with120575ℎ = 0 is globally asymptotically stablewheneverR1198900 gt 1 in Φ and Φ2

Proof As for the case of Theorem 3 it can be shown that theunique endemic equilibrium for this special case exists onlyif 1198771198900 gt 1 Additionally 119873ℎ = Λ ℎ120583ℎ as 119905 rarr infin Letting119878ℎ = Λ ℎ120583ℎ minus 119864ℎ minus 119868ℎ minus 119877ℎ and 119878V = Λ V120583V minus 119864V minus 119868V andsubstituting into (1) give the limiting system

119889119864ℎ

119889119905= (1 minus 1199061) 120582ℎ (

Λ ℎ

120583ℎ

minus 119864ℎ minus 119868ℎ minus 119877ℎ) +1205811120579Λ ℎ

120583ℎ

minus (120572ℎ + 120583ℎ) 119864ℎ

119889119868ℎ

119889119905= 120572ℎ119864ℎ minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ

119889119864V

119889119905= 120582V (

Λ V

120583Vminus 119864V minus 119868V) minus (120572V + 120583V + 1205911199063) 119864V

119889119868V

119889119905= 120572V119864V minus (120583V + 1205911199063) 119868V

(39)

Dulacrsquos multiplier 1119864ℎ119868ℎ119864V119868V (see [19]) gives

120597

120597119864ℎ

[(1 minus 1199061) 120573Vℎ120599

119864ℎ119868ℎ119864VΛ ℎ120583ℎ(Λ ℎ

120583ℎ

minus 119864ℎ minus 119868ℎ minus 119877ℎ)

+1205811120579Λ ℎ

119864ℎ119868ℎ119864V119868V120583ℎminus(120572ℎ + 120583ℎ)

119868ℎ119864V119868V]

+120597

120597119868ℎ

[120572ℎ

119868ℎ119864V119868Vminus(120601 + 1205781199062 + 120583ℎ + 120575ℎ)

119864ℎ119864V119868V]


+120597

120597119864V[

120573ℎV120599

119864ℎ119864V119868VΛ ℎ120583ℎ(Λ V

120583Vminus 119864V minus 119868V)

minus(120572V + 120583V + 1205911199063)

119864ℎ119868ℎ119864V]

+120597

120597119868V[

120572V

119864ℎ119868ℎ119868Vminus(120583V + 1205911199063)

119864ℎ119868ℎ119864V]

= minus120573Vℎ120599

1198642

ℎ119868ℎ119864V

minus120573Vℎ120599120583ℎ

1198642

ℎ119864V

(1 minus119877ℎ

Λ ℎ120583ℎ

) minus1205811120579Λ ℎ

1198642

ℎ119868ℎ119864V119868V120583ℎ

minus120572ℎ

1198682

ℎ119864V119868V

minus120573ℎV120599

119864ℎ1198642V119868V

[1 minus120583ℎ

Λ ℎ

] minus120572V

119864ℎ119868ℎ1198682V

lt 0

(40)

since 119864ℎ + 119868ℎ + 119877ℎ lt Λ ℎ120583ℎ and 119864V + 119868V lt Λ V120583V in Φ2Thus by Dulacrsquos criterion there are no periodic orbits

in Φ and Φ2 Since Φ2 is positively invariant and theendemic equilibrium exists whenever R1198900 gt 1 then fromthe Poincare-Bendixson Theorem [20] it follows that allsolutions of the limiting system originating in Φ remain inΦ for all 119905 Furthermore the absence of periodic orbits in Φimplies that the unique endemic equilibrium of the specialcase of the malaria model is globally asymptotically stablewheneverR1198900 gt 1

The malaria model has a locally asymptotically stabledisease-free equilibrium whenever R119890 lt 1 and a uniqueendemic equilibrium whenever R119890 gt 1 In addition theunique endemic equilibrium is globally asymptotically stablefor the case 120575ℎ = 0 ifR1198900 gt 1

In the next section we apply the optimal control methodusing Pontryaginrsquos Maximum Principle to determine thenecessary conditions for the combined optimal control ofITNs IRS and treatment effort which are being practised inKaronga District Malawi

3 Analysis of Optimal Control ofthe Malaria Model

The force of infection in the human population is reduced bya factor of (1 minus 1199061(119905)) where 1199061(119905) represents the use of ITNsas a means of minimizing or eliminating mosquito-humancontact 1199062(119905) isin [0 1] represents the control effort on treat-ment of infectious individuals This indeed represents thesituation when individuals in the community seek treatmentafter visiting the hospitals or dispensary in their areas For themosquito population we have a third control variable 1199063(119905)IRS affects the whole mosquito population by increasing itsmortality rate by 1199063(119905) We will use an approach similar tothat in Lashari and Zaman [21] which consists of applyingPontryaginrsquosMaximumPrincipal to determine the conditionsunder which eradication of the disease can be achieved infinite time Following the dynamics of the model system (1)

with appropriate initial conditions the bounded Lebesguemeasurable control is used with the objective functionaldefined as

Γ (1199061 1199062 1199063) = int

119879119891

0

(1198621119864ℎ + 1198622119868ℎ + 1198623119873V

+1

2(11986011199062

1+ 1198602119906

2

2+ 1198603119906

2

3)) 119889119905

(41)

subject to the differential equations in (1) where 1198621 1198622 11986231198601 1198602 1198603 are positive weights We choose a quadratic coston the controls in line with what is known in the literature onepidemic controls [21ndash23]

The purpose of this section is to minimize the costfunctional (41) This functional includes the exposed andinfectious human population and the total mosquito pop-ulation In addition it has the cost of implementing per-sonal protection using ITNs 1198601119906

2

1 treatment of infected

individuals 11986021199062

2 and spraying of houses 1198603119906

2

3 A lin-

ear function has been chosen for the cost incurred byexposed individuals1198621119864ℎ infected individuals1198622119868ℎ and themosquito population 1198623119873V A quadratic form is used for thecost on the controls 1198601119906

2

1 1198602119906

2

2 and 1198603119906

2

3 such that the

terms (12)11986011199062

1 (12)1198602119906

2

2 and (12)1198603119906

2

3describe the cost

associated with the ITNs treatment and IRS respectivelyWeselect to model the control efforts via a linear combinationof quadratic terms 1199062

119894(119905) (119894 = 1 2 3) and the constants

119862119894 and 119860 119894 (119894 = 1 2 3) representing a measure of therelative cost of the interventions over the time horizon (0 119879119891)We seek an optimal control 119906lowast

1(119905) 119906lowast

2(119905) and 119906

lowast

3(119905) such

that

Γ (119906lowast

1 119906lowast

2 119906lowast

3) = min(119906111990621199063)isinΦ2

Γ (1199061 1199062 1199063) (42)

where

Φ2 = 119906 = (1199061 1199062 1199063) | 119906119894 (119905) is Lebesguemeasurable

0 le 119906 (119905) le 1 for 119905 isin [0 119879119891] 997888rarr [0 1] 119894 = 1 2 3

(43)

is the control set subject to the system (1) and appropriate ini-tial conditions We develop the optimal system for which thenecessary conditions it must satisfy come from PontryaginrsquosMaximum Principle [24]

31 Existence of an Optimal Control Problem PontryaginrsquosMaximum Principal converts (1) and (41) into a problem ofminimizing pointwise the Lagrangian 119871 and Hamiltonian119867 with respect to 1199061 1199062 and 1199063 The Lagrangian of thecontrol problem is given by

119871 = 1198621119864ℎ + 1198622119868ℎ + 1198623119873V +1

2(11986011199062

1+ 1198602119906

2

2+ 1198603119906

2

3)

(44)


We search for the minimal value of the Lagrangian Thiscan be done by defining the Hamiltonian 119867 for the controlproblem as

119867 = 119871 + 1205821

119889119878ℎ

119889119905+ 1205822

119889119864ℎ

119889119905+ 1205823

119889119868ℎ

119889119905+ 1205824

119889119877ℎ

119889119905

+ 1205825

119889119878V

119889119905+ 1205826

119889119864V

119889119905+ 1205827

119889119868V

119889119905

= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V +1

2(11986011199062

1+ 1198602119906

2

2+ 1198603119906

2

3)

+ 1205821 [Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119868ℎ

minus (1 minus 1199061) 120573Vℎ120599119868V119878ℎ minus 120583ℎ119878ℎ + 120595119877ℎ]

+ 1205822 [(1 minus 1199061) 120573Vℎ120599119868V119878ℎ + 1205811120579119873ℎ minus (120572ℎ + 120583ℎ) 119864ℎ]

+ 1205823 [120572ℎ119864ℎ minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ]

+ 1205824 [(120601 + 1205781199062) 120588119868ℎ minus (120583ℎ + 120595)119877ℎ]

+ 1205825 [Λ V minus 120573ℎV120599119868ℎ119878V minus (120583V + 1205911199063) 119878V]

+ 1205826 [120573ℎV120599119868ℎ119878V minus (120572V + 120583V + 1205911199063) 119864V]

+ 1205827 [120572V119864V minus (120583V + 1205911199063) 119868V]

(45)

Next we prove the existence of an optimal control for themodel system (1)

Theorem 5 Themodel system (1)with the initial conditions at119905 = 0 has control strategies and there exists an optimal controllowast= (119906lowast

1 119906lowast

2 119906lowast

3) isin Φ2 such that

min(119906111990621199063)isinΦ2

Γ (1199061 1199062 1199063) = Γ (119906lowast

1 119906lowast

2 119906lowast

3) (46)

Proof The state and the control variables of the system (1) arenonnegative values The control set Φ2 is closed and convexThe integrand of the objective cost function Γ expressed by (1)is a convex function of (1199061 1199062 1199063) on the control set Φ2 TheLipschitz property of the state systemwith respect to the statevariables is satisfied since the state solutions are bounded Itcan easily be shown that there exist positive numbers 1205851 1205852and a constant 120598 gt 1 such that

Γ (1199061 1199062 1199063) ge 1205851(100381610038161003816100381611990611003816100381610038161003816

2+100381610038161003816100381611990621003816100381610038161003816

2100381610038161003816100381611990631003816100381610038161003816

2)1205982

minus 1205852(47)

This concludes the existence of an optimal control becausethe state variables are bounded

32 Classification of the Optimal Control Problem We usePontryaginrsquos Maximum Principle to develop the necessaryconditions for this optimal control since there exists anoptimal control for minimizing the functional (41) subjectto the system of equations in (1) From Lashari and Zaman[21] if (120594 119906) is an optimal solution of an optimal control

problem then there exists a nontrivial vector function 120582 =

(1205821 1205822 1205823 120582119899) satisfying the following equations

0 =120597119867 (119905 120594 119906 120582)

120597119906

1205821015840=120597119867 (119905 120594 119906 120582)

120597120594

119889120594

119889119905= minus

120597119867 (119905 120594 119906 120582)

120597120582

(48)

Hence the necessary conditions of the Hamiltonian 119867 canbe applied in (45)

Theorem 6 For the optimal control triple (119906lowast1 119906lowast

2 119906lowast

3) with

their optimal state solutions (119878lowastℎ 119864lowast

ℎ 119868lowast

ℎ 119877lowast

ℎ 119878lowast

V 119864lowast

V 119868lowast

V ) thatminimizes Γ(1199061 1199062 1199063) over Φ2 then there exist adjointvariables (1205821 1205822 1205823 1205824 1205825 1205826 1205827) satisfying

minus1205821015840

1= (1 minus 1205811) 1205791205821 + (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119868V

minus 120583ℎ1205821 + 12057912058111205822

minus1205821015840

2= (1 minus 1205811) 1205791205821 + 12057912058111205822 + 120572ℎ (1205823 minus 1205822) minus 120583ℎ1205822 + 1198621

minus1205821015840

3= (1 minus 1205811) 1205791205821 + (120601 + 1205781199062) [(1 minus 120588) 1205821 + 1205881205824]

+ 12057912058111205822 minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 1205823

+ 120573ℎV120599 (1205826 minus 1205825) 119878V + 1198622

minus1205821015840

4= (1 minus 1205811) 1205791205821 + 1205951205821 + 12057912058111205822 minus (120583ℎ + 120595) 1205824

minus1205821015840

5= 120573ℎV120599 (1205826 minus 1205825) 119868ℎ minus (120583V + 1205911199063) 1205825 + 1198623

minus1205821015840

6= 120572V (1205827 minus 1205826) minus (120583V + 1205911199063) 1205826 + 1198623

minus1205821015840

7= (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119878ℎ minus (120583V + 1205911199063) 1205827 + 1198623

(49)

with transversality conditions

1205821 (119879119891) = 1205822 (119879119891) = 1205823 (119879119891) = 1205824 (119879119891)

= 1205825 (119879119891) = 1205826 (119879119891) = 1205827 (119879119891) = 0

(50)

Additionally the optimal control triple (119906lowast1 119906lowast

2 119906lowast

3) that mini-

mizes Γ over Φ2 satisfies the optimality condition

119906lowast

1= max0min(1

120573Vℎ120599 (1205822 minus 1205821) 119868lowast

V 119878lowast

ℎ

1198601

)

119906lowast

2= max0min(1

120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868lowast

ℎ

1198602

)

119906lowast

3= max0min(1

120591 (1205825119878lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

)

(51)

Proof The adjoint equations can be determined by usingthe differential equations governing the adjoint variables


The Hamiltonian function 119867 is differentiated with respectto 119878ℎ 119864ℎ 119868ℎ 119877ℎ 119878V 119864V and 119868V The adjoint equation is givenby

minus1198891205821

119889119905=120597119867

120597119878ℎ

= (1 minus 1205811) 1205791205821 + (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119868V

minus 120583ℎ1205821 + 12057912058111205822

minus1198891205822

119889119905=120597119867

120597119864ℎ

= (1 minus 1205811) 1205791205821 + 12057912058111205822

+ 120572ℎ (1205823 minus 1205822) minus 120583ℎ1205822 + 1198621

minus1198891205823

119889119905=120597119867

120597119868ℎ

= (1 minus 1205811) 1205791205821

+ (120601 + 1205781199062) [(1 minus 120588) 1205821 + 1205881205824] + 12057912058111205822

minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 1205823 + 120573ℎV120599 (1205826 minus 1205825) 119878V + 1198622

minus1198891205824

119889119905=120597119867

120597119877ℎ

= (1 minus 1205811) 1205791205821 + 1205951205821 + 12057912058111205822 minus (120583ℎ + 120595) 1205824

minus1198891205825

119889119905=120597119867

120597119878V= 120573ℎV120599 (1205826 minus 1205825) 119868ℎ minus (120583V + 1205911199063) 1205825 + 1198623

minus1198891205826

119889119905=120597119867

120597119864V= 120572V (1205827 minus 1205826) minus (120583V + 1205911199063) 1205826 + 1198623

minus1198891205827

119889119905=120597119867

120597119868V= (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119878ℎ

minus (120583V + 1205911199063) 1205827 + 1198623

(52)

with the transversality conditions

1205821 (119879119891) = 1205822 (119879119891) = 1205823 (119879119891) = 1205824 (119879119891)

= 1205825 (119879119891) = 1205826 (119879119891) = 1205827 (119879119891) = 0

(53)

Solving 1205971198671205971199061 = 0 1205971198671205971199062 = 0 and 1205971198671205971199063 = 0evaluating at the optimal control on the interior of the controlset where 0 lt 119906119894 lt 1 for 119894 = 1 2 3 and letting 119878ℎ = 119878

lowast

ℎ

119864ℎ = 119864lowast

ℎ 119868ℎ = 119868

lowast

ℎ 119877ℎ = 119877

lowast

ℎ 119878V = 119878

lowast

V 119864V = 119864lowast

V and 119868V = 119868lowast

Vyields

120597119867

1205971199061

= 119860119906lowast

1+ 120573Vℎ1205991205821119868

lowast

V 119878lowast

ℎminus 120573Vℎ1205991205822119868

lowast

V 119878lowast

ℎ= 0

120597119867

1205971199062

= 11986021199062 + (1 minus 120588) 1205781205821119868lowast

ℎminus 1205781205823119868

lowast

ℎ+ 1205781205881205824119868

lowast

ℎ= 0

120597119867

1205971199063

= 1198603119906lowast

3minus 1205911205825119878

lowast

V minus 1205911205826119864lowast

V minus 1205911205827119868lowast

V = 0

(54)

for which

119906lowast

1=120573Vℎ120599 (1205822 minus 1205821) 119868

lowast

V 119878lowast

ℎ

1198601

119906lowast

2=120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868

lowast

ℎ

1198602

1199063 =120591 (1205825119878

lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

(55)

Then

119906lowast

1= max0min(1

120573Vℎ120599 (1205822 minus 1205821) 119868lowast

V 119878lowast

ℎ

1198601

)

119906lowast

2= max0min(1


ℎ

1198602

)

119906lowast

3= max0min(1

120591 (1205825119878lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

)

(56)

We achieve the uniqueness of the optimal control for small119879119891 due to the prior boundedness of the state and adjointfunctions and the resulting Lipschitz structure of the ordinarydifferential equations The uniqueness of the optimal controltriple trails from the uniqueness of the optimal system whichconsists of (1) (49) and (50) with characterization of theoptimal control (51)

The optimality system is comprised of the state system(1) the adjoint system (49) initial conditions at 119905 = 0boundary conditions (50) and the characterization of theoptimal control (51) Hence the state and optimal control canbe calculated using the optimality system Hence using thefact that the second derivatives of the Lagrangianwith respectto 1199061 1199062 and 1199063 respectively are positive indicates that theoptimal problem is a minimum at controls 119906lowast

1 119906lowast2 and 119906lowast

3

Substituting 119906lowast1 119906lowast2 and 119906lowast

3in the system (1) we obtain

119889119878lowast

ℎ

119889119905= Λ ℎ + (1 minus 1205811) 120579119873

lowast

ℎ

+ (120601 + 1205781199062) (1 minus 120588) 119868lowast

ℎminus 120573Vℎ120599119868

lowast

V 119878lowast

ℎ

times (1minusmax0min(1120573Vℎ120599 (1205822minus1205821) 119868

lowast

V 119878lowast

ℎ

1198601

))

minus 120583ℎ119878lowast

ℎ+ 120595119877lowast

ℎ

119889119864lowast

ℎ

119889119905= 120573Vℎ120599119868

lowast

V 119878lowast

ℎ

times(1 minusmax0min(1120573Vℎ120599 (1205822minus1205821) 119868

lowast

V 119878lowast

ℎ

1198601

))

+ 1205811120579119873lowast

ℎminus 120572ℎ119864

lowast

ℎminus 120583ℎ119864

lowast

ℎ


119889119868lowast

ℎ

119889119905

= 120572ℎ119864lowast

ℎ

minus (120601 + 120578

times(max0min(1120578 (1205823+120588 (1205821minus1205824)minus1205821) 119868

lowast

ℎ

1198602

))

+120583ℎ + 120575ℎ) 119868lowast

ℎ

119889119877lowast

ℎ

119889119905= (120601 + 120578 (max 0min (1 120578 (1205823 + 120588 (1205821minus1205824)minus1205821)

times 119868lowast

ℎtimes (1198602)

minus1))) 120588119868ℎ

minus (120583ℎ + 120595)119877ℎ

119889119878lowast

V

119889119905

= Λ V minus (120583V + 120591

times (max 0min (1 120591 (1205825119878lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

times(1198603)minus1))) 119878

lowast

V minus 120573ℎV120599119868lowast

ℎ119878lowast

V

119889119864lowast

V

119889119905= 120573ℎV120599119868

lowast

ℎ119878lowast

V

minus (120583V + 120591 (max 0min (1 120591 (1205825119878lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

times(1198603)minus1))) 119864

lowast

V minus 120572V119864lowast

V

119889119868lowast

V

119889119905= 120572V119864V


V + 1205826119864lowast

V +1205827119868lowast

V )

times(1198603)minus1))) 119868

lowast

V

(57)

with119867lowast at (119905 119878lowastℎ 119864lowast

ℎ 119868lowast

ℎ 119877lowast

ℎ 119906lowast

1 119906lowast

2 119906lowast

3 1205821 1205822 1205827)

119867lowast

= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V

+1

2(1198601(max0min(1

120573Vℎ120599 (1205822 minus 1205821) 119868lowast

V 119878lowast

ℎ

1198601

))

2

+ 1198602 (max 0min (1 120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868lowast

ℎ

times(1198602)minus1))2

+1198603(max0min(1120591(1205825119878

lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

))

2

)

+ 1205821

119889119878lowast

ℎ

119889119905+ 1205822

119889119864lowast

ℎ

119889119905+ 1205823

119889119868lowast

ℎ

119889119905+ 1205824

119889119877lowast

ℎ

119889119905+ 1205825

119889119878lowast

V

119889119905

+ 1205826

119889119864lowast

V

119889119905+ 1205827

119889119868lowast

V

119889119905

(58)We solve the systems (57) and (58) numerically to determinethe optimal control and the state

33 Numerical Results on Optimal Control Analysis In thissection we discuss the method and present the results ob-tained from solving the optimality system numerically usingthe parameter values in Table 11

Here we consider the optimal control values of the threeintervention strategies namely ITNs IRS and treatmentwhich are common strategies in Karonga District Malawi InFigure 2(a) we see that if the three intervention strategies areeffectively implemented and used they have a positive impactcompared to having ITNs and IRS (1199061 and 1199063) respectivelyas the only intervention strategies in the community Theinitial increase in the infected human population in thegraph with interventions of ITNs and IRS may be due to thefact that some people refused to have their houses sprayedwith insecticide chemicals due to their primitive traditionalbeliefs In addition as Karonga District is along the shore ofLake Malawi some members of the community do not useITNs owing to negative beliefs in the chemicals used and alsodue to hot weather in the districts On the other hand thedistrict is waterlogged and hence this leads to an increase inmosquitoes breeding sites

Similar results appear in Figure 2(b) where the impact ofthe intervention strategies are compared with the use of ITNs(1199061) and treatment (1199062) The results show that the concurrentadministered intervention strategies lead to a decrease inthe number of infected human population much faster thanwhen ITNs and treatment are used as the only interventionstrategies in the community A similar occurrence is observedwhen IRS (1199063) and treatment (1199062) are used as the only meansof intervention strategies in the community (see Figure 2(c))

In addition we also looked at the effects of these inter-vention measures as a stand-alone approach of preventingor controlling malaria disease in the community Figure 2(d)depicts the comparison of the effects of each interventionstrategy and the effect of a multi-intervention strategy Thefigure indicates that if the three combined intervention mea-sures are effectively practised the infected human populationis much lower compared to a situation where only oneintervention strategy is used The graph of treatment (1199062)practised as the only means of intervention measure showsa high number of infected human population owing to anumber of reasonsOne of the reasons is that in this situationthe mosquito population is unaffected hence the infectedmosquitoes will still be available in the community causingmore infections to susceptible humans Furthermore mostpeople in the area do not visit the dispensary or hospitalfor medication when they observe signs or symptoms ofmalaria disease since they need to cover long distances toreach the hospital The interviews conducted revealed that


u1 ne 0 u3 ne 0 u2 = 0

u1 ne 0 u2 ne ne0 u3 0

0

10

20

30

40

50

60

70

times102In

fect

ed h

uman

pop

ulat

ion

0 10 20 30 40 50 60 70 80 90 100

Time (years)

(a)

Constant control u1 ne 0 u2 ne ne0 u3 0

=Constant control u1 ne 0 u2 ne 0 u3 0In

fect

ed h

uman

pop

ulat

ion

0

10

20

30

40

50

60

70

0 10 20 30 40 50 60 70 80 90 100

times102

Time (days)

(b)

u2 ne 0 u3 ne 0 u1 = 0


0 10 20 30 40 50 600

5

10

15

20

25

30

35

40

times102

Infe

cted

hum

an p

opul

atio

n

Time (years)

(c)

u1 ne 0 u2 ne 0 u3 0ne ==u2 ne 0 u1 0 u3 0

=u1 ne 0 u2 0 u3 = 0 = =u3 ne 0 u1 0 u2 0

0

05

1

15

2

25

3

times104

0 10 20 30 40 50 60 70

Infe

cted

hum

an p

opul

atio

n

Time (years)

(d)

Figure 2 Simulations of model (1) showing the effects of intervention measures

some individuals opt to simply use the medication left by theprevious patient or they buy medicines from the shops anduse it before being diagnosed Consequently treatment needsto be consolidatedwith preventivemeasures such as ITNs andIRS for optimal control

The epidemiological implication of the above result isthat malaria could be eliminated from the community ifprevention and treatment can lead to a situation where R119890is less than unity However other factors need to be consid-ered


Table 4 Major diseases in the community-malaria (malaria transmission-bite from an infected mosquito cross tabulation)

Malaria transmission-bite from an infected mosquito TotalNo Yes

Major diseases in the community-malariaNo Count 11 15 26

of total 22 30 53

Yes Count 73 393 466 of total 148 799 94

Total Count 84 408 492 of total 171 829 1000

4 Analysis of Malaria Data fromKaronga District

A structured questionnaire was developed and administeredin KarongaDistrictMalawi in order to determine how inter-vention strategies of malaria disease are being practised andtheir effectiveness The questionnaire was used to conducta directed one to one interview on the respondents whowere randomly sampled with total size of 502 which meansthat 502 questionnaires were administered The enumeratorsreceived training before they went to the field where thequestionnaire was discussed with the respondents

We consider statistical results of how intervention strate-gies are practised Different graphs and tables are depicted forall the prevention and treatment strategies

Knowledge of malaria transmission is a key to malariaprevention [25] Figure 3 shows that 409 respondents (815)stated that malaria transmission occurs when bitten by aninfected mosquito contrary to popular belief in developingcountries (UNICEF (2000) [26]) that someonemay be infect-ed with malaria when soaked in water where most respon-dents (968) answered ldquonordquo Table 4 shows the number ofindividuals who mentioned that malaria disease and malariatransmission are through a bite from an infected mosquito799 responded ldquoyesrdquo to both malaria being a major diseaseandmalaria transmission being acquired through a bite froman infected mosquito

According to the Center for Disease Control and Pre-vention [27] prevention is better than cure and there are anumber of methods which people can use to prevent malariaMost of the respondents (904) stated that they use ITNsor long lasting insecticide treated bed nets (LLITNs) as amethod of preventing occurrence of malaria Only 165of the respondents mentioned that their houses are sprayed(IRS) (see Figure 4) From Table 5 there are about 1 casesof malaria occurrences and about 22 respondents had notexperienced any malaria cases after spraying their houseswith IRS Some of the respondents used nets as well as spray-ing their houses Table 6 shows that 355 of the respondentshad experienced malaria occurrence after their houses weresprayed and also used the bed nets From the respondentswho had been using ITNs or LLITNs 449 did not haveany malaria occurrence However about 459 had had anoccurrence ofmalaria despite the use of bednets (seeTable 7)

409

93 53 47 7 4484

400440 446

486449

9 9 9 9 9 90

100

200

300

400

500

600

Bite

from

anin

fect

edm

osqu

ito

Fem

ale

mos

quito

Stag

nant

wat

er

Envi

ronm

ent

man

agem

ent

Oth

er

YesNo

Malaria transmission

wat

erso

aked

in ra

inRa

ined

on

Anopheles

Do not know

Figure 3 Knowledge of malaria transmission

Seek treatment

Methods of preventing malaria

47

418

493

462

409

454

83

8

38

92

0 100 200 300 400 500 600

Other

YesNo

IPT for pregnantwomen

House sprayed(IRS)

Use of ITNor LLN

Figure 4 Methods of preventing malaria

A chi-square test of independence was conducted witha chi-square value of 1454 since the cross tabulation is atwo by two table The assumption of no cells having anexpected value less than 5 was not violated hence the useof chi-square test Table 8 shows that a calculated value of1454 was obtained The level of significance used for the chi-square test was 005 which was compared to a 119875 value of0228 Since 0228 gt 005 the null hypothesis was rejected


Table 5 Malaria occurrence after spraying and without using net

Frequency PercentNo 11 22Yes 5 10No response 486 968Total 502 1000

Table 6 Malaria occurrence after spraying and using net


hence there was an association between malaria occurrenceand preventive methods (use of ITNs or LLITNs) and theirproportions are not significantly different In Table 10 119886indicates the number of cells with expected count less than 5while 119887 shows Yates continuity correction which is calculatedfor 2 by 2 table The relationship of indoor residual spraying(IRS) against the occurrence of malaria after the house wassprayed and use of bed nets was examined From Table 9for the respondents who had not had their houses sprayed425 did not have an occurrence of malaria while for thoserespondents who had their houses sprayed 384 had anoccurrence ofmalaria A chi-square test of independence wasconductedwith a chi-square value of 0019The assumption ofno cells having an expected value less than 5 was not violatedhence the use of chi-square testThe level of significance usedfor the chi-square test was 005 which was compared to a 119875value of 0889 Table 10 shows a 119875 value of 0889 gt 005which means there was an association between preventivemethod (IRS) and malaria occurrence after spraying andusing ITNs but however there were no significant differencesin proportions between preventive methods and malariaoccurrence after spraying

Despite knowledge of malaria transmission and preven-tion malaria cases still occur [28] About 50 of the respon-dents who used ITNs mentioned that malaria still occursThis is probably because bed nets are only used when goingto bed hence the vulnerability Furthermore the proportionof those respondents who used ITNs or LLITNs and sufferedfrom malaria was not significantly different from those whodid not use ITNs or LLITNs but suffer from malaria (1205942 =1454 calculated 119875 value = 0228 level of significance of005) Of the respondents who had had their houses sprayed(about 32) 1 experienced an occurrence of malariaThose respondentswhohad their houses sprayed and sufferedfrom malaria even though they had used ITNs or LLITNsshowed no significant difference with those respondents whohad had their houses sprayed and had not suffered frommalaria even though they had used ITNs or LLITNs (1205942 =0019 calculated119875 value = 0889 level of significance of 005)

5 Numerical Results

The numerical simulations and analysis were carried outusing a fourth order Runge-Kutta scheme in Matlab Ouraim was to determine and verify the analytic results and thestability of themodel system (1) Someof the parameter valueswere calculated from the data collected in Karonga DistrictMalawi between themonths of January and September 2013The other parameter values were obtained from the NationalStatistical Office (NSO) in Zomba Malawi some have beenassumed and very few have been taken from the literatureThe Government of Malawi has organized a number ofintervention strategies in order to fight against malaria inthe country through the National Malaria Control Program(NMCP) [29]

51 Dynamics of Human State Variables for theMalariaModelwithout Intervention Strategies The analysis of the modelwithout intervention strategies was carried out in order todetermine the dynamics of the disease in the populationThe simulation was generated in a four-year time framesince the first campaign of malaria intervention strategiesin Karonga District was performed in the year 2010 Thesusceptible human population is decreasing exponentially(see Figure 5(a)) showing that most susceptible humans areexposed to the disease due to unavailability of interventionstrategies This has led to an exponential increase in theexposed human population (Figure 5(b)) and the infectedpopulation (Figure 5(c)) with R0 = 18894 The infectedhuman population increases due to an increase in the expo-sure of susceptible individuals to Plasmodium falciparumThis means that the Plasmodium falciparum will continueto multiply in the human and mosquito populations sincethere are no intervention strategies to reduce or eradicatethe disease Hence there is a need of having interventionstrategies in order to reduce or eradicate the disease

52 Prevalence in the Malaria Model without InterventionStrategies Prevalence is defined as the ratio of the numberof cases of the disease in a population to the total numberof individuals in population at a given time The diseaseprevalence in Figure 6 shows a steady increase during the firstdays of infection due to a high number of individuals withoutPlasmodium falciparum The graph drops asymptotically dueto a reduced number of susceptible human populationshowing evidence that communities can be wiped out withmalaria disease if none of the intervention methods arespeedily put into place

53 Dynamical System of the Individual Population StateVariables in the Model with Intervention Strategies We nowconsider the effects of the three intervention strategies (ITNsIRS and treatment) which are campaigned concurrentlyin Karonga District Malawi It appears that if the threeintervention strategies are effectively monitored and imple-mented then there is a positive impact of combating malariain the community In Figures 7(a) and 7(b) an increase isobserved in the human population recovering comparedwith


R0 = 18894

0 05 1 15 2 25 3 35 4

Time (years)

0

20

40

60

80

100

120

Susc

eptib

le in

divi

dual

s

Susceptible individualstimes102

(a)

R0 = 18894

0 05 1 15 2 25 3 35 4

Time (years)

0

20

40

60

80

100

120

140

times102

Expo

sed

indi

vidu

als

Exposed individuals

(b)

R0 = 18894

0 05 1 15 2 25 3 35 49

10

11

12

13

14

15

16

17

Time (years)

Infe

cted

indi

vidu

als

Infected individuals without any intervention strategiestimes102

(c)

0 100 200 300 400 500 600 7000

1

2

3

4

5

6

7

8

Time (days)

Reco

vere

d in

divi

dual

s

Recovered individuals without any intervention strategies

R0 = 18894

times102

(d)

Figure 5The changes in the four state variables of the malaria model without intervention strategies illustrating the dynamics with time of(a) susceptible individuals (b) exposed individuals (c) infected individuals and (d) recovered human individuals

the exposed and infected individuals This steady increasein the recovery class is due to the readily available effec-tive prevention strategies and treatment Figure 7(a) wherehuman population is plotted against time on a four-yearperiod shows a steady increase in the recovery of the humanpopulation It is evidenced that during this period chemicalswhich are available in the mosquito nets and sprayed in thehouses are effective in reducing the mosquito population and

at the same time reducing the contact rate between the humanpopulation and the mosquito population Treatment strategyhas also played an important role in reducing the number ofinfected individuals thus leading to an increase in recoveredindividuals

A slight decrease in the gradient of the recovered humanpopulation is observed in Figure 7(b) as the period of usingprevention and treatment strategies available is increased


Table 7 Method of preventing malaria use of ITN or LLITN (use of net and malaria occurrence cross tabulation)

Use of net and malariaTotalOccurrence

No Yes

Method of preventing use of ITN or LLITNNo Count 17 27 44

of total 36 57 92



Time (days)

R0 = 18894

Prev

alen

ce

0 20 40 60 80 100 120 140 160 180 2000

01

02

03

04

05

06

07

Figure 6 Exposed humans without any intervention strategies

Despite being given ITNs and spraying long lasting chemicalsin the houses the campaign needs to be revisited to confirmwhether the chemicals are still effective The decrease of thegraph explains that the effectiveness of the treated bed netsand the residual spraying deteriorates with time Hence thereis need to assess the right time interval to carry out therespraying of the houses and the resupply of ITNs

6 Conclusions

An optimal control model (using a deterministic systemof nonlinear ordinary differential equations) for the trans-mission dynamics of malaria in Karonga District Malawiwas presented The model considered a varying total humanpopulation that incorporated recruitment of new individualsinto the susceptible class through birth or immigrationand those immigrant individuals who were exposed to thedisease were recruited into the exposed individual class Theprevention (IRS and ITNs) and other treatment interventionstrategies were included in the model to assess the potential

Table 8 Chi-square test of use of ITN or LLITN and use of net andmalaria occurrence

Value df Asymp sig (2-sided)Pearson chi-square 1861

119886 1 0173Continuity correctionb 1454 1 0228

impact of these strategies on the transmission dynamics ofthe disease

Our model incorporated features that were potentiallyeffective to control or reduce the transmission of malariadisease in Malawi Analysis of the optimal control modelrevealed that there exists a domain where the model isepidemiologically and mathematically well-posed We alsocomputed the effective reproduction number R119890 and thenqualitatively analyzed the existence and stability of theirmodel equilibria The basic reproduction number R0 wasobtained from the threshold reproduction number by elimi-nating all the intervention strategies Then it was proved thatifR119890 lt 1 the disease cannot survive in the district Hence theeffective reproduction numberR119890 is an essential indicationof the effort required to eliminate the disease It was alsofound thatR119890 le R0 which implied that increased preventiveand control intervention practices had a positive impact onthe reduction of R119890 Thus malaria can be eradicated inthe district by deployment of a combination of interventionstrategies such as effective mass drug administration andvector control (LLITNs and IRS) to combat and eventuallyeliminate the disease

Analysis of the model supported that effective control oreradication of malaria can be achieved by the combinationof protection and treatment measures We have seen thatwhen the three intervention strategies are combined thereis a greater reduction in the number of exposed and infectedindividuals The prevention strategies played a greater role inreducing the number of infected individuals by lowering thecontact rate between the mosquito and human populationsfor instance through the use of ITNs On the other hand bothprevention strategies led to the reduction of the mosquitopopulation hence lowering the infectedmosquito populationEffective treatment consolidated the prevention strategiesThis study provides useful tools for assessing the effectiveness


Table 9 Method of preventing malaria house sprayed IRS (malaria occurrence after spraying and using net cross tabulation)

Malaria occurrence after spraying and using net TotalNo Yes

Method of preventing house spread (IRS)No Count 158 143 301

of total 425 384 809



Hum

an p

opul

atio

n

Recovered individualsExposed individualsInfected individuals

times104

Time (years)0 05 1 15 2 25 3 35 4

0

05

1

15

2

25

(a)

times105

Hum

an p

opul

atio

n


Time (years)0 10 20 30 40 50 60

0

02

04

06

08

1

12

14

16

18

2

(b)

Figure 7 The dynamics of the exposed and infected individuals in the model with intervention strategies forR119890 = 06922

of a combination of the three intervention strategies andanalyzing the potential impact of prevention with treatment

Demographic findings also showed that the preventivemeasures namely ITNs and IRS if effectively practised canhelp to reduce malaria transmission These primary healthintervention strategies are very important as they reduce themosquito population and contact between the human andmosquito populationsThese practiceswill lead to a reductionin the transfer of Plasmodium between the host and thevectorHoweverDzinjalamala [30] states thatmalaria controlin Malawi is still heavily reliant on chemotherapy Hence theapproach needs to change and effectively accommodate thecampaign strategy taking place inKarongaDistrict in order tocombat the disease Therefore the presumptive treatment forfever and the primary health intervention practices (LLITNsand IRS) should both be effectively implemented or practisedin order to reduce or eliminate malaria disease

Using the optimal values of the three intervention prac-tices (LLITNs IRS and treatment) the results showed thatthe combination of the three intervention strategies has apositive and greater impact in eliminating or reducing theepidemic of malariaThis can be achieved when themeasuresare effectively implemented by the suppliers and effectivelypractised by the beneficiaries (the community members)

To effectively control and potentially eradicate the spreadof malaria treatment programs must be complemented withother intervention strategies such as vector reduction andpersonal protection Intervention practices that involve bothprevention and treatment controls yield relatively betterresultsThe combination of these strategies can play a positiverole in Karonga District in reducing or eradicating malariadisease Therefore control and prevention efforts aimed atlowering the infectivity of infected individuals to themosqui-to vector will contribute greatly to the reduction of malaria


Table 10 House sprayed IRS malaria occurrence after spraying andusing net



Table 11 Table for parameter values of the optimal malaria model

Symbol Value SourceΛ ℎ 002326 Calculated120579 03 Calculated120583119907 01429 [6]120572ℎ

117 [7]120573ℎ119907 00375 Assumed120588 0035 AssumedΨ 00018 Assumed1199061(119905) 00904 Calculated1199063(119905) 0076 Calculated120578 04 Assumed119873ℎ 20000 [8]1205811 0003 Calculated120583ℎ 004326 [8]120575ℎ

003454 [8]120573119907ℎ 00833 [9]120599 035 Assumed120601 03 Assumed120572119907 01 Assumed1199062(119905) 0165 Calculated120591 001 AssumedΛ 119907 1000 dayminus1 [7]119873119907 3500 Assumed

transmission and this will eventually lower the prevalence ofmalaria and the incidence of the disease in the community

The proposed model has some limitations We did notconsider infective immigrants Also the population was notstratified by age as it is well-known that malaria dispropor-tionately affects children under the age of 5 years Finallythere are various control measures out there and only threebasic ones were considered herein

Conflict of Interests

The authors declare that there is no conflict of interests re-garding the publication of this paper

Acknowledgments

P M Mwamtobe acknowledges with thanks the financialsupport of the Consortium for Advanced Research Trainingin Africa (CARTA) and theUniversity ofMalawiTheMalawiPolytechnic for study leave The authors thank the reviewersfor their insightful comments which led to the improvementof the paper

References

[1] Health Protection Agency (HPA) ldquoForeign travel-associatedillness England Wales and Northern Irelandrdquo Annual ReportHPA London UK 2007 httpwwwhpaorgukwebcHPAw-ebFileHPAweb C1204186182561

[2] WHO ldquoMalaria Fact Sheetsrdquo June 2009 httpwwwwhointmalariaworld malaria report 2009factsheeten

[3] U K Bupa ldquoMalaria symptoms causes and treatment ofdiseaserdquo June 2009 httpwwwbupa-intlcomhealthhealth-informationfactsheetsmmalaria-the-disease

[4] Medicinenet ldquoMalariardquo May 2012 httpwwwonhealthcommalariaarticlehtm

[5] M Faal ldquoWhat is malariardquo Daily Observer 2011 httpwwwobservergmafricagambiaarticlewhat-is-malaria

[6] ODMakinde andKOOkosun ldquoImpact of chemo-therapy onoptimal control of malaria disease with infected immigrantsrdquoBioSystems vol 104 no 1 pp 32ndash41 2011

[7] K Blayneh Y Cao and H-D Kwon ldquoOptimal control ofvector-borne diseases treatment and preventionrdquo Discrete andContinuous Dynamical Systems Series B vol 11 no 3 pp 587ndash611 2009

[8] National Statistical Office (NSO) and UNICEF ldquoMalawi multi-ple indicator cluster survey 2006rdquo Final Report National Statis-tical Office (NSO) and UNICEF Lilongwe Malawi 2008

[9] M I Teboh-Ewungkem C N Podder and A B GumelldquoMathematical study of the role of gametocytes and an imper-fect vaccine on malaria transmission dynamicsrdquo Bulletin ofMathematical Biology vol 72 no 1 pp 63ndash93 2010

[10] N Chitnis J M Hyman and J M Cushing ldquoDeterminingimportant parameters in the spread of malaria through thesensitivity analysis of a mathematical modelrdquo Bulletin of Math-ematical Biology vol 70 no 5 pp 1272ndash1296 2008

[11] G A Ngwa andW S Shu ldquoA mathematical model for endemicmalaria with variable human andmosquito populationsrdquoMath-ematical and Computer Modelling vol 32 no 7-8 pp 747ndash7632000

[12] J Tumwiine J Y T Mugisha and L S Luboobi ldquoOn oscillatorypattern of malaria dynamics in a population with tempo-rary immunityrdquo Computational and Mathematical Methods inMedicine vol 8 no 3 pp 191ndash203 2007

[13] C Chiyaka Z Mukandavire P Das F Nyabadza S D Hove-Musekwa and H Mwambi ldquoTheoretical analysis of mixedPlasmodium malariae and Plasmodium falciparum infectionswith partial cross-immunityrdquo Journal ofTheoretical Biology vol263 no 2 pp 169ndash178 2010

[14] C Chiyaka Z Mukandavire S Dube G Musuka and J MTchuenche ldquoA review of mathematical modelling of the epi-demiology of malariardquo in Infectious Disease Modelling ResearchProgress Public Health in the 21st Century pp 151ndash176 NovaScience Publishers New York NY USA 2009

[15] D L Smith J M Cohen C Chiyaka et al ldquoA sticky situationthe unexpected stability of malaria eliminationrdquo PhilosophicalTransactions of the Royal Society B vol 368 no 1623 2013

[16] G Birkhoff and G-C Rota Ordinary Differential EquationsGinn and Company Boston Mass USA 1982

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


[18] O Diekmann J A Heesterbeek and J A Metz ldquoOn the defini-tion and the computation of the basic reproduction ratio R0 inmodels for infectious diseases in heterogeneous populationsrdquoJournal of Mathematical Biology vol 28 no 4 pp 365ndash3821990

[19] Z Mukandavire A B Gumel W Garira and J M TchuencheldquoMathematical analysis of a model for HIV-malaria co-infec-tionrdquo Mathematical Biosciences and Engineering vol 6 no 2pp 333ndash362 2009

[20] L Perko Differential Equations and Dynamics Systems in Ap-plied Mathematics vol 7 Springer Berlin Germany 2000

[21] A A Lashari and G Zaman ldquoOptimal control of a vector bornedisease with horizontal transmissionrdquo Nonlinear Analysis RealWorld Applications vol 13 no 1 pp 203ndash212 2012

[22] K O Okosun R Ouifki and N Marcus ldquoOptimal controlanalysis of a malaria disease transmission model that includestreatment and vaccination with waning immunityrdquo BioSystemsvol 106 no 2-3 pp 136ndash145 2011

[23] R C A Thome H M Yang and L Esteva ldquoOptimal controlof Aedes aegypti mosquitoes by the sterile insect technique andinsecticiderdquoMathematical Biosciences vol 223 no 1 pp 12ndash232010

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

[25] S O Akinleye and I O Ajayi ldquoKnowledge of malaria andpreventive measures among pregnant women attending ante-natal clinics in a rural local government area in SouthwesternNigeriardquo World Health amp Population vol 12 no 3 pp 13ndash222011

[26] UNICEF Malaria Prevention and Treatment UNICEFrsquos Pro-grammeDivision in cooperation with theWorldHealth Organ-isation 2000 httpwwwuniceforgprescribereng p18pdf

[27] CDC ldquoMalariardquo May 2012 httpwwwcdcgovmalariatrav-elerscountry tableahtml

[28] S Mali S P Kachur and P M Arguin ldquoMalaria surveillancemdashUnited States 2010rdquoMorbidity andMortalityWeekly Report vol61 no 2 pp 1ndash17 2012

[29] National Malaria Control Programme(NMCP) [Malawi] andICF InternationalMalawiMalaria Indicator Survey (MIS) 2012NMCP and ICF International Lilongwe Malawi 2012

[30] F Dzinjalamala ldquoEpidemiology of malaria in Malawirdquo in Epi-demiology of Malawi vol 203 p 21 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


of public health expenditures 30ndash50 of in-patient hospitaladmissions and up to 60 of out-patient health clinic visits[2] The Government of Malawi has put in place severalcontrol strategies through the National Malaria ControlProgramme to reduce and possibly (ultimately) eliminatemalaria The main strategic areas that have been identifiedfor scaling-up of malaria control activities include malariacase management intermittent preventive treatment (IPT) ofpregnantwomenusing sulfadoxine-pyrimethamine (SP) andmalaria prevention with special emphasis on the use of ITNsas well as IRS [8]

Various compartmental models for the spread of malariahave been proposed [10ndash12] Extensive review of mathemat-ical models of malaria dynamics using 119878119864119868119877119878-type modelsand their variants can be found in Chiyaka et al [13 14]For information on the unexpected stability of malaria elimi-nation and eradication and additional literature on malariadynamics see Smith et al [15] who also showed that ifmosquito birth rate is increased by 25 then the minimaleffective spraying period is reduced by half and if doubledthe period is reduced by three-quarters Our goal is to assessthe role of optimal control of preventive measures namelyITNs and IRS as well as treatment of the transmission dynam-ics of malaria in Karonga District Malawi without actuallytargeting a certain high-risk group Despite a plethora ofstudies on the dynamics of malaria and its control (seeChiyaka et al [14] and Smith et al [15]) to the best of ourknowledge the proposed model with exposed immigrants isseemingly new the use of optimal values of a combinationof three intervention strategies namely indoor residualspraying (IRS) insecticide treated nets (ITNs) and treatmentto investigate the effectiveness and optimal control strategiesin the transmission dynamics of malaria in a locality in aresource constrained setting

2 Model Formulation and Analysis

We formulate an optimal control model for malaria with thepopulation under study being subdivided into compartmentsaccording to individualsrsquo disease status We consider thetotal population sizes denoted by 119873ℎ(119905) and 119873V(119905) for thehuman hosts and Anopheles female mosquitoes respectivelyWe employ the 119878119864119868119877119878 framework to describe a disease withtemporary immunity on recovery from infection The 119878119864119868119877119878model indicates that the passage of individuals is from thesusceptible class 119878ℎ to the exposed class 119864ℎ then to theinfectious class 119868ℎ and finally to the recovery class 119877ℎ 119878ℎ(119905)represents the number of individuals not yet infectedwith themalaria parasite at time 119905 The latent or exposed class 119864ℎ(119905)represents individuals who are infected but not yet infectiousIndividuals in the 119868ℎ(119905) class are infected with malariaand are capable of transmitting the disease to susceptiblemosquitoes 119877ℎ(119905) represents the class of individuals whohave temporarily recovered from the disease The susceptiblehuman population is increased by recruitment (birth) ata constant rate Λ ℎ while others are generated throughmigration by (1 minus 1205811)120579119873ℎ where 1205811 is the proportion ofinfected immigrants into the exposed class and 120579 is the rate

at which people migrate into Karonga District All the re-cruited individuals are assumed to be naive when joining thecommunity

There is some finite probability 120573Vℎ that the parasites(in the form of sporozoites) will be passed onto the humanswhen an infectious female Anopheles mosquito bites a sus-ceptible human The parasite then moves to the liver whereit develops into its next life stage merozoites Using theapproach adopted in [6] susceptible individuals acquiremalaria through contact with infectious mosquitoes at therate 120599 The infected person moves to the exposed class atthe rate (1 minus 1199061)120582ℎ119868V119878ℎ The preventive variable 1199061(119905) isin

[0 1] represents the use of ITNs as a means of minimizingor eliminating mosquito-human contacts After a certainperiod of time the parasite (in the form of merozoites)enters the blood stream usually signaling the clinical onsetof malaria Then the exposed individuals become infectiousand progress to the infected state at a constant rate 120572ℎ Theindividuals who have experienced infectionmay recover withtemporary immunity at a constant rate 120588 and move to therecovery class while some infectious humans after recoverywithout immunity become immediately susceptible again atthe rate (1minus120588) Infectious individuals recover due to treatmentat a rate 1205781199062 with 1199062(119905) isin [0 1] representing the controleffort on treatment and 120601 being the proportion of indi-viduals who recover spontaneously Recovered individualslose immunity at a rate 120595 The natural and disease induceddeath rates are 120583ℎ and 120575ℎ respectively The disease induceddeath rate is very small in comparison with the recoveryrate

The mosquito population 119873V is divided into three com-partments susceptible 119878V(119905) exposed 119864V(119905) and infectious119868V(119905) FemaleAnophelesmosquitoes enter the susceptible classthrough birth at a rate Λ V The parasites in the form ofgametocytes enter the mosquito population with probability120573ℎV This happens when the mosquito bites an infectioushuman and the mosquito moves from the susceptible tothe exposed class Mosquitoes are assumed to suffer deathdue to natural causes at a rate 120583V The exposed mosquitoesprogress to the class of symptomatic mosquitoes 119868V at a rate120572V It is assumed that the disease does not induce deathto the mosquito population Finally the mortality rate ofthe mosquito population increases at a rate proportional to1205911199063(119905) where 120591 gt 0 is a rate constant when using IRS and1199063(119905) isin [0 1] is a constant rate of IRS Figure 1 represents aschematic flow diagram of the proposed model

The model state variables are represented in Table 1Table 2 represents prevention and control strategies practisedin the district Table 3 shows parameters of the model

The state variables in Table 1 the prevention and controlparameters in Table 2 and the model parameters in Table 3for the malaria model satisfy (1) It is assumed that allstate variables and parameters of the model which moni-tors human and mosquito populations are positive for all119905 ge 0 We will therefore analyse the model in a suitableregion

The assumptions lead to the following deterministicsystem of nonlinear ordinary differential equations which


Sh

S

E

Rh

Ih

I

Eh

120583hEh

120572hEh

Λh

120595Rh120583hSh

120583hIh 120575hIh

120583hRh

Λ

120583I

120583E

120583S

120572E 120582S

120591u3I

120591u3E

120591u3S


1( (120601 + 120578u2) minus 120588)Ih

(120601 + 120578u2)120588Ih

1205811120579Nh




119889119878ℎ

119889119905= Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119868ℎ


119889119864ℎ


119889119868ℎ


minus (120583ℎ + 120575ℎ) 119868ℎ

119889119877ℎ

119889119905= (120601 + 1205781199062) 120588119868ℎ minus (120583ℎ + 120595)119877ℎ

119889119878V


119889119864V


119889119868V

119889119905= 120572V119864V minus (120583V + 1205911199063) 119868V

(1)









Symbol Description









120572ℎ



120573ℎV








The feasible region

Φ = (119878ℎ 119864ℎ 119868ℎ 119877ℎ 119878V 119864V 119868V) isin R7

+119873ℎ le

Λ ℎ

120583ℎ

= 119873lowast

ℎ

119873V leΛ V

120583V= 119873lowast

V

(2)





119889119878ℎ

119889119905= Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119868ℎ



(3)


119889

119889119905[119878ℎ (119905) 119890

int119905

0(1minus1199061)120582ℎ(119904)119889119904+120583

ℎ119905]

ge Λ ℎ119890int119905

0(1minus1199061)120582ℎ(119904)119889119904+120583

ℎ119905

(4)

Therefore

119878ℎ (119905) 119890int119905

0(1minus119906

1)120582ℎ(119904)119889119904+120583

ℎ119905minus 119878ℎ (0)

ge int

119905

0

Λ ℎ119890int119906

0(1minus1199061)120582ℎ(119908)119889119908+119906

ℎ119906119889119906

(5)

so that

119878ℎ (119905) ge 119878ℎ (0) 119890minus(int119905

0(1minus1199061)120582ℎ(119904)119889119904+120583

ℎ119905)

+ 119890minus(int119905

0(1minus1199061)120582ℎ(119904)119889119904+120583

ℎ119905)

times int

119905

0

Λ ℎ119890int119906

0(1minus1199061)120582ℎ(119908)119889119908+120583

ℎ119906119889119906 gt 0

(6)



ℎ 119864lowast

ℎ 119868lowast

ℎ 119877lowast

ℎ 119878lowast

V 119864lowast

V 119868lowast


1198781015840

ℎ(119905) = 119864

1015840

ℎ(119905) = 119868

1015840

ℎ(119905) = 119877

1015840

ℎ(119905) = 119878

1015840

V (119905) = 1198641015840

V (119905) = 1198681015840

V (119905) = 0

(7)


ℎ= 119868lowast

ℎ= 119877lowast

ℎ= 119864lowast

V = 119868lowast


1198640 = (Λ ℎ

120583ℎ

0 0 0Λ ℎ

120583V 0 0) (8)


F119894 =

[[[[[[[[[

[

(1 minus 1199061) 120573Vℎ120599119868V119878ℎ

119873ℎ

+ 1205811120579119873ℎ

0

120573ℎV120599119868ℎ119878V

119873ℎ

0

]]]]]]]]]

]

V119894 =[[[[

[

(120572ℎ + 120583ℎ) 119864ℎ

(120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ minus 120572ℎ119864ℎ

(120572V + 120583V + 1205911199063) 119864V(120583V + 1205911199063) 119868V minus 120572V119864V

]]]]

]

(9)


119865 =[[[

[

0 0 0 1199031

0 0 0 0

0 119895 0 0

0 0 0 0

]]]

]

119881 =[[[

[

119898 0 0 0

minus120572ℎ 119899 0 0

0 0 119886 0

0 0 minus120572V b

]]]

]

(10)

where

1199031 = (1 minus 1199061) 120573Vℎ120599 119895 =120573ℎV120599Λ V120583ℎ

Λ ℎ120583V

119898 = 120572ℎ + 120583ℎ 119899 = 120601 + 1205781199062 + 120583ℎ + 120575ℎ

119886 = 120572V + 120583V + 1205911199063 119887 = 120583V + 1205911199063

(11)



times ( (120572ℎ + 120583ℎ) (120601 + 1205781199062 + 120583ℎ + 120575ℎ)


)

12

(12)

where

R119890V =120573ℎV120572V120599Λ V

120583V (120572V + 120583V + 1205911199063) (120583V + 1205911199063)(13)



R119890ℎ =120573Vℎ120599120583ℎ120572ℎ (1 minus 1199061)

Λ ℎ (120572ℎ + 120583ℎ) (120601 + 1205781199062 + 120583ℎ + 120575ℎ)(14)






R119890119905 = radic120573ℎV120573Vℎ120599

2120572ℎ120572VΛ V120583ℎ

(120572ℎ + 120583ℎ) (120601 + 1205781199062 + 120583ℎ + 120575ℎ) (120572V + 120583V) 1205832VΛ ℎ

(15)


R119890119873 = radic120573ℎV120573Vℎ120599

2120572ℎ120572VΛ V120583ℎ (1 minus 1199061)

(120572ℎ + 120583ℎ) (120601 + 120583ℎ + 120575ℎ) (120572V + 120583V) 1205832VΛ ℎ

(16)


R119890119904 = (120573ℎV120573Vℎ1205992120572ℎ120572V120583ℎΛ V

times ((120572ℎ + 120583ℎ) (120601 + 120583ℎ + 120575ℎ) (120572V + 120583V + 1205911199063)

times (120583V + 1205911199063) 120583VΛ ℎ)minus1)12

(17)


R0 = radic120573ℎV120573119906ℎ120599

2120572ℎ120572VΛ V120583ℎ

(120572ℎ + 120583ℎ) (120601 + 120583ℎ + 120575ℎ) (120572V + 120583V) 1205832VΛ ℎ

(18)


R119890 le R0 (19)




119889119904

119889119905=

1

119873ℎ

[119889119878ℎ

119889119905minus 119904

119889119873ℎ

119889119905]

=1

119873ℎ

[Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119894119873ℎ


(20)


119889119904

119889119905=

1

119873ℎ


minus119904

119873ℎ


=Λ ℎ

119873ℎ

minus [Λ ℎ

119873ℎ

+ 120579 minus 120575ℎ119894] 119904 + (1 minus 1205811) 120579

+ (120601 + 1205781199062) (1 minus 120588) 119894

minus (1 minus 1199061) 120573Vℎ120599119911119904 + 120595119903

119889119890

119889119905=

1

119873ℎ

[119889119864

119889119905minus 119890

119889119873ℎ

119889119905]

=1

119873ℎ


minus119890

119873ℎ


= (1 minus 1199061) 120573Vℎ120599119911119904 + 1205811120579 minus [Λ ℎ

119873ℎ

minus 120575ℎ119894 + 120572ℎ + 120579] 119890


119889119894

119889119905=

1

119873ℎ

[119889119868

119889119905minus 119894119889119873ℎ

119889119905]

=1

119873ℎ

[120572ℎ119890119873ℎ minus (120601 + 1205781199062) (1 minus 120588) 119894119873ℎ

minus (120601 + 1205781199062) 120588119894119873ℎ minus (120583ℎ + 120575ℎ) 119894119873ℎ]

minus119894

119873ℎ


= 120572ℎ119890 minus [Λ ℎ

119873ℎ

+ 120601 + 1205781199062 + 120575ℎ + 120579 minus 120575ℎ119894] 119894

119889119903

119889119905=

1

119873ℎ

[119889119877

119889119905minus 119903

119889119873ℎ

119889119905]

=1

119873ℎ

[(120601 + 1205781199062) 120588119894119873ℎ minus (120583ℎ + 120595) 119903119873ℎ]

minus119903

119873ℎ


= (120601 + 1205781199062) 120588119894 minus [Λ ℎ

119873ℎ

+ 120595 + 120579 minus 120575ℎ119894] 119903

119889119909

119889119905=

1

119873V[119889119878V

119889119905minus 119909

119889119873V

119889119905]

=1


minus119909

119873V[Λ V minus 120583V119873V]

= [Λ V

119873Vminus 119909

Λ V

119873V] minus 120573ℎV120599119894119909 minus 1205911199063119909

119889119910

119889119905=

1

119873V[119889119864V

119889119905minus 119910

119889119873V

119889119905]

=1


minus119910

119873V[Λ V minus 120583V119873V]

= 120573ℎV120599119894119909 minus [Λ V

119873V+ 120572V + 1205911199063]119910

119889119911

119889119905=

1

119873V[119889119868V

119889119905minus 119885

119889119873V

119889119905]

=1

119873V[120572V119910119873V minus (120583V + 1205911199063) 119911119873V]

minus119911

119873V[Λ V minus 120583V119873V]

= 120572V119910 minus [Λ V

119873V+ 1205911199063] 119911

119889119873ℎ

119889119905= [

Λ ℎ

119873ℎ

minus 120583ℎ minus 120575ℎ119894]119873ℎ

119889119873V

119889119905= [

Λ V

119873Vminus 120583V]119873V

(21)


Φ1 = 119890 119894 119903119873ℎ 119910 119911119873V isin R7

+| 119890 ge 0 119894 ge 0 119903 ge 0

119890 + 119894 + 119903 le 1 119873ℎ leΛ ℎ

120583ℎ

119910 ge 0 119911 ge 0 119910 + 119911 le 1

119873V leΛ V

120583V

(22)



119889119890


minus [Λ ℎ

119873ℎ

minus 120575ℎ119894 + 120572ℎ + 120579] 119890

119889119894

119889119905= 120572ℎ119890 minus [

Λ ℎ

119873ℎ

+ 120601 + 1205781199062 + 120575ℎ + 120579 minus 120575ℎ119894] 119894

119889119903

119889119905= (120601 + 1205781199062) 120588119894 minus [

Λ ℎ

119873ℎ

+ 120595 + 120579 + minus120575ℎ119894] 119903

119889119873ℎ


119889119910


Λ V

119873V+ 120572V + 1205911199063] 119910

119889119911

119889119905= 120572V minus [

Λ V

119873V+ 1205911199063] 119911

119889119873V

119889119905= Λ V minus 120583V119873V

(23)






119873ℎ

minus 120575ℎ119894 + 120572ℎ + 120579] 119890

120572ℎ119890 = [Λ ℎ

119873ℎ

+ 120601 + 1205781199062 + 120575ℎ + 120579 minus 120575ℎ119894] 119894

(120601 + 1205781199062) 120588119894 = [Λ ℎ

119873ℎ

+ 120595 + 120579 minus 120575ℎ119894] 119903

Λ ℎ

119873ℎ

= 120575ℎ119894 + 120583ℎ

(1 minus 119910 minus 119911) 120573ℎV120599119894 = [Λ V

119873V+ 120572V + 1205911199063]119910

120572V119910 = [Λ V

119873V+ 1205781199063] 119911

Λ V

119873V= 120583V

(24)



(120583ℎ + 120572ℎ) 119890 + (1 minus 1199061) 120573Vℎ120599119911119890


(120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ119911) 119890



120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911

(25)


120572ℎ119890 = (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894

120572ℎ119890 = (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894

119890 =(120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894

120572ℎ

(26)



=(120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894

120572ℎ

(1 minus 1199061) 120573Vℎ120599120572ℎ119911119894

+ [120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911 (120583ℎ + 120601 + 1205781199062 + 120575ℎ)] 119894

= (1 minus 1199061) (1 minus 119903) 120573Vℎ120599120572ℎ119911 + 1205811120579120572ℎ

+ 120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911

(27)


119894 = ((1 minus 1199061) (1 minus 119903) 120573Vℎ120599120572ℎ119911 + 1205811120579120572ℎ

+120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911)

times ((1 minus 1199061) 120573Vℎ120599120572ℎ119911119894


(28)


(120601 + 1205781199062) 120588119894 = (120583ℎ + 120595) 119903

119903 =(120601 + 1205781199062) 119894

120583ℎ + 120595

(29)


119894 =1198871 (1 minus 1198861119894) 119888119911 + 119889 + 119892 + 119887ℎ119911

1198871119888119911 + 119892 + 1198871ℎ119896119911

(1198871119888119911 + 119892 + 1198871ℎ119896119911 + 11988711198881198861119911

1198871119888119911 + 119892 + 1198871ℎ119896119911) 119894 =

1198871119888119911 + 119889 + 119892 + 1198871ℎ119911

1198871119888119911 + 119892 + 1198871ℎ119896119911

119894 =1198871119888119911 + 119889 + 119892 + 1198871ℎ119911

1198871119888119911 + 119892 + 1198871ℎ119896119911 + 11988711198881198861119911

(30)


(120583V + 120572V + 1205911199063) 119910 + 120573ℎV119894119910 = 120573ℎV120599 (1 minus 119911) 119894

119910 =(1 minus 119911) 120573ℎV120599119894

120583V + 120572V + 1205911199063 + 120573ℎV120599119894

(31)


120572V119910 = (120583V + 1205911199063) 119911

119911 =120572V119910

120583V + 1205911199063

(32)


119911 =(1 minus 119911) 120572V120573ℎV120599119894

(120583V + 1205911199063) (120583V + 120572V + 1205911199063 + 120573ℎV120599119894)

=1198982119894

1198992 (1199011 + 1199021119894)minus

1198982119894119911

1198992 (1199011 + 1199021119894)

=1198982119894

11989921199011 + 11989921199021119894 + 1198982119894

(33)


119894 =119889 + 119892 + (1198871119888 + 1198871ℎ) (1198982119894 (11989921199011 + 11989921199021119894 + 1198982119894))

119892 + (1198871119888 + 1198871ℎ119896 + 11988711198881198861) (1198982119894 (11989921199011 + 11989921199021119894 + 1198982119894))

(34)



1198601198942+ 119861119894 + 119862 = 0 (35)

where119860 = 11989211989921199021 + 1198921198982 + 11988711198881198982 + 1198871ℎ1198961198982 + 119887111988811988611198982

= (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 120572V120573ℎV120599

+ (1 minus 1199061) 120573Vℎ120573ℎV120572ℎ120572V1205992

+ (1 minus 1199061) (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 120573Vℎ120573ℎV120572V1205992

+ (1 minus 1199061) (120601 + 1205781199062

120583ℎ + 120595)120573Vℎ120573ℎV120572ℎ120572V120599

2

gt 0

119862 = minus11988911989921199011 minus 11989921199011

= minus (120583V + 1205911199063) (120583V + 120572V + 1205911199063) [(1 minus 1199061)]

lt 0

119861 = 11988911989921199021 + 1198891198982 + 11989211989921199021 + 1198921198982 + 1198871ℎ1198982 + 11989211989921199011 minus 11988711198881198982

= (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599

+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 120572V120573ℎV120599

+ (1 minus 1199061) 120573Vℎ120573ℎV120572V1205992

+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) (120583V + 120572V + 1205911199063)


= (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599

+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599


+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) (120583V + 120572V + 1205911199063) (1 minusR2

119890)

(36)


119861 = (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599

+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 1205812120579120572V120573ℎV120599

+ (1 minus 1199061) 120573Vℎ120573ℎV120572V1205992+ (120583ℎ + 120572ℎ) (120583V + 1205911199063)

times (120583V + 120572V + 1205911199063) (1 minusR2

119890)

lt 0

(37)








= 119864V = 119868V = 0 with R1198900 = R119890 | 120575ℎ = 0

(38)




119889119864ℎ

119889119905= (1 minus 1199061) 120582ℎ (

Λ ℎ

120583ℎ


120583ℎ

minus (120572ℎ + 120583ℎ) 119864ℎ

119889119868ℎ

119889119905= 120572ℎ119864ℎ minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ

119889119864V

119889119905= 120582V (

Λ V


119889119868V

119889119905= 120572V119864V minus (120583V + 1205911199063) 119868V

(39)


120597

120597119864ℎ

[(1 minus 1199061) 120573Vℎ120599

119864ℎ119868ℎ119864VΛ ℎ120583ℎ(Λ ℎ

120583ℎ


+1205811120579Λ ℎ

119864ℎ119868ℎ119864V119868V120583ℎminus(120572ℎ + 120583ℎ)

119868ℎ119864V119868V]

+120597

120597119868ℎ

[120572ℎ

119868ℎ119864V119868Vminus(120601 + 1205781199062 + 120583ℎ + 120575ℎ)

119864ℎ119864V119868V]


+120597

120597119864V[

120573ℎV120599

119864ℎ119864V119868VΛ ℎ120583ℎ(Λ V

120583Vminus 119864V minus 119868V)

minus(120572V + 120583V + 1205911199063)

119864ℎ119868ℎ119864V]

+120597

120597119868V[

120572V

119864ℎ119868ℎ119868Vminus(120583V + 1205911199063)

119864ℎ119868ℎ119864V]

= minus120573Vℎ120599

1198642

ℎ119868ℎ119864V

minus120573Vℎ120599120583ℎ

1198642

ℎ119864V

(1 minus119877ℎ

Λ ℎ120583ℎ

) minus1205811120579Λ ℎ

1198642

ℎ119868ℎ119864V119868V120583ℎ

minus120572ℎ

1198682

ℎ119864V119868V

minus120573ℎV120599

119864ℎ1198642V119868V

[1 minus120583ℎ

Λ ℎ

] minus120572V

119864ℎ119868ℎ1198682V

lt 0

(40)








Γ (1199061 1199062 1199063) = int

119879119891

0

(1198621119864ℎ + 1198622119868ℎ + 1198623119873V

+1

2(11986011199062

1+ 1198602119906

2

2+ 1198603119906

2

3)) 119889119905

(41)



2




2

3 A lin-


2

1 1198602119906

2

2 and 1198603119906

2

3 such that the

terms (12)11986011199062

1 (12)1198602119906

2

2 and (12)1198603119906

2

3describe the cost




1(119905) 119906lowast

2(119905) and 119906

lowast

3(119905) such

that

Γ (119906lowast

1 119906lowast

2 119906lowast

3) = min(119906111990621199063)isinΦ2

Γ (1199061 1199062 1199063) (42)

where



(43)



119871 = 1198621119864ℎ + 1198622119868ℎ + 1198623119873V +1

2(11986011199062

1+ 1198602119906

2

2+ 1198603119906

2

3)

(44)



119867 = 119871 + 1205821

119889119878ℎ

119889119905+ 1205822

119889119864ℎ

119889119905+ 1205823

119889119868ℎ

119889119905+ 1205824

119889119877ℎ

119889119905

+ 1205825

119889119878V

119889119905+ 1205826

119889119864V

119889119905+ 1205827

119889119868V

119889119905

= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V +1

2(11986011199062

1+ 1198602119906

2

2+ 1198603119906

2

3)

+ 1205821 [Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119868ℎ



+ 1205823 [120572ℎ119864ℎ minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ]

+ 1205824 [(120601 + 1205781199062) 120588119868ℎ minus (120583ℎ + 120595)119877ℎ]


+ 1205826 [120573ℎV120599119868ℎ119878V minus (120572V + 120583V + 1205911199063) 119864V]

+ 1205827 [120572V119864V minus (120583V + 1205911199063) 119868V]

(45)



1 119906lowast

2 119906lowast


min(119906111990621199063)isinΦ2

Γ (1199061 1199062 1199063) = Γ (119906lowast

1 119906lowast

2 119906lowast

3) (46)


Γ (1199061 1199062 1199063) ge 1205851(100381610038161003816100381611990611003816100381610038161003816

2+100381610038161003816100381611990621003816100381610038161003816

2100381610038161003816100381611990631003816100381610038161003816

2)1205982

minus 1205852(47)





0 =120597119867 (119905 120594 119906 120582)

120597119906

1205821015840=120597119867 (119905 120594 119906 120582)

120597120594

119889120594

119889119905= minus

120597119867 (119905 120594 119906 120582)

120597120582

(48)



2 119906lowast

3) with


ℎ 119868lowast

ℎ 119877lowast

ℎ 119878lowast

V 119864lowast

V 119868lowast


minus1205821015840


minus 120583ℎ1205821 + 12057912058111205822

minus1205821015840


minus1205821015840

3= (1 minus 1205811) 1205791205821 + (120601 + 1205781199062) [(1 minus 120588) 1205821 + 1205881205824]

+ 12057912058111205822 minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 1205823

+ 120573ℎV120599 (1205826 minus 1205825) 119878V + 1198622

minus1205821015840

4= (1 minus 1205811) 1205791205821 + 1205951205821 + 12057912058111205822 minus (120583ℎ + 120595) 1205824

minus1205821015840

5= 120573ℎV120599 (1205826 minus 1205825) 119868ℎ minus (120583V + 1205911199063) 1205825 + 1198623

minus1205821015840

6= 120572V (1205827 minus 1205826) minus (120583V + 1205911199063) 1205826 + 1198623

minus1205821015840


(49)


1205821 (119879119891) = 1205822 (119879119891) = 1205823 (119879119891) = 1205824 (119879119891)

= 1205825 (119879119891) = 1205826 (119879119891) = 1205827 (119879119891) = 0

(50)


2 119906lowast

3) that mini-


119906lowast

1= max0min(1

120573Vℎ120599 (1205822 minus 1205821) 119868lowast

V 119878lowast

ℎ

1198601

)

119906lowast

2= max0min(1


ℎ

1198602

)

119906lowast

3= max0min(1

120591 (1205825119878lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

)

(51)




minus1198891205821

119889119905=120597119867

120597119878ℎ


minus 120583ℎ1205821 + 12057912058111205822

minus1198891205822

119889119905=120597119867

120597119864ℎ

= (1 minus 1205811) 1205791205821 + 12057912058111205822

+ 120572ℎ (1205823 minus 1205822) minus 120583ℎ1205822 + 1198621

minus1198891205823

119889119905=120597119867

120597119868ℎ

= (1 minus 1205811) 1205791205821

+ (120601 + 1205781199062) [(1 minus 120588) 1205821 + 1205881205824] + 12057912058111205822


minus1198891205824

119889119905=120597119867

120597119877ℎ

= (1 minus 1205811) 1205791205821 + 1205951205821 + 12057912058111205822 minus (120583ℎ + 120595) 1205824

minus1198891205825

119889119905=120597119867


minus1198891205826

119889119905=120597119867

120597119864V= 120572V (1205827 minus 1205826) minus (120583V + 1205911199063) 1205826 + 1198623

minus1198891205827

119889119905=120597119867

120597119868V= (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119878ℎ

minus (120583V + 1205911199063) 1205827 + 1198623

(52)


1205821 (119879119891) = 1205822 (119879119891) = 1205823 (119879119891) = 1205824 (119879119891)

= 1205825 (119879119891) = 1205826 (119879119891) = 1205827 (119879119891) = 0

(53)


lowast

ℎ

119864ℎ = 119864lowast

ℎ 119868ℎ = 119868

lowast

ℎ 119877ℎ = 119877

lowast

ℎ 119878V = 119878

lowast

V 119864V = 119864lowast

V and 119868V = 119868lowast

Vyields

120597119867

1205971199061

= 119860119906lowast

1+ 120573Vℎ1205991205821119868

lowast

V 119878lowast

ℎminus 120573Vℎ1205991205822119868

lowast

V 119878lowast

ℎ= 0

120597119867

1205971199062

= 11986021199062 + (1 minus 120588) 1205781205821119868lowast

ℎminus 1205781205823119868

lowast

ℎ+ 1205781205881205824119868

lowast

ℎ= 0

120597119867

1205971199063

= 1198603119906lowast

3minus 1205911205825119878

lowast

V minus 1205911205826119864lowast

V minus 1205911205827119868lowast

V = 0

(54)

for which

119906lowast

1=120573Vℎ120599 (1205822 minus 1205821) 119868

lowast

V 119878lowast

ℎ

1198601

119906lowast

2=120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868

lowast

ℎ

1198602

1199063 =120591 (1205825119878

lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

(55)

Then

119906lowast

1= max0min(1

120573Vℎ120599 (1205822 minus 1205821) 119868lowast

V 119878lowast

ℎ

1198601

)

119906lowast

2= max0min(1


ℎ

1198602

)

119906lowast

3= max0min(1

120591 (1205825119878lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

)

(56)




3



119889119878lowast

ℎ

119889119905= Λ ℎ + (1 minus 1205811) 120579119873

lowast

ℎ

+ (120601 + 1205781199062) (1 minus 120588) 119868lowast

ℎminus 120573Vℎ120599119868

lowast

V 119878lowast

ℎ


lowast

V 119878lowast

ℎ

1198601

))


ℎ+ 120595119877lowast

ℎ

119889119864lowast

ℎ

119889119905= 120573Vℎ120599119868

lowast

V 119878lowast

ℎ


lowast

V 119878lowast

ℎ

1198601

))

+ 1205811120579119873lowast

ℎminus 120572ℎ119864

lowast

ℎminus 120583ℎ119864

lowast

ℎ


119889119868lowast

ℎ

119889119905

= 120572ℎ119864lowast

ℎ

minus (120601 + 120578


lowast

ℎ

1198602

))

+120583ℎ + 120575ℎ) 119868lowast

ℎ

119889119877lowast

ℎ


times 119868lowast

ℎtimes (1198602)

minus1))) 120588119868ℎ

minus (120583ℎ + 120595)119877ℎ

119889119878lowast

V

119889119905

= Λ V minus (120583V + 120591


V + 1205826119864lowast

V + 1205827119868lowast

V )

times(1198603)minus1))) 119878

lowast


ℎ119878lowast

V

119889119864lowast

V

119889119905= 120573ℎV120599119868

lowast

ℎ119878lowast

V


V + 1205826119864lowast

V + 1205827119868lowast

V )

times(1198603)minus1))) 119864

lowast


V

119889119868lowast

V

119889119905= 120572V119864V


V + 1205826119864lowast

V +1205827119868lowast

V )

times(1198603)minus1))) 119868

lowast

V

(57)


ℎ 119868lowast

ℎ 119877lowast

ℎ 119906lowast

1 119906lowast

2 119906lowast

3 1205821 1205822 1205827)

119867lowast

= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V

+1

2(1198601(max0min(1

120573Vℎ120599 (1205822 minus 1205821) 119868lowast

V 119878lowast

ℎ

1198601

))

2


ℎ


+1198603(max0min(1120591(1205825119878

lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

))

2

)

+ 1205821

119889119878lowast

ℎ

119889119905+ 1205822

119889119864lowast

ℎ

119889119905+ 1205823

119889119868lowast

ℎ

119889119905+ 1205824

119889119877lowast

ℎ

119889119905+ 1205825

119889119878lowast

V

119889119905

+ 1205826

119889119864lowast

V

119889119905+ 1205827

119889119868lowast

V

119889119905







u1 ne 0 u3 ne 0 u2 = 0


0

10

20

30

40

50

60

70

times102In

fect

ed h

uman

pop

ulat

ion

0 10 20 30 40 50 60 70 80 90 100

Time (years)

(a)



fect

ed h

uman

pop

ulat

ion

0

10

20

30

40

50

60

70

0 10 20 30 40 50 60 70 80 90 100

times102

Time (days)

(b)

u2 ne 0 u3 ne 0 u1 = 0


0 10 20 30 40 50 600

5

10

15

20

25

30

35

40

times102

Infe

cted

hum

an p

opul

atio

n

Time (years)

(c)

u1 ne 0 u2 ne 0 u3 0ne ==u2 ne 0 u1 0 u3 0

=u1 ne 0 u2 0 u3 = 0 = =u3 ne 0 u1 0 u2 0

0

05

1

15

2

25

3

times104

0 10 20 30 40 50 60 70

Infe

cted

hum

an p

opul

atio

n

Time (years)

(d)








of total 22 30 53








409

93 53 47 7 4484

400440 446

486449

9 9 9 9 9 90

100

200

300

400

500

600

Bite

from

anin

fect

edm

osqu

ito

Fem

ale

mos

quito

Stag

nant

wat

er

Envi

ronm

ent

man

agem

ent

Oth

er

YesNo


wat

erso

aked

in ra

inRa

ined

on

Anopheles

Do not know


Seek treatment


47

418

493

462

409

454

83

8

38

92

0 100 200 300 400 500 600

Other

YesNo


House sprayed(IRS)

Use of ITNor LLN










5 Numerical Results






R0 = 18894

0 05 1 15 2 25 3 35 4

Time (years)

0

20

40

60

80

100

120

Susc

eptib

le in

divi

dual

s


(a)

R0 = 18894

0 05 1 15 2 25 3 35 4

Time (years)

0

20

40

60

80

100

120

140

times102

Expo

sed

indi

vidu

als

Exposed individuals

(b)

R0 = 18894

0 05 1 15 2 25 3 35 49

10

11

12

13

14

15

16

17

Time (years)

Infe

cted

indi

vidu

als


(c)

0 100 200 300 400 500 600 7000

1

2

3

4

5

6

7

8

Time (days)

Reco

vere

d in

divi

dual

s


R0 = 18894

times102

(d)








No Yes


of total 36 57 92



Time (days)

R0 = 18894

Prev

alen

ce

0 20 40 60 80 100 120 140 160 180 2000

01

02

03

04

05

06

07



6 Conclusions












of total 425 384 809



Hum

an p

opul

atio

n


times104

Time (years)0 05 1 15 2 25 3 35 4

0

05

1

15

2

25

(a)

times105

Hum

an p

opul

atio

n


Time (years)0 10 20 30 40 50 60

0

02

04

06

08

1

12

14

16

18

2

(b)


















Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Sh

S

E

Rh

Ih

I

Eh

120583hEh

120572hEh

Λh

120595Rh120583hSh

120583hIh 120575hIh

120583hRh

Λ

120583I

120583E

120583S

120572E 120582S

120591u3I

120591u3E

120591u3S


1( (120601 + 120578u2) minus 120588)Ih

(120601 + 120578u2)120588Ih

1205811120579Nh




119889119878ℎ

119889119905= Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119868ℎ


119889119864ℎ


119889119868ℎ


minus (120583ℎ + 120575ℎ) 119868ℎ

119889119877ℎ

119889119905= (120601 + 1205781199062) 120588119868ℎ minus (120583ℎ + 120595)119877ℎ

119889119878V


119889119864V


119889119868V

119889119905= 120572V119864V minus (120583V + 1205911199063) 119868V

(1)









Symbol Description









120572ℎ



120573ℎV








The feasible region

Φ = (119878ℎ 119864ℎ 119868ℎ 119877ℎ 119878V 119864V 119868V) isin R7

+119873ℎ le

Λ ℎ

120583ℎ

= 119873lowast

ℎ

119873V leΛ V

120583V= 119873lowast

V

(2)





119889119878ℎ

119889119905= Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119868ℎ



(3)


119889

119889119905[119878ℎ (119905) 119890

int119905

0(1minus1199061)120582ℎ(119904)119889119904+120583

ℎ119905]

ge Λ ℎ119890int119905

0(1minus1199061)120582ℎ(119904)119889119904+120583

ℎ119905

(4)

Therefore

119878ℎ (119905) 119890int119905

0(1minus119906

1)120582ℎ(119904)119889119904+120583

ℎ119905minus 119878ℎ (0)

ge int

119905

0

Λ ℎ119890int119906

0(1minus1199061)120582ℎ(119908)119889119908+119906

ℎ119906119889119906

(5)

so that

119878ℎ (119905) ge 119878ℎ (0) 119890minus(int119905

0(1minus1199061)120582ℎ(119904)119889119904+120583

ℎ119905)

+ 119890minus(int119905

0(1minus1199061)120582ℎ(119904)119889119904+120583

ℎ119905)

times int

119905

0

Λ ℎ119890int119906

0(1minus1199061)120582ℎ(119908)119889119908+120583

ℎ119906119889119906 gt 0

(6)



ℎ 119864lowast

ℎ 119868lowast

ℎ 119877lowast

ℎ 119878lowast

V 119864lowast

V 119868lowast


1198781015840

ℎ(119905) = 119864

1015840

ℎ(119905) = 119868

1015840

ℎ(119905) = 119877

1015840

ℎ(119905) = 119878

1015840

V (119905) = 1198641015840

V (119905) = 1198681015840

V (119905) = 0

(7)


ℎ= 119868lowast

ℎ= 119877lowast

ℎ= 119864lowast

V = 119868lowast


1198640 = (Λ ℎ

120583ℎ

0 0 0Λ ℎ

120583V 0 0) (8)


F119894 =

[[[[[[[[[

[

(1 minus 1199061) 120573Vℎ120599119868V119878ℎ

119873ℎ

+ 1205811120579119873ℎ

0

120573ℎV120599119868ℎ119878V

119873ℎ

0

]]]]]]]]]

]

V119894 =[[[[

[

(120572ℎ + 120583ℎ) 119864ℎ

(120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ minus 120572ℎ119864ℎ

(120572V + 120583V + 1205911199063) 119864V(120583V + 1205911199063) 119868V minus 120572V119864V

]]]]

]

(9)


119865 =[[[

[

0 0 0 1199031

0 0 0 0

0 119895 0 0

0 0 0 0

]]]

]

119881 =[[[

[

119898 0 0 0

minus120572ℎ 119899 0 0

0 0 119886 0

0 0 minus120572V b

]]]

]

(10)

where

1199031 = (1 minus 1199061) 120573Vℎ120599 119895 =120573ℎV120599Λ V120583ℎ

Λ ℎ120583V

119898 = 120572ℎ + 120583ℎ 119899 = 120601 + 1205781199062 + 120583ℎ + 120575ℎ

119886 = 120572V + 120583V + 1205911199063 119887 = 120583V + 1205911199063

(11)



times ( (120572ℎ + 120583ℎ) (120601 + 1205781199062 + 120583ℎ + 120575ℎ)


)

12

(12)

where

R119890V =120573ℎV120572V120599Λ V

120583V (120572V + 120583V + 1205911199063) (120583V + 1205911199063)(13)



R119890ℎ =120573Vℎ120599120583ℎ120572ℎ (1 minus 1199061)

Λ ℎ (120572ℎ + 120583ℎ) (120601 + 1205781199062 + 120583ℎ + 120575ℎ)(14)






R119890119905 = radic120573ℎV120573Vℎ120599

2120572ℎ120572VΛ V120583ℎ

(120572ℎ + 120583ℎ) (120601 + 1205781199062 + 120583ℎ + 120575ℎ) (120572V + 120583V) 1205832VΛ ℎ

(15)


R119890119873 = radic120573ℎV120573Vℎ120599

2120572ℎ120572VΛ V120583ℎ (1 minus 1199061)

(120572ℎ + 120583ℎ) (120601 + 120583ℎ + 120575ℎ) (120572V + 120583V) 1205832VΛ ℎ

(16)


R119890119904 = (120573ℎV120573Vℎ1205992120572ℎ120572V120583ℎΛ V

times ((120572ℎ + 120583ℎ) (120601 + 120583ℎ + 120575ℎ) (120572V + 120583V + 1205911199063)

times (120583V + 1205911199063) 120583VΛ ℎ)minus1)12

(17)


R0 = radic120573ℎV120573119906ℎ120599

2120572ℎ120572VΛ V120583ℎ

(120572ℎ + 120583ℎ) (120601 + 120583ℎ + 120575ℎ) (120572V + 120583V) 1205832VΛ ℎ

(18)


R119890 le R0 (19)




119889119904

119889119905=

1

119873ℎ

[119889119878ℎ

119889119905minus 119904

119889119873ℎ

119889119905]

=1

119873ℎ

[Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119894119873ℎ


(20)


119889119904

119889119905=

1

119873ℎ


minus119904

119873ℎ


=Λ ℎ

119873ℎ

minus [Λ ℎ

119873ℎ

+ 120579 minus 120575ℎ119894] 119904 + (1 minus 1205811) 120579

+ (120601 + 1205781199062) (1 minus 120588) 119894

minus (1 minus 1199061) 120573Vℎ120599119911119904 + 120595119903

119889119890

119889119905=

1

119873ℎ

[119889119864

119889119905minus 119890

119889119873ℎ

119889119905]

=1

119873ℎ


minus119890

119873ℎ


= (1 minus 1199061) 120573Vℎ120599119911119904 + 1205811120579 minus [Λ ℎ

119873ℎ

minus 120575ℎ119894 + 120572ℎ + 120579] 119890


119889119894

119889119905=

1

119873ℎ

[119889119868

119889119905minus 119894119889119873ℎ

119889119905]

=1

119873ℎ

[120572ℎ119890119873ℎ minus (120601 + 1205781199062) (1 minus 120588) 119894119873ℎ

minus (120601 + 1205781199062) 120588119894119873ℎ minus (120583ℎ + 120575ℎ) 119894119873ℎ]

minus119894

119873ℎ


= 120572ℎ119890 minus [Λ ℎ

119873ℎ

+ 120601 + 1205781199062 + 120575ℎ + 120579 minus 120575ℎ119894] 119894

119889119903

119889119905=

1

119873ℎ

[119889119877

119889119905minus 119903

119889119873ℎ

119889119905]

=1

119873ℎ

[(120601 + 1205781199062) 120588119894119873ℎ minus (120583ℎ + 120595) 119903119873ℎ]

minus119903

119873ℎ


= (120601 + 1205781199062) 120588119894 minus [Λ ℎ

119873ℎ

+ 120595 + 120579 minus 120575ℎ119894] 119903

119889119909

119889119905=

1

119873V[119889119878V

119889119905minus 119909

119889119873V

119889119905]

=1


minus119909

119873V[Λ V minus 120583V119873V]

= [Λ V

119873Vminus 119909

Λ V

119873V] minus 120573ℎV120599119894119909 minus 1205911199063119909

119889119910

119889119905=

1

119873V[119889119864V

119889119905minus 119910

119889119873V

119889119905]

=1


minus119910

119873V[Λ V minus 120583V119873V]

= 120573ℎV120599119894119909 minus [Λ V

119873V+ 120572V + 1205911199063]119910

119889119911

119889119905=

1

119873V[119889119868V

119889119905minus 119885

119889119873V

119889119905]

=1

119873V[120572V119910119873V minus (120583V + 1205911199063) 119911119873V]

minus119911

119873V[Λ V minus 120583V119873V]

= 120572V119910 minus [Λ V

119873V+ 1205911199063] 119911

119889119873ℎ

119889119905= [

Λ ℎ

119873ℎ

minus 120583ℎ minus 120575ℎ119894]119873ℎ

119889119873V

119889119905= [

Λ V

119873Vminus 120583V]119873V

(21)


Φ1 = 119890 119894 119903119873ℎ 119910 119911119873V isin R7

+| 119890 ge 0 119894 ge 0 119903 ge 0

119890 + 119894 + 119903 le 1 119873ℎ leΛ ℎ

120583ℎ

119910 ge 0 119911 ge 0 119910 + 119911 le 1

119873V leΛ V

120583V

(22)



119889119890


minus [Λ ℎ

119873ℎ

minus 120575ℎ119894 + 120572ℎ + 120579] 119890

119889119894

119889119905= 120572ℎ119890 minus [

Λ ℎ

119873ℎ

+ 120601 + 1205781199062 + 120575ℎ + 120579 minus 120575ℎ119894] 119894

119889119903

119889119905= (120601 + 1205781199062) 120588119894 minus [

Λ ℎ

119873ℎ

+ 120595 + 120579 + minus120575ℎ119894] 119903

119889119873ℎ


119889119910


Λ V

119873V+ 120572V + 1205911199063] 119910

119889119911

119889119905= 120572V minus [

Λ V

119873V+ 1205911199063] 119911

119889119873V

119889119905= Λ V minus 120583V119873V

(23)






119873ℎ

minus 120575ℎ119894 + 120572ℎ + 120579] 119890

120572ℎ119890 = [Λ ℎ

119873ℎ

+ 120601 + 1205781199062 + 120575ℎ + 120579 minus 120575ℎ119894] 119894

(120601 + 1205781199062) 120588119894 = [Λ ℎ

119873ℎ

+ 120595 + 120579 minus 120575ℎ119894] 119903

Λ ℎ

119873ℎ

= 120575ℎ119894 + 120583ℎ

(1 minus 119910 minus 119911) 120573ℎV120599119894 = [Λ V

119873V+ 120572V + 1205911199063]119910

120572V119910 = [Λ V

119873V+ 1205781199063] 119911

Λ V

119873V= 120583V

(24)



(120583ℎ + 120572ℎ) 119890 + (1 minus 1199061) 120573Vℎ120599119911119890


(120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ119911) 119890



120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911

(25)


120572ℎ119890 = (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894

120572ℎ119890 = (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894

119890 =(120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894

120572ℎ

(26)



=(120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894

120572ℎ

(1 minus 1199061) 120573Vℎ120599120572ℎ119911119894

+ [120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911 (120583ℎ + 120601 + 1205781199062 + 120575ℎ)] 119894

= (1 minus 1199061) (1 minus 119903) 120573Vℎ120599120572ℎ119911 + 1205811120579120572ℎ

+ 120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911

(27)


119894 = ((1 minus 1199061) (1 minus 119903) 120573Vℎ120599120572ℎ119911 + 1205811120579120572ℎ

+120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911)

times ((1 minus 1199061) 120573Vℎ120599120572ℎ119911119894


(28)


(120601 + 1205781199062) 120588119894 = (120583ℎ + 120595) 119903

119903 =(120601 + 1205781199062) 119894

120583ℎ + 120595

(29)


119894 =1198871 (1 minus 1198861119894) 119888119911 + 119889 + 119892 + 119887ℎ119911

1198871119888119911 + 119892 + 1198871ℎ119896119911

(1198871119888119911 + 119892 + 1198871ℎ119896119911 + 11988711198881198861119911

1198871119888119911 + 119892 + 1198871ℎ119896119911) 119894 =

1198871119888119911 + 119889 + 119892 + 1198871ℎ119911

1198871119888119911 + 119892 + 1198871ℎ119896119911

119894 =1198871119888119911 + 119889 + 119892 + 1198871ℎ119911

1198871119888119911 + 119892 + 1198871ℎ119896119911 + 11988711198881198861119911

(30)


(120583V + 120572V + 1205911199063) 119910 + 120573ℎV119894119910 = 120573ℎV120599 (1 minus 119911) 119894

119910 =(1 minus 119911) 120573ℎV120599119894

120583V + 120572V + 1205911199063 + 120573ℎV120599119894

(31)


120572V119910 = (120583V + 1205911199063) 119911

119911 =120572V119910

120583V + 1205911199063

(32)


119911 =(1 minus 119911) 120572V120573ℎV120599119894

(120583V + 1205911199063) (120583V + 120572V + 1205911199063 + 120573ℎV120599119894)

=1198982119894

1198992 (1199011 + 1199021119894)minus

1198982119894119911

1198992 (1199011 + 1199021119894)

=1198982119894

11989921199011 + 11989921199021119894 + 1198982119894

(33)


119894 =119889 + 119892 + (1198871119888 + 1198871ℎ) (1198982119894 (11989921199011 + 11989921199021119894 + 1198982119894))

119892 + (1198871119888 + 1198871ℎ119896 + 11988711198881198861) (1198982119894 (11989921199011 + 11989921199021119894 + 1198982119894))

(34)



1198601198942+ 119861119894 + 119862 = 0 (35)

where119860 = 11989211989921199021 + 1198921198982 + 11988711198881198982 + 1198871ℎ1198961198982 + 119887111988811988611198982

= (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 120572V120573ℎV120599

+ (1 minus 1199061) 120573Vℎ120573ℎV120572ℎ120572V1205992

+ (1 minus 1199061) (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 120573Vℎ120573ℎV120572V1205992

+ (1 minus 1199061) (120601 + 1205781199062

120583ℎ + 120595)120573Vℎ120573ℎV120572ℎ120572V120599

2

gt 0

119862 = minus11988911989921199011 minus 11989921199011

= minus (120583V + 1205911199063) (120583V + 120572V + 1205911199063) [(1 minus 1199061)]

lt 0

119861 = 11988911989921199021 + 1198891198982 + 11989211989921199021 + 1198921198982 + 1198871ℎ1198982 + 11989211989921199011 minus 11988711198881198982

= (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599

+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 120572V120573ℎV120599

+ (1 minus 1199061) 120573Vℎ120573ℎV120572V1205992

+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) (120583V + 120572V + 1205911199063)


= (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599

+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599


+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) (120583V + 120572V + 1205911199063) (1 minusR2

119890)

(36)


119861 = (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599

+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 1205812120579120572V120573ℎV120599

+ (1 minus 1199061) 120573Vℎ120573ℎV120572V1205992+ (120583ℎ + 120572ℎ) (120583V + 1205911199063)

times (120583V + 120572V + 1205911199063) (1 minusR2

119890)

lt 0

(37)








= 119864V = 119868V = 0 with R1198900 = R119890 | 120575ℎ = 0

(38)




119889119864ℎ

119889119905= (1 minus 1199061) 120582ℎ (

Λ ℎ

120583ℎ


120583ℎ

minus (120572ℎ + 120583ℎ) 119864ℎ

119889119868ℎ

119889119905= 120572ℎ119864ℎ minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ

119889119864V

119889119905= 120582V (

Λ V


119889119868V

119889119905= 120572V119864V minus (120583V + 1205911199063) 119868V

(39)


120597

120597119864ℎ

[(1 minus 1199061) 120573Vℎ120599

119864ℎ119868ℎ119864VΛ ℎ120583ℎ(Λ ℎ

120583ℎ


+1205811120579Λ ℎ

119864ℎ119868ℎ119864V119868V120583ℎminus(120572ℎ + 120583ℎ)

119868ℎ119864V119868V]

+120597

120597119868ℎ

[120572ℎ

119868ℎ119864V119868Vminus(120601 + 1205781199062 + 120583ℎ + 120575ℎ)

119864ℎ119864V119868V]


+120597

120597119864V[

120573ℎV120599

119864ℎ119864V119868VΛ ℎ120583ℎ(Λ V

120583Vminus 119864V minus 119868V)

minus(120572V + 120583V + 1205911199063)

119864ℎ119868ℎ119864V]

+120597

120597119868V[

120572V

119864ℎ119868ℎ119868Vminus(120583V + 1205911199063)

119864ℎ119868ℎ119864V]

= minus120573Vℎ120599

1198642

ℎ119868ℎ119864V

minus120573Vℎ120599120583ℎ

1198642

ℎ119864V

(1 minus119877ℎ

Λ ℎ120583ℎ

) minus1205811120579Λ ℎ

1198642

ℎ119868ℎ119864V119868V120583ℎ

minus120572ℎ

1198682

ℎ119864V119868V

minus120573ℎV120599

119864ℎ1198642V119868V

[1 minus120583ℎ

Λ ℎ

] minus120572V

119864ℎ119868ℎ1198682V

lt 0

(40)








Γ (1199061 1199062 1199063) = int

119879119891

0

(1198621119864ℎ + 1198622119868ℎ + 1198623119873V

+1

2(11986011199062

1+ 1198602119906

2

2+ 1198603119906

2

3)) 119889119905

(41)



2




2

3 A lin-


2

1 1198602119906

2

2 and 1198603119906

2

3 such that the

terms (12)11986011199062

1 (12)1198602119906

2

2 and (12)1198603119906

2

3describe the cost




1(119905) 119906lowast

2(119905) and 119906

lowast

3(119905) such

that

Γ (119906lowast

1 119906lowast

2 119906lowast

3) = min(119906111990621199063)isinΦ2

Γ (1199061 1199062 1199063) (42)

where



(43)



119871 = 1198621119864ℎ + 1198622119868ℎ + 1198623119873V +1

2(11986011199062

1+ 1198602119906

2

2+ 1198603119906

2

3)

(44)



119867 = 119871 + 1205821

119889119878ℎ

119889119905+ 1205822

119889119864ℎ

119889119905+ 1205823

119889119868ℎ

119889119905+ 1205824

119889119877ℎ

119889119905

+ 1205825

119889119878V

119889119905+ 1205826

119889119864V

119889119905+ 1205827

119889119868V

119889119905

= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V +1

2(11986011199062

1+ 1198602119906

2

2+ 1198603119906

2

3)

+ 1205821 [Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119868ℎ



+ 1205823 [120572ℎ119864ℎ minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ]

+ 1205824 [(120601 + 1205781199062) 120588119868ℎ minus (120583ℎ + 120595)119877ℎ]


+ 1205826 [120573ℎV120599119868ℎ119878V minus (120572V + 120583V + 1205911199063) 119864V]

+ 1205827 [120572V119864V minus (120583V + 1205911199063) 119868V]

(45)



1 119906lowast

2 119906lowast


min(119906111990621199063)isinΦ2

Γ (1199061 1199062 1199063) = Γ (119906lowast

1 119906lowast

2 119906lowast

3) (46)


Γ (1199061 1199062 1199063) ge 1205851(100381610038161003816100381611990611003816100381610038161003816

2+100381610038161003816100381611990621003816100381610038161003816

2100381610038161003816100381611990631003816100381610038161003816

2)1205982

minus 1205852(47)





0 =120597119867 (119905 120594 119906 120582)

120597119906

1205821015840=120597119867 (119905 120594 119906 120582)

120597120594

119889120594

119889119905= minus

120597119867 (119905 120594 119906 120582)

120597120582

(48)



2 119906lowast

3) with


ℎ 119868lowast

ℎ 119877lowast

ℎ 119878lowast

V 119864lowast

V 119868lowast


minus1205821015840


minus 120583ℎ1205821 + 12057912058111205822

minus1205821015840


minus1205821015840

3= (1 minus 1205811) 1205791205821 + (120601 + 1205781199062) [(1 minus 120588) 1205821 + 1205881205824]

+ 12057912058111205822 minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 1205823

+ 120573ℎV120599 (1205826 minus 1205825) 119878V + 1198622

minus1205821015840

4= (1 minus 1205811) 1205791205821 + 1205951205821 + 12057912058111205822 minus (120583ℎ + 120595) 1205824

minus1205821015840

5= 120573ℎV120599 (1205826 minus 1205825) 119868ℎ minus (120583V + 1205911199063) 1205825 + 1198623

minus1205821015840

6= 120572V (1205827 minus 1205826) minus (120583V + 1205911199063) 1205826 + 1198623

minus1205821015840


(49)


1205821 (119879119891) = 1205822 (119879119891) = 1205823 (119879119891) = 1205824 (119879119891)

= 1205825 (119879119891) = 1205826 (119879119891) = 1205827 (119879119891) = 0

(50)


2 119906lowast

3) that mini-


119906lowast

1= max0min(1

120573Vℎ120599 (1205822 minus 1205821) 119868lowast

V 119878lowast

ℎ

1198601

)

119906lowast

2= max0min(1


ℎ

1198602

)

119906lowast

3= max0min(1

120591 (1205825119878lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

)

(51)




minus1198891205821

119889119905=120597119867

120597119878ℎ


minus 120583ℎ1205821 + 12057912058111205822

minus1198891205822

119889119905=120597119867

120597119864ℎ

= (1 minus 1205811) 1205791205821 + 12057912058111205822

+ 120572ℎ (1205823 minus 1205822) minus 120583ℎ1205822 + 1198621

minus1198891205823

119889119905=120597119867

120597119868ℎ

= (1 minus 1205811) 1205791205821

+ (120601 + 1205781199062) [(1 minus 120588) 1205821 + 1205881205824] + 12057912058111205822


minus1198891205824

119889119905=120597119867

120597119877ℎ

= (1 minus 1205811) 1205791205821 + 1205951205821 + 12057912058111205822 minus (120583ℎ + 120595) 1205824

minus1198891205825

119889119905=120597119867


minus1198891205826

119889119905=120597119867

120597119864V= 120572V (1205827 minus 1205826) minus (120583V + 1205911199063) 1205826 + 1198623

minus1198891205827

119889119905=120597119867

120597119868V= (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119878ℎ

minus (120583V + 1205911199063) 1205827 + 1198623

(52)


1205821 (119879119891) = 1205822 (119879119891) = 1205823 (119879119891) = 1205824 (119879119891)

= 1205825 (119879119891) = 1205826 (119879119891) = 1205827 (119879119891) = 0

(53)


lowast

ℎ

119864ℎ = 119864lowast

ℎ 119868ℎ = 119868

lowast

ℎ 119877ℎ = 119877

lowast

ℎ 119878V = 119878

lowast

V 119864V = 119864lowast

V and 119868V = 119868lowast

Vyields

120597119867

1205971199061

= 119860119906lowast

1+ 120573Vℎ1205991205821119868

lowast

V 119878lowast

ℎminus 120573Vℎ1205991205822119868

lowast

V 119878lowast

ℎ= 0

120597119867

1205971199062

= 11986021199062 + (1 minus 120588) 1205781205821119868lowast

ℎminus 1205781205823119868

lowast

ℎ+ 1205781205881205824119868

lowast

ℎ= 0

120597119867

1205971199063

= 1198603119906lowast

3minus 1205911205825119878

lowast

V minus 1205911205826119864lowast

V minus 1205911205827119868lowast

V = 0

(54)

for which

119906lowast

1=120573Vℎ120599 (1205822 minus 1205821) 119868

lowast

V 119878lowast

ℎ

1198601

119906lowast

2=120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868

lowast

ℎ

1198602

1199063 =120591 (1205825119878

lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

(55)

Then

119906lowast

1= max0min(1

120573Vℎ120599 (1205822 minus 1205821) 119868lowast

V 119878lowast

ℎ

1198601

)

119906lowast

2= max0min(1


ℎ

1198602

)

119906lowast

3= max0min(1

120591 (1205825119878lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

)

(56)




3



119889119878lowast

ℎ

119889119905= Λ ℎ + (1 minus 1205811) 120579119873

lowast

ℎ

+ (120601 + 1205781199062) (1 minus 120588) 119868lowast

ℎminus 120573Vℎ120599119868

lowast

V 119878lowast

ℎ


lowast

V 119878lowast

ℎ

1198601

))


ℎ+ 120595119877lowast

ℎ

119889119864lowast

ℎ

119889119905= 120573Vℎ120599119868

lowast

V 119878lowast

ℎ


lowast

V 119878lowast

ℎ

1198601

))

+ 1205811120579119873lowast

ℎminus 120572ℎ119864

lowast

ℎminus 120583ℎ119864

lowast

ℎ


119889119868lowast

ℎ

119889119905

= 120572ℎ119864lowast

ℎ

minus (120601 + 120578


lowast

ℎ

1198602

))

+120583ℎ + 120575ℎ) 119868lowast

ℎ

119889119877lowast

ℎ


times 119868lowast

ℎtimes (1198602)

minus1))) 120588119868ℎ

minus (120583ℎ + 120595)119877ℎ

119889119878lowast

V

119889119905

= Λ V minus (120583V + 120591


V + 1205826119864lowast

V + 1205827119868lowast

V )

times(1198603)minus1))) 119878

lowast


ℎ119878lowast

V

119889119864lowast

V

119889119905= 120573ℎV120599119868

lowast

ℎ119878lowast

V


V + 1205826119864lowast

V + 1205827119868lowast

V )

times(1198603)minus1))) 119864

lowast


V

119889119868lowast

V

119889119905= 120572V119864V


V + 1205826119864lowast

V +1205827119868lowast

V )

times(1198603)minus1))) 119868

lowast

V

(57)


ℎ 119868lowast

ℎ 119877lowast

ℎ 119906lowast

1 119906lowast

2 119906lowast

3 1205821 1205822 1205827)

119867lowast

= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V

+1

2(1198601(max0min(1

120573Vℎ120599 (1205822 minus 1205821) 119868lowast

V 119878lowast

ℎ

1198601

))

2


ℎ


+1198603(max0min(1120591(1205825119878

lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

))

2

)

+ 1205821

119889119878lowast

ℎ

119889119905+ 1205822

119889119864lowast

ℎ

119889119905+ 1205823

119889119868lowast

ℎ

119889119905+ 1205824

119889119877lowast

ℎ

119889119905+ 1205825

119889119878lowast

V

119889119905

+ 1205826

119889119864lowast

V

119889119905+ 1205827

119889119868lowast

V

119889119905







u1 ne 0 u3 ne 0 u2 = 0


0

10

20

30

40

50

60

70

times102In

fect

ed h

uman

pop

ulat

ion

0 10 20 30 40 50 60 70 80 90 100

Time (years)

(a)



fect

ed h

uman

pop

ulat

ion

0

10

20

30

40

50

60

70

0 10 20 30 40 50 60 70 80 90 100

times102

Time (days)

(b)

u2 ne 0 u3 ne 0 u1 = 0


0 10 20 30 40 50 600

5

10

15

20

25

30

35

40

times102

Infe

cted

hum

an p

opul

atio

n

Time (years)

(c)

u1 ne 0 u2 ne 0 u3 0ne ==u2 ne 0 u1 0 u3 0

=u1 ne 0 u2 0 u3 = 0 = =u3 ne 0 u1 0 u2 0

0

05

1

15

2

25

3

times104

0 10 20 30 40 50 60 70

Infe

cted

hum

an p

opul

atio

n

Time (years)

(d)








of total 22 30 53








409

93 53 47 7 4484

400440 446

486449

9 9 9 9 9 90

100

200

300

400

500

600

Bite

from

anin

fect

edm

osqu

ito

Fem

ale

mos

quito

Stag

nant

wat

er

Envi

ronm

ent

man

agem

ent

Oth

er

YesNo


wat

erso

aked

in ra

inRa

ined

on

Anopheles

Do not know


Seek treatment


47

418

493

462

409

454

83

8

38

92

0 100 200 300 400 500 600

Other

YesNo


House sprayed(IRS)

Use of ITNor LLN










5 Numerical Results






R0 = 18894

0 05 1 15 2 25 3 35 4

Time (years)

0

20

40

60

80

100

120

Susc

eptib

le in

divi

dual

s


(a)

R0 = 18894

0 05 1 15 2 25 3 35 4

Time (years)

0

20

40

60

80

100

120

140

times102

Expo

sed

indi

vidu

als

Exposed individuals

(b)

R0 = 18894

0 05 1 15 2 25 3 35 49

10

11

12

13

14

15

16

17

Time (years)

Infe

cted

indi

vidu

als


(c)

0 100 200 300 400 500 600 7000

1

2

3

4

5

6

7

8

Time (days)

Reco

vere

d in

divi

dual

s


R0 = 18894

times102

(d)








No Yes


of total 36 57 92



Time (days)

R0 = 18894

Prev

alen

ce

0 20 40 60 80 100 120 140 160 180 2000

01

02

03

04

05

06

07



6 Conclusions












of total 425 384 809



Hum

an p

opul

atio

n


times104

Time (years)0 05 1 15 2 25 3 35 4

0

05

1

15

2

25

(a)

times105

Hum

an p

opul

atio

n


Time (years)0 10 20 30 40 50 60

0

02

04

06

08

1

12

14

16

18

2

(b)


















Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










The feasible region

Φ = (119878ℎ 119864ℎ 119868ℎ 119877ℎ 119878V 119864V 119868V) isin R7

+119873ℎ le

Λ ℎ

120583ℎ

= 119873lowast

ℎ

119873V leΛ V

120583V= 119873lowast

V

(2)





119889119878ℎ

119889119905= Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119868ℎ



(3)


119889

119889119905[119878ℎ (119905) 119890

int119905

0(1minus1199061)120582ℎ(119904)119889119904+120583

ℎ119905]

ge Λ ℎ119890int119905

0(1minus1199061)120582ℎ(119904)119889119904+120583

ℎ119905

(4)

Therefore

119878ℎ (119905) 119890int119905

0(1minus119906

1)120582ℎ(119904)119889119904+120583

ℎ119905minus 119878ℎ (0)

ge int

119905

0

Λ ℎ119890int119906

0(1minus1199061)120582ℎ(119908)119889119908+119906

ℎ119906119889119906

(5)

so that

119878ℎ (119905) ge 119878ℎ (0) 119890minus(int119905

0(1minus1199061)120582ℎ(119904)119889119904+120583

ℎ119905)

+ 119890minus(int119905

0(1minus1199061)120582ℎ(119904)119889119904+120583

ℎ119905)

times int

119905

0

Λ ℎ119890int119906

0(1minus1199061)120582ℎ(119908)119889119908+120583

ℎ119906119889119906 gt 0

(6)



ℎ 119864lowast

ℎ 119868lowast

ℎ 119877lowast

ℎ 119878lowast

V 119864lowast

V 119868lowast


1198781015840

ℎ(119905) = 119864

1015840

ℎ(119905) = 119868

1015840

ℎ(119905) = 119877

1015840

ℎ(119905) = 119878

1015840

V (119905) = 1198641015840

V (119905) = 1198681015840

V (119905) = 0

(7)


ℎ= 119868lowast

ℎ= 119877lowast

ℎ= 119864lowast

V = 119868lowast


1198640 = (Λ ℎ

120583ℎ

0 0 0Λ ℎ

120583V 0 0) (8)


F119894 =

[[[[[[[[[

[

(1 minus 1199061) 120573Vℎ120599119868V119878ℎ

119873ℎ

+ 1205811120579119873ℎ

0

120573ℎV120599119868ℎ119878V

119873ℎ

0

]]]]]]]]]

]

V119894 =[[[[

[

(120572ℎ + 120583ℎ) 119864ℎ

(120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ minus 120572ℎ119864ℎ

(120572V + 120583V + 1205911199063) 119864V(120583V + 1205911199063) 119868V minus 120572V119864V

]]]]

]

(9)


119865 =[[[

[

0 0 0 1199031

0 0 0 0

0 119895 0 0

0 0 0 0

]]]

]

119881 =[[[

[

119898 0 0 0

minus120572ℎ 119899 0 0

0 0 119886 0

0 0 minus120572V b

]]]

]

(10)

where

1199031 = (1 minus 1199061) 120573Vℎ120599 119895 =120573ℎV120599Λ V120583ℎ

Λ ℎ120583V

119898 = 120572ℎ + 120583ℎ 119899 = 120601 + 1205781199062 + 120583ℎ + 120575ℎ

119886 = 120572V + 120583V + 1205911199063 119887 = 120583V + 1205911199063

(11)



times ( (120572ℎ + 120583ℎ) (120601 + 1205781199062 + 120583ℎ + 120575ℎ)


)

12

(12)

where

R119890V =120573ℎV120572V120599Λ V

120583V (120572V + 120583V + 1205911199063) (120583V + 1205911199063)(13)



R119890ℎ =120573Vℎ120599120583ℎ120572ℎ (1 minus 1199061)

Λ ℎ (120572ℎ + 120583ℎ) (120601 + 1205781199062 + 120583ℎ + 120575ℎ)(14)






R119890119905 = radic120573ℎV120573Vℎ120599

2120572ℎ120572VΛ V120583ℎ

(120572ℎ + 120583ℎ) (120601 + 1205781199062 + 120583ℎ + 120575ℎ) (120572V + 120583V) 1205832VΛ ℎ

(15)


R119890119873 = radic120573ℎV120573Vℎ120599

2120572ℎ120572VΛ V120583ℎ (1 minus 1199061)

(120572ℎ + 120583ℎ) (120601 + 120583ℎ + 120575ℎ) (120572V + 120583V) 1205832VΛ ℎ

(16)


R119890119904 = (120573ℎV120573Vℎ1205992120572ℎ120572V120583ℎΛ V

times ((120572ℎ + 120583ℎ) (120601 + 120583ℎ + 120575ℎ) (120572V + 120583V + 1205911199063)

times (120583V + 1205911199063) 120583VΛ ℎ)minus1)12

(17)


R0 = radic120573ℎV120573119906ℎ120599

2120572ℎ120572VΛ V120583ℎ

(120572ℎ + 120583ℎ) (120601 + 120583ℎ + 120575ℎ) (120572V + 120583V) 1205832VΛ ℎ

(18)


R119890 le R0 (19)




119889119904

119889119905=

1

119873ℎ

[119889119878ℎ

119889119905minus 119904

119889119873ℎ

119889119905]

=1

119873ℎ

[Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119894119873ℎ


(20)


119889119904

119889119905=

1

119873ℎ


minus119904

119873ℎ


=Λ ℎ

119873ℎ

minus [Λ ℎ

119873ℎ

+ 120579 minus 120575ℎ119894] 119904 + (1 minus 1205811) 120579

+ (120601 + 1205781199062) (1 minus 120588) 119894

minus (1 minus 1199061) 120573Vℎ120599119911119904 + 120595119903

119889119890

119889119905=

1

119873ℎ

[119889119864

119889119905minus 119890

119889119873ℎ

119889119905]

=1

119873ℎ


minus119890

119873ℎ


= (1 minus 1199061) 120573Vℎ120599119911119904 + 1205811120579 minus [Λ ℎ

119873ℎ

minus 120575ℎ119894 + 120572ℎ + 120579] 119890


119889119894

119889119905=

1

119873ℎ

[119889119868

119889119905minus 119894119889119873ℎ

119889119905]

=1

119873ℎ

[120572ℎ119890119873ℎ minus (120601 + 1205781199062) (1 minus 120588) 119894119873ℎ

minus (120601 + 1205781199062) 120588119894119873ℎ minus (120583ℎ + 120575ℎ) 119894119873ℎ]

minus119894

119873ℎ


= 120572ℎ119890 minus [Λ ℎ

119873ℎ

+ 120601 + 1205781199062 + 120575ℎ + 120579 minus 120575ℎ119894] 119894

119889119903

119889119905=

1

119873ℎ

[119889119877

119889119905minus 119903

119889119873ℎ

119889119905]

=1

119873ℎ

[(120601 + 1205781199062) 120588119894119873ℎ minus (120583ℎ + 120595) 119903119873ℎ]

minus119903

119873ℎ


= (120601 + 1205781199062) 120588119894 minus [Λ ℎ

119873ℎ

+ 120595 + 120579 minus 120575ℎ119894] 119903

119889119909

119889119905=

1

119873V[119889119878V

119889119905minus 119909

119889119873V

119889119905]

=1


minus119909

119873V[Λ V minus 120583V119873V]

= [Λ V

119873Vminus 119909

Λ V

119873V] minus 120573ℎV120599119894119909 minus 1205911199063119909

119889119910

119889119905=

1

119873V[119889119864V

119889119905minus 119910

119889119873V

119889119905]

=1


minus119910

119873V[Λ V minus 120583V119873V]

= 120573ℎV120599119894119909 minus [Λ V

119873V+ 120572V + 1205911199063]119910

119889119911

119889119905=

1

119873V[119889119868V

119889119905minus 119885

119889119873V

119889119905]

=1

119873V[120572V119910119873V minus (120583V + 1205911199063) 119911119873V]

minus119911

119873V[Λ V minus 120583V119873V]

= 120572V119910 minus [Λ V

119873V+ 1205911199063] 119911

119889119873ℎ

119889119905= [

Λ ℎ

119873ℎ

minus 120583ℎ minus 120575ℎ119894]119873ℎ

119889119873V

119889119905= [

Λ V

119873Vminus 120583V]119873V

(21)


Φ1 = 119890 119894 119903119873ℎ 119910 119911119873V isin R7

+| 119890 ge 0 119894 ge 0 119903 ge 0

119890 + 119894 + 119903 le 1 119873ℎ leΛ ℎ

120583ℎ

119910 ge 0 119911 ge 0 119910 + 119911 le 1

119873V leΛ V

120583V

(22)



119889119890


minus [Λ ℎ

119873ℎ

minus 120575ℎ119894 + 120572ℎ + 120579] 119890

119889119894

119889119905= 120572ℎ119890 minus [

Λ ℎ

119873ℎ

+ 120601 + 1205781199062 + 120575ℎ + 120579 minus 120575ℎ119894] 119894

119889119903

119889119905= (120601 + 1205781199062) 120588119894 minus [

Λ ℎ

119873ℎ

+ 120595 + 120579 + minus120575ℎ119894] 119903

119889119873ℎ


119889119910


Λ V

119873V+ 120572V + 1205911199063] 119910

119889119911

119889119905= 120572V minus [

Λ V

119873V+ 1205911199063] 119911

119889119873V

119889119905= Λ V minus 120583V119873V

(23)






119873ℎ

minus 120575ℎ119894 + 120572ℎ + 120579] 119890

120572ℎ119890 = [Λ ℎ

119873ℎ

+ 120601 + 1205781199062 + 120575ℎ + 120579 minus 120575ℎ119894] 119894

(120601 + 1205781199062) 120588119894 = [Λ ℎ

119873ℎ

+ 120595 + 120579 minus 120575ℎ119894] 119903

Λ ℎ

119873ℎ

= 120575ℎ119894 + 120583ℎ

(1 minus 119910 minus 119911) 120573ℎV120599119894 = [Λ V

119873V+ 120572V + 1205911199063]119910

120572V119910 = [Λ V

119873V+ 1205781199063] 119911

Λ V

119873V= 120583V

(24)



(120583ℎ + 120572ℎ) 119890 + (1 minus 1199061) 120573Vℎ120599119911119890


(120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ119911) 119890



120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911

(25)


120572ℎ119890 = (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894

120572ℎ119890 = (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894

119890 =(120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894

120572ℎ

(26)



=(120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894

120572ℎ

(1 minus 1199061) 120573Vℎ120599120572ℎ119911119894

+ [120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911 (120583ℎ + 120601 + 1205781199062 + 120575ℎ)] 119894

= (1 minus 1199061) (1 minus 119903) 120573Vℎ120599120572ℎ119911 + 1205811120579120572ℎ

+ 120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911

(27)


119894 = ((1 minus 1199061) (1 minus 119903) 120573Vℎ120599120572ℎ119911 + 1205811120579120572ℎ

+120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911)

times ((1 minus 1199061) 120573Vℎ120599120572ℎ119911119894


(28)


(120601 + 1205781199062) 120588119894 = (120583ℎ + 120595) 119903

119903 =(120601 + 1205781199062) 119894

120583ℎ + 120595

(29)


119894 =1198871 (1 minus 1198861119894) 119888119911 + 119889 + 119892 + 119887ℎ119911

1198871119888119911 + 119892 + 1198871ℎ119896119911

(1198871119888119911 + 119892 + 1198871ℎ119896119911 + 11988711198881198861119911

1198871119888119911 + 119892 + 1198871ℎ119896119911) 119894 =

1198871119888119911 + 119889 + 119892 + 1198871ℎ119911

1198871119888119911 + 119892 + 1198871ℎ119896119911

119894 =1198871119888119911 + 119889 + 119892 + 1198871ℎ119911

1198871119888119911 + 119892 + 1198871ℎ119896119911 + 11988711198881198861119911

(30)


(120583V + 120572V + 1205911199063) 119910 + 120573ℎV119894119910 = 120573ℎV120599 (1 minus 119911) 119894

119910 =(1 minus 119911) 120573ℎV120599119894

120583V + 120572V + 1205911199063 + 120573ℎV120599119894

(31)


120572V119910 = (120583V + 1205911199063) 119911

119911 =120572V119910

120583V + 1205911199063

(32)


119911 =(1 minus 119911) 120572V120573ℎV120599119894

(120583V + 1205911199063) (120583V + 120572V + 1205911199063 + 120573ℎV120599119894)

=1198982119894

1198992 (1199011 + 1199021119894)minus

1198982119894119911

1198992 (1199011 + 1199021119894)

=1198982119894

11989921199011 + 11989921199021119894 + 1198982119894

(33)


119894 =119889 + 119892 + (1198871119888 + 1198871ℎ) (1198982119894 (11989921199011 + 11989921199021119894 + 1198982119894))

119892 + (1198871119888 + 1198871ℎ119896 + 11988711198881198861) (1198982119894 (11989921199011 + 11989921199021119894 + 1198982119894))

(34)



1198601198942+ 119861119894 + 119862 = 0 (35)

where119860 = 11989211989921199021 + 1198921198982 + 11988711198881198982 + 1198871ℎ1198961198982 + 119887111988811988611198982

= (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 120572V120573ℎV120599

+ (1 minus 1199061) 120573Vℎ120573ℎV120572ℎ120572V1205992

+ (1 minus 1199061) (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 120573Vℎ120573ℎV120572V1205992

+ (1 minus 1199061) (120601 + 1205781199062

120583ℎ + 120595)120573Vℎ120573ℎV120572ℎ120572V120599

2

gt 0

119862 = minus11988911989921199011 minus 11989921199011

= minus (120583V + 1205911199063) (120583V + 120572V + 1205911199063) [(1 minus 1199061)]

lt 0

119861 = 11988911989921199021 + 1198891198982 + 11989211989921199021 + 1198921198982 + 1198871ℎ1198982 + 11989211989921199011 minus 11988711198881198982

= (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599

+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 120572V120573ℎV120599

+ (1 minus 1199061) 120573Vℎ120573ℎV120572V1205992

+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) (120583V + 120572V + 1205911199063)


= (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599

+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599


+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) (120583V + 120572V + 1205911199063) (1 minusR2

119890)

(36)


119861 = (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599

+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 1205812120579120572V120573ℎV120599

+ (1 minus 1199061) 120573Vℎ120573ℎV120572V1205992+ (120583ℎ + 120572ℎ) (120583V + 1205911199063)

times (120583V + 120572V + 1205911199063) (1 minusR2

119890)

lt 0

(37)








= 119864V = 119868V = 0 with R1198900 = R119890 | 120575ℎ = 0

(38)




119889119864ℎ

119889119905= (1 minus 1199061) 120582ℎ (

Λ ℎ

120583ℎ


120583ℎ

minus (120572ℎ + 120583ℎ) 119864ℎ

119889119868ℎ

119889119905= 120572ℎ119864ℎ minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ

119889119864V

119889119905= 120582V (

Λ V


119889119868V

119889119905= 120572V119864V minus (120583V + 1205911199063) 119868V

(39)


120597

120597119864ℎ

[(1 minus 1199061) 120573Vℎ120599

119864ℎ119868ℎ119864VΛ ℎ120583ℎ(Λ ℎ

120583ℎ


+1205811120579Λ ℎ

119864ℎ119868ℎ119864V119868V120583ℎminus(120572ℎ + 120583ℎ)

119868ℎ119864V119868V]

+120597

120597119868ℎ

[120572ℎ

119868ℎ119864V119868Vminus(120601 + 1205781199062 + 120583ℎ + 120575ℎ)

119864ℎ119864V119868V]


+120597

120597119864V[

120573ℎV120599

119864ℎ119864V119868VΛ ℎ120583ℎ(Λ V

120583Vminus 119864V minus 119868V)

minus(120572V + 120583V + 1205911199063)

119864ℎ119868ℎ119864V]

+120597

120597119868V[

120572V

119864ℎ119868ℎ119868Vminus(120583V + 1205911199063)

119864ℎ119868ℎ119864V]

= minus120573Vℎ120599

1198642

ℎ119868ℎ119864V

minus120573Vℎ120599120583ℎ

1198642

ℎ119864V

(1 minus119877ℎ

Λ ℎ120583ℎ

) minus1205811120579Λ ℎ

1198642

ℎ119868ℎ119864V119868V120583ℎ

minus120572ℎ

1198682

ℎ119864V119868V

minus120573ℎV120599

119864ℎ1198642V119868V

[1 minus120583ℎ

Λ ℎ

] minus120572V

119864ℎ119868ℎ1198682V

lt 0

(40)








Γ (1199061 1199062 1199063) = int

119879119891

0

(1198621119864ℎ + 1198622119868ℎ + 1198623119873V

+1

2(11986011199062

1+ 1198602119906

2

2+ 1198603119906

2

3)) 119889119905

(41)



2




2

3 A lin-


2

1 1198602119906

2

2 and 1198603119906

2

3 such that the

terms (12)11986011199062

1 (12)1198602119906

2

2 and (12)1198603119906

2

3describe the cost




1(119905) 119906lowast

2(119905) and 119906

lowast

3(119905) such

that

Γ (119906lowast

1 119906lowast

2 119906lowast

3) = min(119906111990621199063)isinΦ2

Γ (1199061 1199062 1199063) (42)

where



(43)



119871 = 1198621119864ℎ + 1198622119868ℎ + 1198623119873V +1

2(11986011199062

1+ 1198602119906

2

2+ 1198603119906

2

3)

(44)



119867 = 119871 + 1205821

119889119878ℎ

119889119905+ 1205822

119889119864ℎ

119889119905+ 1205823

119889119868ℎ

119889119905+ 1205824

119889119877ℎ

119889119905

+ 1205825

119889119878V

119889119905+ 1205826

119889119864V

119889119905+ 1205827

119889119868V

119889119905

= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V +1

2(11986011199062

1+ 1198602119906

2

2+ 1198603119906

2

3)

+ 1205821 [Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119868ℎ



+ 1205823 [120572ℎ119864ℎ minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ]

+ 1205824 [(120601 + 1205781199062) 120588119868ℎ minus (120583ℎ + 120595)119877ℎ]


+ 1205826 [120573ℎV120599119868ℎ119878V minus (120572V + 120583V + 1205911199063) 119864V]

+ 1205827 [120572V119864V minus (120583V + 1205911199063) 119868V]

(45)



1 119906lowast

2 119906lowast


min(119906111990621199063)isinΦ2

Γ (1199061 1199062 1199063) = Γ (119906lowast

1 119906lowast

2 119906lowast

3) (46)


Γ (1199061 1199062 1199063) ge 1205851(100381610038161003816100381611990611003816100381610038161003816

2+100381610038161003816100381611990621003816100381610038161003816

2100381610038161003816100381611990631003816100381610038161003816

2)1205982

minus 1205852(47)





0 =120597119867 (119905 120594 119906 120582)

120597119906

1205821015840=120597119867 (119905 120594 119906 120582)

120597120594

119889120594

119889119905= minus

120597119867 (119905 120594 119906 120582)

120597120582

(48)



2 119906lowast

3) with


ℎ 119868lowast

ℎ 119877lowast

ℎ 119878lowast

V 119864lowast

V 119868lowast


minus1205821015840


minus 120583ℎ1205821 + 12057912058111205822

minus1205821015840


minus1205821015840

3= (1 minus 1205811) 1205791205821 + (120601 + 1205781199062) [(1 minus 120588) 1205821 + 1205881205824]

+ 12057912058111205822 minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 1205823

+ 120573ℎV120599 (1205826 minus 1205825) 119878V + 1198622

minus1205821015840

4= (1 minus 1205811) 1205791205821 + 1205951205821 + 12057912058111205822 minus (120583ℎ + 120595) 1205824

minus1205821015840

5= 120573ℎV120599 (1205826 minus 1205825) 119868ℎ minus (120583V + 1205911199063) 1205825 + 1198623

minus1205821015840

6= 120572V (1205827 minus 1205826) minus (120583V + 1205911199063) 1205826 + 1198623

minus1205821015840


(49)


1205821 (119879119891) = 1205822 (119879119891) = 1205823 (119879119891) = 1205824 (119879119891)

= 1205825 (119879119891) = 1205826 (119879119891) = 1205827 (119879119891) = 0

(50)


2 119906lowast

3) that mini-


119906lowast

1= max0min(1

120573Vℎ120599 (1205822 minus 1205821) 119868lowast

V 119878lowast

ℎ

1198601

)

119906lowast

2= max0min(1


ℎ

1198602

)

119906lowast

3= max0min(1

120591 (1205825119878lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

)

(51)




minus1198891205821

119889119905=120597119867

120597119878ℎ


minus 120583ℎ1205821 + 12057912058111205822

minus1198891205822

119889119905=120597119867

120597119864ℎ

= (1 minus 1205811) 1205791205821 + 12057912058111205822

+ 120572ℎ (1205823 minus 1205822) minus 120583ℎ1205822 + 1198621

minus1198891205823

119889119905=120597119867

120597119868ℎ

= (1 minus 1205811) 1205791205821

+ (120601 + 1205781199062) [(1 minus 120588) 1205821 + 1205881205824] + 12057912058111205822


minus1198891205824

119889119905=120597119867

120597119877ℎ

= (1 minus 1205811) 1205791205821 + 1205951205821 + 12057912058111205822 minus (120583ℎ + 120595) 1205824

minus1198891205825

119889119905=120597119867


minus1198891205826

119889119905=120597119867

120597119864V= 120572V (1205827 minus 1205826) minus (120583V + 1205911199063) 1205826 + 1198623

minus1198891205827

119889119905=120597119867

120597119868V= (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119878ℎ

minus (120583V + 1205911199063) 1205827 + 1198623

(52)


1205821 (119879119891) = 1205822 (119879119891) = 1205823 (119879119891) = 1205824 (119879119891)

= 1205825 (119879119891) = 1205826 (119879119891) = 1205827 (119879119891) = 0

(53)


lowast

ℎ

119864ℎ = 119864lowast

ℎ 119868ℎ = 119868

lowast

ℎ 119877ℎ = 119877

lowast

ℎ 119878V = 119878

lowast

V 119864V = 119864lowast

V and 119868V = 119868lowast

Vyields

120597119867

1205971199061

= 119860119906lowast

1+ 120573Vℎ1205991205821119868

lowast

V 119878lowast

ℎminus 120573Vℎ1205991205822119868

lowast

V 119878lowast

ℎ= 0

120597119867

1205971199062

= 11986021199062 + (1 minus 120588) 1205781205821119868lowast

ℎminus 1205781205823119868

lowast

ℎ+ 1205781205881205824119868

lowast

ℎ= 0

120597119867

1205971199063

= 1198603119906lowast

3minus 1205911205825119878

lowast

V minus 1205911205826119864lowast

V minus 1205911205827119868lowast

V = 0

(54)

for which

119906lowast

1=120573Vℎ120599 (1205822 minus 1205821) 119868

lowast

V 119878lowast

ℎ

1198601

119906lowast

2=120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868

lowast

ℎ

1198602

1199063 =120591 (1205825119878

lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

(55)

Then

119906lowast

1= max0min(1

120573Vℎ120599 (1205822 minus 1205821) 119868lowast

V 119878lowast

ℎ

1198601

)

119906lowast

2= max0min(1


ℎ

1198602

)

119906lowast

3= max0min(1

120591 (1205825119878lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

)

(56)




3



119889119878lowast

ℎ

119889119905= Λ ℎ + (1 minus 1205811) 120579119873

lowast

ℎ

+ (120601 + 1205781199062) (1 minus 120588) 119868lowast

ℎminus 120573Vℎ120599119868

lowast

V 119878lowast

ℎ


lowast

V 119878lowast

ℎ

1198601

))


ℎ+ 120595119877lowast

ℎ

119889119864lowast

ℎ

119889119905= 120573Vℎ120599119868

lowast

V 119878lowast

ℎ


lowast

V 119878lowast

ℎ

1198601

))

+ 1205811120579119873lowast

ℎminus 120572ℎ119864

lowast

ℎminus 120583ℎ119864

lowast

ℎ


119889119868lowast

ℎ

119889119905

= 120572ℎ119864lowast

ℎ

minus (120601 + 120578


lowast

ℎ

1198602

))

+120583ℎ + 120575ℎ) 119868lowast

ℎ

119889119877lowast

ℎ


times 119868lowast

ℎtimes (1198602)

minus1))) 120588119868ℎ

minus (120583ℎ + 120595)119877ℎ

119889119878lowast

V

119889119905

= Λ V minus (120583V + 120591


V + 1205826119864lowast

V + 1205827119868lowast

V )

times(1198603)minus1))) 119878

lowast


ℎ119878lowast

V

119889119864lowast

V

119889119905= 120573ℎV120599119868

lowast

ℎ119878lowast

V


V + 1205826119864lowast

V + 1205827119868lowast

V )

times(1198603)minus1))) 119864

lowast


V

119889119868lowast

V

119889119905= 120572V119864V


V + 1205826119864lowast

V +1205827119868lowast

V )

times(1198603)minus1))) 119868

lowast

V

(57)


ℎ 119868lowast

ℎ 119877lowast

ℎ 119906lowast

1 119906lowast

2 119906lowast

3 1205821 1205822 1205827)

119867lowast

= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V

+1

2(1198601(max0min(1

120573Vℎ120599 (1205822 minus 1205821) 119868lowast

V 119878lowast

ℎ

1198601

))

2


ℎ


+1198603(max0min(1120591(1205825119878

lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

))

2

)

+ 1205821

119889119878lowast

ℎ

119889119905+ 1205822

119889119864lowast

ℎ

119889119905+ 1205823

119889119868lowast

ℎ

119889119905+ 1205824

119889119877lowast

ℎ

119889119905+ 1205825

119889119878lowast

V

119889119905

+ 1205826

119889119864lowast

V

119889119905+ 1205827

119889119868lowast

V

119889119905







u1 ne 0 u3 ne 0 u2 = 0


0

10

20

30

40

50

60

70

times102In

fect

ed h

uman

pop

ulat

ion

0 10 20 30 40 50 60 70 80 90 100

Time (years)

(a)



fect

ed h

uman

pop

ulat

ion

0

10

20

30

40

50

60

70

0 10 20 30 40 50 60 70 80 90 100

times102

Time (days)

(b)

u2 ne 0 u3 ne 0 u1 = 0


0 10 20 30 40 50 600

5

10

15

20

25

30

35

40

times102

Infe

cted

hum

an p

opul

atio

n

Time (years)

(c)

u1 ne 0 u2 ne 0 u3 0ne ==u2 ne 0 u1 0 u3 0

=u1 ne 0 u2 0 u3 = 0 = =u3 ne 0 u1 0 u2 0

0

05

1

15

2

25

3

times104

0 10 20 30 40 50 60 70

Infe

cted

hum

an p

opul

atio

n

Time (years)

(d)








of total 22 30 53








409

93 53 47 7 4484

400440 446

486449

9 9 9 9 9 90

100

200

300

400

500

600

Bite

from

anin

fect

edm

osqu

ito

Fem

ale

mos

quito

Stag

nant

wat

er

Envi

ronm

ent

man

agem

ent

Oth

er

YesNo


wat

erso

aked

in ra

inRa

ined

on

Anopheles

Do not know


Seek treatment


47

418

493

462

409

454

83

8

38

92

0 100 200 300 400 500 600

Other

YesNo


House sprayed(IRS)

Use of ITNor LLN










5 Numerical Results






R0 = 18894

0 05 1 15 2 25 3 35 4

Time (years)

0

20

40

60

80

100

120

Susc

eptib

le in

divi

dual

s


(a)

R0 = 18894

0 05 1 15 2 25 3 35 4

Time (years)

0

20

40

60

80

100

120

140

times102

Expo

sed

indi

vidu

als

Exposed individuals

(b)

R0 = 18894

0 05 1 15 2 25 3 35 49

10

11

12

13

14

15

16

17

Time (years)

Infe

cted

indi

vidu

als


(c)

0 100 200 300 400 500 600 7000

1

2

3

4

5

6

7

8

Time (days)

Reco

vere

d in

divi

dual

s


R0 = 18894

times102

(d)








No Yes


of total 36 57 92



Time (days)

R0 = 18894

Prev

alen

ce

0 20 40 60 80 100 120 140 160 180 2000

01

02

03

04

05

06

07



6 Conclusions












of total 425 384 809



Hum

an p

opul

atio

n


times104

Time (years)0 05 1 15 2 25 3 35 4

0

05

1

15

2

25

(a)

times105

Hum

an p

opul

atio

n


Time (years)0 10 20 30 40 50 60

0

02

04

06

08

1

12

14

16

18

2

(b)


















Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











R119890ℎ =120573Vℎ120599120583ℎ120572ℎ (1 minus 1199061)

Λ ℎ (120572ℎ + 120583ℎ) (120601 + 1205781199062 + 120583ℎ + 120575ℎ)(14)






R119890119905 = radic120573ℎV120573Vℎ120599

2120572ℎ120572VΛ V120583ℎ

(120572ℎ + 120583ℎ) (120601 + 1205781199062 + 120583ℎ + 120575ℎ) (120572V + 120583V) 1205832VΛ ℎ

(15)


R119890119873 = radic120573ℎV120573Vℎ120599

2120572ℎ120572VΛ V120583ℎ (1 minus 1199061)

(120572ℎ + 120583ℎ) (120601 + 120583ℎ + 120575ℎ) (120572V + 120583V) 1205832VΛ ℎ

(16)


R119890119904 = (120573ℎV120573Vℎ1205992120572ℎ120572V120583ℎΛ V

times ((120572ℎ + 120583ℎ) (120601 + 120583ℎ + 120575ℎ) (120572V + 120583V + 1205911199063)

times (120583V + 1205911199063) 120583VΛ ℎ)minus1)12

(17)


R0 = radic120573ℎV120573119906ℎ120599

2120572ℎ120572VΛ V120583ℎ

(120572ℎ + 120583ℎ) (120601 + 120583ℎ + 120575ℎ) (120572V + 120583V) 1205832VΛ ℎ

(18)


R119890 le R0 (19)




119889119904

119889119905=

1

119873ℎ

[119889119878ℎ

119889119905minus 119904

119889119873ℎ

119889119905]

=1

119873ℎ

[Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119894119873ℎ


(20)


119889119904

119889119905=

1

119873ℎ


minus119904

119873ℎ


=Λ ℎ

119873ℎ

minus [Λ ℎ

119873ℎ

+ 120579 minus 120575ℎ119894] 119904 + (1 minus 1205811) 120579

+ (120601 + 1205781199062) (1 minus 120588) 119894

minus (1 minus 1199061) 120573Vℎ120599119911119904 + 120595119903

119889119890

119889119905=

1

119873ℎ

[119889119864

119889119905minus 119890

119889119873ℎ

119889119905]

=1

119873ℎ


minus119890

119873ℎ


= (1 minus 1199061) 120573Vℎ120599119911119904 + 1205811120579 minus [Λ ℎ

119873ℎ

minus 120575ℎ119894 + 120572ℎ + 120579] 119890


119889119894

119889119905=

1

119873ℎ

[119889119868

119889119905minus 119894119889119873ℎ

119889119905]

=1

119873ℎ

[120572ℎ119890119873ℎ minus (120601 + 1205781199062) (1 minus 120588) 119894119873ℎ

minus (120601 + 1205781199062) 120588119894119873ℎ minus (120583ℎ + 120575ℎ) 119894119873ℎ]

minus119894

119873ℎ


= 120572ℎ119890 minus [Λ ℎ

119873ℎ

+ 120601 + 1205781199062 + 120575ℎ + 120579 minus 120575ℎ119894] 119894

119889119903

119889119905=

1

119873ℎ

[119889119877

119889119905minus 119903

119889119873ℎ

119889119905]

=1

119873ℎ

[(120601 + 1205781199062) 120588119894119873ℎ minus (120583ℎ + 120595) 119903119873ℎ]

minus119903

119873ℎ


= (120601 + 1205781199062) 120588119894 minus [Λ ℎ

119873ℎ

+ 120595 + 120579 minus 120575ℎ119894] 119903

119889119909

119889119905=

1

119873V[119889119878V

119889119905minus 119909

119889119873V

119889119905]

=1


minus119909

119873V[Λ V minus 120583V119873V]

= [Λ V

119873Vminus 119909

Λ V

119873V] minus 120573ℎV120599119894119909 minus 1205911199063119909

119889119910

119889119905=

1

119873V[119889119864V

119889119905minus 119910

119889119873V

119889119905]

=1


minus119910

119873V[Λ V minus 120583V119873V]

= 120573ℎV120599119894119909 minus [Λ V

119873V+ 120572V + 1205911199063]119910

119889119911

119889119905=

1

119873V[119889119868V

119889119905minus 119885

119889119873V

119889119905]

=1

119873V[120572V119910119873V minus (120583V + 1205911199063) 119911119873V]

minus119911

119873V[Λ V minus 120583V119873V]

= 120572V119910 minus [Λ V

119873V+ 1205911199063] 119911

119889119873ℎ

119889119905= [

Λ ℎ

119873ℎ

minus 120583ℎ minus 120575ℎ119894]119873ℎ

119889119873V

119889119905= [

Λ V

119873Vminus 120583V]119873V

(21)


Φ1 = 119890 119894 119903119873ℎ 119910 119911119873V isin R7

+| 119890 ge 0 119894 ge 0 119903 ge 0

119890 + 119894 + 119903 le 1 119873ℎ leΛ ℎ

120583ℎ

119910 ge 0 119911 ge 0 119910 + 119911 le 1

119873V leΛ V

120583V

(22)



119889119890


minus [Λ ℎ

119873ℎ

minus 120575ℎ119894 + 120572ℎ + 120579] 119890

119889119894

119889119905= 120572ℎ119890 minus [

Λ ℎ

119873ℎ

+ 120601 + 1205781199062 + 120575ℎ + 120579 minus 120575ℎ119894] 119894

119889119903

119889119905= (120601 + 1205781199062) 120588119894 minus [

Λ ℎ

119873ℎ

+ 120595 + 120579 + minus120575ℎ119894] 119903

119889119873ℎ


119889119910


Λ V

119873V+ 120572V + 1205911199063] 119910

119889119911

119889119905= 120572V minus [

Λ V

119873V+ 1205911199063] 119911

119889119873V

119889119905= Λ V minus 120583V119873V

(23)






119873ℎ

minus 120575ℎ119894 + 120572ℎ + 120579] 119890

120572ℎ119890 = [Λ ℎ

119873ℎ

+ 120601 + 1205781199062 + 120575ℎ + 120579 minus 120575ℎ119894] 119894

(120601 + 1205781199062) 120588119894 = [Λ ℎ

119873ℎ

+ 120595 + 120579 minus 120575ℎ119894] 119903

Λ ℎ

119873ℎ

= 120575ℎ119894 + 120583ℎ

(1 minus 119910 minus 119911) 120573ℎV120599119894 = [Λ V

119873V+ 120572V + 1205911199063]119910

120572V119910 = [Λ V

119873V+ 1205781199063] 119911

Λ V

119873V= 120583V

(24)



(120583ℎ + 120572ℎ) 119890 + (1 minus 1199061) 120573Vℎ120599119911119890


(120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ119911) 119890



120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911

(25)


120572ℎ119890 = (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894

120572ℎ119890 = (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894

119890 =(120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894

120572ℎ

(26)



=(120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894

120572ℎ

(1 minus 1199061) 120573Vℎ120599120572ℎ119911119894

+ [120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911 (120583ℎ + 120601 + 1205781199062 + 120575ℎ)] 119894

= (1 minus 1199061) (1 minus 119903) 120573Vℎ120599120572ℎ119911 + 1205811120579120572ℎ

+ 120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911

(27)


119894 = ((1 minus 1199061) (1 minus 119903) 120573Vℎ120599120572ℎ119911 + 1205811120579120572ℎ

+120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911)

times ((1 minus 1199061) 120573Vℎ120599120572ℎ119911119894


(28)


(120601 + 1205781199062) 120588119894 = (120583ℎ + 120595) 119903

119903 =(120601 + 1205781199062) 119894

120583ℎ + 120595

(29)


119894 =1198871 (1 minus 1198861119894) 119888119911 + 119889 + 119892 + 119887ℎ119911

1198871119888119911 + 119892 + 1198871ℎ119896119911

(1198871119888119911 + 119892 + 1198871ℎ119896119911 + 11988711198881198861119911

1198871119888119911 + 119892 + 1198871ℎ119896119911) 119894 =

1198871119888119911 + 119889 + 119892 + 1198871ℎ119911

1198871119888119911 + 119892 + 1198871ℎ119896119911

119894 =1198871119888119911 + 119889 + 119892 + 1198871ℎ119911

1198871119888119911 + 119892 + 1198871ℎ119896119911 + 11988711198881198861119911

(30)


(120583V + 120572V + 1205911199063) 119910 + 120573ℎV119894119910 = 120573ℎV120599 (1 minus 119911) 119894

119910 =(1 minus 119911) 120573ℎV120599119894

120583V + 120572V + 1205911199063 + 120573ℎV120599119894

(31)


120572V119910 = (120583V + 1205911199063) 119911

119911 =120572V119910

120583V + 1205911199063

(32)


119911 =(1 minus 119911) 120572V120573ℎV120599119894

(120583V + 1205911199063) (120583V + 120572V + 1205911199063 + 120573ℎV120599119894)

=1198982119894

1198992 (1199011 + 1199021119894)minus

1198982119894119911

1198992 (1199011 + 1199021119894)

=1198982119894

11989921199011 + 11989921199021119894 + 1198982119894

(33)


119894 =119889 + 119892 + (1198871119888 + 1198871ℎ) (1198982119894 (11989921199011 + 11989921199021119894 + 1198982119894))

119892 + (1198871119888 + 1198871ℎ119896 + 11988711198881198861) (1198982119894 (11989921199011 + 11989921199021119894 + 1198982119894))

(34)



1198601198942+ 119861119894 + 119862 = 0 (35)

where119860 = 11989211989921199021 + 1198921198982 + 11988711198881198982 + 1198871ℎ1198961198982 + 119887111988811988611198982

= (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 120572V120573ℎV120599

+ (1 minus 1199061) 120573Vℎ120573ℎV120572ℎ120572V1205992

+ (1 minus 1199061) (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 120573Vℎ120573ℎV120572V1205992

+ (1 minus 1199061) (120601 + 1205781199062

120583ℎ + 120595)120573Vℎ120573ℎV120572ℎ120572V120599

2

gt 0

119862 = minus11988911989921199011 minus 11989921199011

= minus (120583V + 1205911199063) (120583V + 120572V + 1205911199063) [(1 minus 1199061)]

lt 0

119861 = 11988911989921199021 + 1198891198982 + 11989211989921199021 + 1198921198982 + 1198871ℎ1198982 + 11989211989921199011 minus 11988711198881198982

= (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599

+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 120572V120573ℎV120599

+ (1 minus 1199061) 120573Vℎ120573ℎV120572V1205992

+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) (120583V + 120572V + 1205911199063)


= (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599

+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599


+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) (120583V + 120572V + 1205911199063) (1 minusR2

119890)

(36)


119861 = (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599

+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 1205812120579120572V120573ℎV120599

+ (1 minus 1199061) 120573Vℎ120573ℎV120572V1205992+ (120583ℎ + 120572ℎ) (120583V + 1205911199063)

times (120583V + 120572V + 1205911199063) (1 minusR2

119890)

lt 0

(37)








= 119864V = 119868V = 0 with R1198900 = R119890 | 120575ℎ = 0

(38)




119889119864ℎ

119889119905= (1 minus 1199061) 120582ℎ (

Λ ℎ

120583ℎ


120583ℎ

minus (120572ℎ + 120583ℎ) 119864ℎ

119889119868ℎ

119889119905= 120572ℎ119864ℎ minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ

119889119864V

119889119905= 120582V (

Λ V


119889119868V

119889119905= 120572V119864V minus (120583V + 1205911199063) 119868V

(39)


120597

120597119864ℎ

[(1 minus 1199061) 120573Vℎ120599

119864ℎ119868ℎ119864VΛ ℎ120583ℎ(Λ ℎ

120583ℎ


+1205811120579Λ ℎ

119864ℎ119868ℎ119864V119868V120583ℎminus(120572ℎ + 120583ℎ)

119868ℎ119864V119868V]

+120597

120597119868ℎ

[120572ℎ

119868ℎ119864V119868Vminus(120601 + 1205781199062 + 120583ℎ + 120575ℎ)

119864ℎ119864V119868V]


+120597

120597119864V[

120573ℎV120599

119864ℎ119864V119868VΛ ℎ120583ℎ(Λ V

120583Vminus 119864V minus 119868V)

minus(120572V + 120583V + 1205911199063)

119864ℎ119868ℎ119864V]

+120597

120597119868V[

120572V

119864ℎ119868ℎ119868Vminus(120583V + 1205911199063)

119864ℎ119868ℎ119864V]

= minus120573Vℎ120599

1198642

ℎ119868ℎ119864V

minus120573Vℎ120599120583ℎ

1198642

ℎ119864V

(1 minus119877ℎ

Λ ℎ120583ℎ

) minus1205811120579Λ ℎ

1198642

ℎ119868ℎ119864V119868V120583ℎ

minus120572ℎ

1198682

ℎ119864V119868V

minus120573ℎV120599

119864ℎ1198642V119868V

[1 minus120583ℎ

Λ ℎ

] minus120572V

119864ℎ119868ℎ1198682V

lt 0

(40)








Γ (1199061 1199062 1199063) = int

119879119891

0

(1198621119864ℎ + 1198622119868ℎ + 1198623119873V

+1

2(11986011199062

1+ 1198602119906

2

2+ 1198603119906

2

3)) 119889119905

(41)



2




2

3 A lin-


2

1 1198602119906

2

2 and 1198603119906

2

3 such that the

terms (12)11986011199062

1 (12)1198602119906

2

2 and (12)1198603119906

2

3describe the cost




1(119905) 119906lowast

2(119905) and 119906

lowast

3(119905) such

that

Γ (119906lowast

1 119906lowast

2 119906lowast

3) = min(119906111990621199063)isinΦ2

Γ (1199061 1199062 1199063) (42)

where



(43)



119871 = 1198621119864ℎ + 1198622119868ℎ + 1198623119873V +1

2(11986011199062

1+ 1198602119906

2

2+ 1198603119906

2

3)

(44)



119867 = 119871 + 1205821

119889119878ℎ

119889119905+ 1205822

119889119864ℎ

119889119905+ 1205823

119889119868ℎ

119889119905+ 1205824

119889119877ℎ

119889119905

+ 1205825

119889119878V

119889119905+ 1205826

119889119864V

119889119905+ 1205827

119889119868V

119889119905

= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V +1

2(11986011199062

1+ 1198602119906

2

2+ 1198603119906

2

3)

+ 1205821 [Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119868ℎ



+ 1205823 [120572ℎ119864ℎ minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ]

+ 1205824 [(120601 + 1205781199062) 120588119868ℎ minus (120583ℎ + 120595)119877ℎ]


+ 1205826 [120573ℎV120599119868ℎ119878V minus (120572V + 120583V + 1205911199063) 119864V]

+ 1205827 [120572V119864V minus (120583V + 1205911199063) 119868V]

(45)



1 119906lowast

2 119906lowast


min(119906111990621199063)isinΦ2

Γ (1199061 1199062 1199063) = Γ (119906lowast

1 119906lowast

2 119906lowast

3) (46)


Γ (1199061 1199062 1199063) ge 1205851(100381610038161003816100381611990611003816100381610038161003816

2+100381610038161003816100381611990621003816100381610038161003816

2100381610038161003816100381611990631003816100381610038161003816

2)1205982

minus 1205852(47)





0 =120597119867 (119905 120594 119906 120582)

120597119906

1205821015840=120597119867 (119905 120594 119906 120582)

120597120594

119889120594

119889119905= minus

120597119867 (119905 120594 119906 120582)

120597120582

(48)



2 119906lowast

3) with


ℎ 119868lowast

ℎ 119877lowast

ℎ 119878lowast

V 119864lowast

V 119868lowast


minus1205821015840


minus 120583ℎ1205821 + 12057912058111205822

minus1205821015840


minus1205821015840

3= (1 minus 1205811) 1205791205821 + (120601 + 1205781199062) [(1 minus 120588) 1205821 + 1205881205824]

+ 12057912058111205822 minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 1205823

+ 120573ℎV120599 (1205826 minus 1205825) 119878V + 1198622

minus1205821015840

4= (1 minus 1205811) 1205791205821 + 1205951205821 + 12057912058111205822 minus (120583ℎ + 120595) 1205824

minus1205821015840

5= 120573ℎV120599 (1205826 minus 1205825) 119868ℎ minus (120583V + 1205911199063) 1205825 + 1198623

minus1205821015840

6= 120572V (1205827 minus 1205826) minus (120583V + 1205911199063) 1205826 + 1198623

minus1205821015840


(49)


1205821 (119879119891) = 1205822 (119879119891) = 1205823 (119879119891) = 1205824 (119879119891)

= 1205825 (119879119891) = 1205826 (119879119891) = 1205827 (119879119891) = 0

(50)


2 119906lowast

3) that mini-


119906lowast

1= max0min(1

120573Vℎ120599 (1205822 minus 1205821) 119868lowast

V 119878lowast

ℎ

1198601

)

119906lowast

2= max0min(1


ℎ

1198602

)

119906lowast

3= max0min(1

120591 (1205825119878lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

)

(51)




minus1198891205821

119889119905=120597119867

120597119878ℎ


minus 120583ℎ1205821 + 12057912058111205822

minus1198891205822

119889119905=120597119867

120597119864ℎ

= (1 minus 1205811) 1205791205821 + 12057912058111205822

+ 120572ℎ (1205823 minus 1205822) minus 120583ℎ1205822 + 1198621

minus1198891205823

119889119905=120597119867

120597119868ℎ

= (1 minus 1205811) 1205791205821

+ (120601 + 1205781199062) [(1 minus 120588) 1205821 + 1205881205824] + 12057912058111205822


minus1198891205824

119889119905=120597119867

120597119877ℎ

= (1 minus 1205811) 1205791205821 + 1205951205821 + 12057912058111205822 minus (120583ℎ + 120595) 1205824

minus1198891205825

119889119905=120597119867


minus1198891205826

119889119905=120597119867

120597119864V= 120572V (1205827 minus 1205826) minus (120583V + 1205911199063) 1205826 + 1198623

minus1198891205827

119889119905=120597119867

120597119868V= (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119878ℎ

minus (120583V + 1205911199063) 1205827 + 1198623

(52)


1205821 (119879119891) = 1205822 (119879119891) = 1205823 (119879119891) = 1205824 (119879119891)

= 1205825 (119879119891) = 1205826 (119879119891) = 1205827 (119879119891) = 0

(53)


lowast

ℎ

119864ℎ = 119864lowast

ℎ 119868ℎ = 119868

lowast

ℎ 119877ℎ = 119877

lowast

ℎ 119878V = 119878

lowast

V 119864V = 119864lowast

V and 119868V = 119868lowast

Vyields

120597119867

1205971199061

= 119860119906lowast

1+ 120573Vℎ1205991205821119868

lowast

V 119878lowast

ℎminus 120573Vℎ1205991205822119868

lowast

V 119878lowast

ℎ= 0

120597119867

1205971199062

= 11986021199062 + (1 minus 120588) 1205781205821119868lowast

ℎminus 1205781205823119868

lowast

ℎ+ 1205781205881205824119868

lowast

ℎ= 0

120597119867

1205971199063

= 1198603119906lowast

3minus 1205911205825119878

lowast

V minus 1205911205826119864lowast

V minus 1205911205827119868lowast

V = 0

(54)

for which

119906lowast

1=120573Vℎ120599 (1205822 minus 1205821) 119868

lowast

V 119878lowast

ℎ

1198601

119906lowast

2=120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868

lowast

ℎ

1198602

1199063 =120591 (1205825119878

lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

(55)

Then

119906lowast

1= max0min(1

120573Vℎ120599 (1205822 minus 1205821) 119868lowast

V 119878lowast

ℎ

1198601

)

119906lowast

2= max0min(1


ℎ

1198602

)

119906lowast

3= max0min(1

120591 (1205825119878lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

)

(56)




3



119889119878lowast

ℎ

119889119905= Λ ℎ + (1 minus 1205811) 120579119873

lowast

ℎ

+ (120601 + 1205781199062) (1 minus 120588) 119868lowast

ℎminus 120573Vℎ120599119868

lowast

V 119878lowast

ℎ


lowast

V 119878lowast

ℎ

1198601

))


ℎ+ 120595119877lowast

ℎ

119889119864lowast

ℎ

119889119905= 120573Vℎ120599119868

lowast

V 119878lowast

ℎ


lowast

V 119878lowast

ℎ

1198601

))

+ 1205811120579119873lowast

ℎminus 120572ℎ119864

lowast

ℎminus 120583ℎ119864

lowast

ℎ


119889119868lowast

ℎ

119889119905

= 120572ℎ119864lowast

ℎ

minus (120601 + 120578


lowast

ℎ

1198602

))

+120583ℎ + 120575ℎ) 119868lowast

ℎ

119889119877lowast

ℎ


times 119868lowast

ℎtimes (1198602)

minus1))) 120588119868ℎ

minus (120583ℎ + 120595)119877ℎ

119889119878lowast

V

119889119905

= Λ V minus (120583V + 120591


V + 1205826119864lowast

V + 1205827119868lowast

V )

times(1198603)minus1))) 119878

lowast


ℎ119878lowast

V

119889119864lowast

V

119889119905= 120573ℎV120599119868

lowast

ℎ119878lowast

V


V + 1205826119864lowast

V + 1205827119868lowast

V )

times(1198603)minus1))) 119864

lowast


V

119889119868lowast

V

119889119905= 120572V119864V


V + 1205826119864lowast

V +1205827119868lowast

V )

times(1198603)minus1))) 119868

lowast

V

(57)


ℎ 119868lowast

ℎ 119877lowast

ℎ 119906lowast

1 119906lowast

2 119906lowast

3 1205821 1205822 1205827)

119867lowast

= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V

+1

2(1198601(max0min(1

120573Vℎ120599 (1205822 minus 1205821) 119868lowast

V 119878lowast

ℎ

1198601

))

2


ℎ


+1198603(max0min(1120591(1205825119878

lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

))

2

)

+ 1205821

119889119878lowast

ℎ

119889119905+ 1205822

119889119864lowast

ℎ

119889119905+ 1205823

119889119868lowast

ℎ

119889119905+ 1205824

119889119877lowast

ℎ

119889119905+ 1205825

119889119878lowast

V

119889119905

+ 1205826

119889119864lowast

V

119889119905+ 1205827

119889119868lowast

V

119889119905







u1 ne 0 u3 ne 0 u2 = 0


0

10

20

30

40

50

60

70

times102In

fect

ed h

uman

pop

ulat

ion

0 10 20 30 40 50 60 70 80 90 100

Time (years)

(a)



fect

ed h

uman

pop

ulat

ion

0

10

20

30

40

50

60

70

0 10 20 30 40 50 60 70 80 90 100

times102

Time (days)

(b)

u2 ne 0 u3 ne 0 u1 = 0


0 10 20 30 40 50 600

5

10

15

20

25

30

35

40

times102

Infe

cted

hum

an p

opul

atio

n

Time (years)

(c)

u1 ne 0 u2 ne 0 u3 0ne ==u2 ne 0 u1 0 u3 0

=u1 ne 0 u2 0 u3 = 0 = =u3 ne 0 u1 0 u2 0

0

05

1

15

2

25

3

times104

0 10 20 30 40 50 60 70

Infe

cted

hum

an p

opul

atio

n

Time (years)

(d)








of total 22 30 53








409

93 53 47 7 4484

400440 446

486449

9 9 9 9 9 90

100

200

300

400

500

600

Bite

from

anin

fect

edm

osqu

ito

Fem

ale

mos

quito

Stag

nant

wat

er

Envi

ronm

ent

man

agem

ent

Oth

er

YesNo


wat

erso

aked

in ra

inRa

ined

on

Anopheles

Do not know


Seek treatment


47

418

493

462

409

454

83

8

38

92

0 100 200 300 400 500 600

Other

YesNo


House sprayed(IRS)

Use of ITNor LLN










5 Numerical Results






R0 = 18894

0 05 1 15 2 25 3 35 4

Time (years)

0

20

40

60

80

100

120

Susc

eptib

le in

divi

dual

s


(a)

R0 = 18894

0 05 1 15 2 25 3 35 4

Time (years)

0

20

40

60

80

100

120

140

times102

Expo

sed

indi

vidu

als

Exposed individuals

(b)

R0 = 18894

0 05 1 15 2 25 3 35 49

10

11

12

13

14

15

16

17

Time (years)

Infe

cted

indi

vidu

als


(c)

0 100 200 300 400 500 600 7000

1

2

3

4

5

6

7

8

Time (days)

Reco

vere

d in

divi

dual

s


R0 = 18894

times102

(d)








No Yes


of total 36 57 92



Time (days)

R0 = 18894

Prev

alen

ce

0 20 40 60 80 100 120 140 160 180 2000

01

02

03

04

05

06

07



6 Conclusions












of total 425 384 809



Hum

an p

opul

atio

n


times104

Time (years)0 05 1 15 2 25 3 35 4

0

05

1

15

2

25

(a)

times105

Hum

an p

opul

atio

n


Time (years)0 10 20 30 40 50 60

0

02

04

06

08

1

12

14

16

18

2

(b)


















Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119889119894

119889119905=

1

119873ℎ

[119889119868

119889119905minus 119894119889119873ℎ

119889119905]

=1

119873ℎ

[120572ℎ119890119873ℎ minus (120601 + 1205781199062) (1 minus 120588) 119894119873ℎ

minus (120601 + 1205781199062) 120588119894119873ℎ minus (120583ℎ + 120575ℎ) 119894119873ℎ]

minus119894

119873ℎ


= 120572ℎ119890 minus [Λ ℎ

119873ℎ

+ 120601 + 1205781199062 + 120575ℎ + 120579 minus 120575ℎ119894] 119894

119889119903

119889119905=

1

119873ℎ

[119889119877

119889119905minus 119903

119889119873ℎ

119889119905]

=1

119873ℎ

[(120601 + 1205781199062) 120588119894119873ℎ minus (120583ℎ + 120595) 119903119873ℎ]

minus119903

119873ℎ


= (120601 + 1205781199062) 120588119894 minus [Λ ℎ

119873ℎ

+ 120595 + 120579 minus 120575ℎ119894] 119903

119889119909

119889119905=

1

119873V[119889119878V

119889119905minus 119909

119889119873V

119889119905]

=1


minus119909

119873V[Λ V minus 120583V119873V]

= [Λ V

119873Vminus 119909

Λ V

119873V] minus 120573ℎV120599119894119909 minus 1205911199063119909

119889119910

119889119905=

1

119873V[119889119864V

119889119905minus 119910

119889119873V

119889119905]

=1


minus119910

119873V[Λ V minus 120583V119873V]

= 120573ℎV120599119894119909 minus [Λ V

119873V+ 120572V + 1205911199063]119910

119889119911

119889119905=

1

119873V[119889119868V

119889119905minus 119885

119889119873V

119889119905]

=1

119873V[120572V119910119873V minus (120583V + 1205911199063) 119911119873V]

minus119911

119873V[Λ V minus 120583V119873V]

= 120572V119910 minus [Λ V

119873V+ 1205911199063] 119911

119889119873ℎ

119889119905= [

Λ ℎ

119873ℎ

minus 120583ℎ minus 120575ℎ119894]119873ℎ

119889119873V

119889119905= [

Λ V

119873Vminus 120583V]119873V

(21)


Φ1 = 119890 119894 119903119873ℎ 119910 119911119873V isin R7

+| 119890 ge 0 119894 ge 0 119903 ge 0

119890 + 119894 + 119903 le 1 119873ℎ leΛ ℎ

120583ℎ

119910 ge 0 119911 ge 0 119910 + 119911 le 1

119873V leΛ V

120583V

(22)



119889119890


minus [Λ ℎ

119873ℎ

minus 120575ℎ119894 + 120572ℎ + 120579] 119890

119889119894

119889119905= 120572ℎ119890 minus [

Λ ℎ

119873ℎ

+ 120601 + 1205781199062 + 120575ℎ + 120579 minus 120575ℎ119894] 119894

119889119903

119889119905= (120601 + 1205781199062) 120588119894 minus [

Λ ℎ

119873ℎ

+ 120595 + 120579 + minus120575ℎ119894] 119903

119889119873ℎ


119889119910


Λ V

119873V+ 120572V + 1205911199063] 119910

119889119911

119889119905= 120572V minus [

Λ V

119873V+ 1205911199063] 119911

119889119873V

119889119905= Λ V minus 120583V119873V

(23)






119873ℎ

minus 120575ℎ119894 + 120572ℎ + 120579] 119890

120572ℎ119890 = [Λ ℎ

119873ℎ

+ 120601 + 1205781199062 + 120575ℎ + 120579 minus 120575ℎ119894] 119894

(120601 + 1205781199062) 120588119894 = [Λ ℎ

119873ℎ

+ 120595 + 120579 minus 120575ℎ119894] 119903

Λ ℎ

119873ℎ

= 120575ℎ119894 + 120583ℎ

(1 minus 119910 minus 119911) 120573ℎV120599119894 = [Λ V

119873V+ 120572V + 1205911199063]119910

120572V119910 = [Λ V

119873V+ 1205781199063] 119911

Λ V

119873V= 120583V

(24)



(120583ℎ + 120572ℎ) 119890 + (1 minus 1199061) 120573Vℎ120599119911119890


(120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ119911) 119890



120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911

(25)


120572ℎ119890 = (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894

120572ℎ119890 = (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894

119890 =(120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894

120572ℎ

(26)



=(120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894

120572ℎ

(1 minus 1199061) 120573Vℎ120599120572ℎ119911119894

+ [120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911 (120583ℎ + 120601 + 1205781199062 + 120575ℎ)] 119894

= (1 minus 1199061) (1 minus 119903) 120573Vℎ120599120572ℎ119911 + 1205811120579120572ℎ

+ 120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911

(27)


119894 = ((1 minus 1199061) (1 minus 119903) 120573Vℎ120599120572ℎ119911 + 1205811120579120572ℎ

+120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911)

times ((1 minus 1199061) 120573Vℎ120599120572ℎ119911119894


(28)


(120601 + 1205781199062) 120588119894 = (120583ℎ + 120595) 119903

119903 =(120601 + 1205781199062) 119894

120583ℎ + 120595

(29)


119894 =1198871 (1 minus 1198861119894) 119888119911 + 119889 + 119892 + 119887ℎ119911

1198871119888119911 + 119892 + 1198871ℎ119896119911

(1198871119888119911 + 119892 + 1198871ℎ119896119911 + 11988711198881198861119911

1198871119888119911 + 119892 + 1198871ℎ119896119911) 119894 =

1198871119888119911 + 119889 + 119892 + 1198871ℎ119911

1198871119888119911 + 119892 + 1198871ℎ119896119911

119894 =1198871119888119911 + 119889 + 119892 + 1198871ℎ119911

1198871119888119911 + 119892 + 1198871ℎ119896119911 + 11988711198881198861119911

(30)


(120583V + 120572V + 1205911199063) 119910 + 120573ℎV119894119910 = 120573ℎV120599 (1 minus 119911) 119894

119910 =(1 minus 119911) 120573ℎV120599119894

120583V + 120572V + 1205911199063 + 120573ℎV120599119894

(31)


120572V119910 = (120583V + 1205911199063) 119911

119911 =120572V119910

120583V + 1205911199063

(32)


119911 =(1 minus 119911) 120572V120573ℎV120599119894

(120583V + 1205911199063) (120583V + 120572V + 1205911199063 + 120573ℎV120599119894)

=1198982119894

1198992 (1199011 + 1199021119894)minus

1198982119894119911

1198992 (1199011 + 1199021119894)

=1198982119894

11989921199011 + 11989921199021119894 + 1198982119894

(33)


119894 =119889 + 119892 + (1198871119888 + 1198871ℎ) (1198982119894 (11989921199011 + 11989921199021119894 + 1198982119894))

119892 + (1198871119888 + 1198871ℎ119896 + 11988711198881198861) (1198982119894 (11989921199011 + 11989921199021119894 + 1198982119894))

(34)



1198601198942+ 119861119894 + 119862 = 0 (35)

where119860 = 11989211989921199021 + 1198921198982 + 11988711198881198982 + 1198871ℎ1198961198982 + 119887111988811988611198982

= (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 120572V120573ℎV120599

+ (1 minus 1199061) 120573Vℎ120573ℎV120572ℎ120572V1205992

+ (1 minus 1199061) (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 120573Vℎ120573ℎV120572V1205992

+ (1 minus 1199061) (120601 + 1205781199062

120583ℎ + 120595)120573Vℎ120573ℎV120572ℎ120572V120599

2

gt 0

119862 = minus11988911989921199011 minus 11989921199011

= minus (120583V + 1205911199063) (120583V + 120572V + 1205911199063) [(1 minus 1199061)]

lt 0

119861 = 11988911989921199021 + 1198891198982 + 11989211989921199021 + 1198921198982 + 1198871ℎ1198982 + 11989211989921199011 minus 11988711198881198982

= (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599

+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 120572V120573ℎV120599

+ (1 minus 1199061) 120573Vℎ120573ℎV120572V1205992

+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) (120583V + 120572V + 1205911199063)


= (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599

+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599


+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) (120583V + 120572V + 1205911199063) (1 minusR2

119890)

(36)


119861 = (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599

+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 1205812120579120572V120573ℎV120599

+ (1 minus 1199061) 120573Vℎ120573ℎV120572V1205992+ (120583ℎ + 120572ℎ) (120583V + 1205911199063)

times (120583V + 120572V + 1205911199063) (1 minusR2

119890)

lt 0

(37)








= 119864V = 119868V = 0 with R1198900 = R119890 | 120575ℎ = 0

(38)




119889119864ℎ

119889119905= (1 minus 1199061) 120582ℎ (

Λ ℎ

120583ℎ


120583ℎ

minus (120572ℎ + 120583ℎ) 119864ℎ

119889119868ℎ

119889119905= 120572ℎ119864ℎ minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ

119889119864V

119889119905= 120582V (

Λ V


119889119868V

119889119905= 120572V119864V minus (120583V + 1205911199063) 119868V

(39)


120597

120597119864ℎ

[(1 minus 1199061) 120573Vℎ120599

119864ℎ119868ℎ119864VΛ ℎ120583ℎ(Λ ℎ

120583ℎ


+1205811120579Λ ℎ

119864ℎ119868ℎ119864V119868V120583ℎminus(120572ℎ + 120583ℎ)

119868ℎ119864V119868V]

+120597

120597119868ℎ

[120572ℎ

119868ℎ119864V119868Vminus(120601 + 1205781199062 + 120583ℎ + 120575ℎ)

119864ℎ119864V119868V]


+120597

120597119864V[

120573ℎV120599

119864ℎ119864V119868VΛ ℎ120583ℎ(Λ V

120583Vminus 119864V minus 119868V)

minus(120572V + 120583V + 1205911199063)

119864ℎ119868ℎ119864V]

+120597

120597119868V[

120572V

119864ℎ119868ℎ119868Vminus(120583V + 1205911199063)

119864ℎ119868ℎ119864V]

= minus120573Vℎ120599

1198642

ℎ119868ℎ119864V

minus120573Vℎ120599120583ℎ

1198642

ℎ119864V

(1 minus119877ℎ

Λ ℎ120583ℎ

) minus1205811120579Λ ℎ

1198642

ℎ119868ℎ119864V119868V120583ℎ

minus120572ℎ

1198682

ℎ119864V119868V

minus120573ℎV120599

119864ℎ1198642V119868V

[1 minus120583ℎ

Λ ℎ

] minus120572V

119864ℎ119868ℎ1198682V

lt 0

(40)








Γ (1199061 1199062 1199063) = int

119879119891

0

(1198621119864ℎ + 1198622119868ℎ + 1198623119873V

+1

2(11986011199062

1+ 1198602119906

2

2+ 1198603119906

2

3)) 119889119905

(41)



2




2

3 A lin-


2

1 1198602119906

2

2 and 1198603119906

2

3 such that the

terms (12)11986011199062

1 (12)1198602119906

2

2 and (12)1198603119906

2

3describe the cost




1(119905) 119906lowast

2(119905) and 119906

lowast

3(119905) such

that

Γ (119906lowast

1 119906lowast

2 119906lowast

3) = min(119906111990621199063)isinΦ2

Γ (1199061 1199062 1199063) (42)

where



(43)



119871 = 1198621119864ℎ + 1198622119868ℎ + 1198623119873V +1

2(11986011199062

1+ 1198602119906

2

2+ 1198603119906

2

3)

(44)



119867 = 119871 + 1205821

119889119878ℎ

119889119905+ 1205822

119889119864ℎ

119889119905+ 1205823

119889119868ℎ

119889119905+ 1205824

119889119877ℎ

119889119905

+ 1205825

119889119878V

119889119905+ 1205826

119889119864V

119889119905+ 1205827

119889119868V

119889119905

= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V +1

2(11986011199062

1+ 1198602119906

2

2+ 1198603119906

2

3)

+ 1205821 [Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119868ℎ



+ 1205823 [120572ℎ119864ℎ minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ]

+ 1205824 [(120601 + 1205781199062) 120588119868ℎ minus (120583ℎ + 120595)119877ℎ]


+ 1205826 [120573ℎV120599119868ℎ119878V minus (120572V + 120583V + 1205911199063) 119864V]

+ 1205827 [120572V119864V minus (120583V + 1205911199063) 119868V]

(45)



1 119906lowast

2 119906lowast


min(119906111990621199063)isinΦ2

Γ (1199061 1199062 1199063) = Γ (119906lowast

1 119906lowast

2 119906lowast

3) (46)


Γ (1199061 1199062 1199063) ge 1205851(100381610038161003816100381611990611003816100381610038161003816

2+100381610038161003816100381611990621003816100381610038161003816

2100381610038161003816100381611990631003816100381610038161003816

2)1205982

minus 1205852(47)





0 =120597119867 (119905 120594 119906 120582)

120597119906

1205821015840=120597119867 (119905 120594 119906 120582)

120597120594

119889120594

119889119905= minus

120597119867 (119905 120594 119906 120582)

120597120582

(48)



2 119906lowast

3) with


ℎ 119868lowast

ℎ 119877lowast

ℎ 119878lowast

V 119864lowast

V 119868lowast


minus1205821015840


minus 120583ℎ1205821 + 12057912058111205822

minus1205821015840


minus1205821015840

3= (1 minus 1205811) 1205791205821 + (120601 + 1205781199062) [(1 minus 120588) 1205821 + 1205881205824]

+ 12057912058111205822 minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 1205823

+ 120573ℎV120599 (1205826 minus 1205825) 119878V + 1198622

minus1205821015840

4= (1 minus 1205811) 1205791205821 + 1205951205821 + 12057912058111205822 minus (120583ℎ + 120595) 1205824

minus1205821015840

5= 120573ℎV120599 (1205826 minus 1205825) 119868ℎ minus (120583V + 1205911199063) 1205825 + 1198623

minus1205821015840

6= 120572V (1205827 minus 1205826) minus (120583V + 1205911199063) 1205826 + 1198623

minus1205821015840


(49)


1205821 (119879119891) = 1205822 (119879119891) = 1205823 (119879119891) = 1205824 (119879119891)

= 1205825 (119879119891) = 1205826 (119879119891) = 1205827 (119879119891) = 0

(50)


2 119906lowast

3) that mini-


119906lowast

1= max0min(1

120573Vℎ120599 (1205822 minus 1205821) 119868lowast

V 119878lowast

ℎ

1198601

)

119906lowast

2= max0min(1


ℎ

1198602

)

119906lowast

3= max0min(1

120591 (1205825119878lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

)

(51)




minus1198891205821

119889119905=120597119867

120597119878ℎ


minus 120583ℎ1205821 + 12057912058111205822

minus1198891205822

119889119905=120597119867

120597119864ℎ

= (1 minus 1205811) 1205791205821 + 12057912058111205822

+ 120572ℎ (1205823 minus 1205822) minus 120583ℎ1205822 + 1198621

minus1198891205823

119889119905=120597119867

120597119868ℎ

= (1 minus 1205811) 1205791205821

+ (120601 + 1205781199062) [(1 minus 120588) 1205821 + 1205881205824] + 12057912058111205822


minus1198891205824

119889119905=120597119867

120597119877ℎ

= (1 minus 1205811) 1205791205821 + 1205951205821 + 12057912058111205822 minus (120583ℎ + 120595) 1205824

minus1198891205825

119889119905=120597119867


minus1198891205826

119889119905=120597119867

120597119864V= 120572V (1205827 minus 1205826) minus (120583V + 1205911199063) 1205826 + 1198623

minus1198891205827

119889119905=120597119867

120597119868V= (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119878ℎ

minus (120583V + 1205911199063) 1205827 + 1198623

(52)


1205821 (119879119891) = 1205822 (119879119891) = 1205823 (119879119891) = 1205824 (119879119891)

= 1205825 (119879119891) = 1205826 (119879119891) = 1205827 (119879119891) = 0

(53)


lowast

ℎ

119864ℎ = 119864lowast

ℎ 119868ℎ = 119868

lowast

ℎ 119877ℎ = 119877

lowast

ℎ 119878V = 119878

lowast

V 119864V = 119864lowast

V and 119868V = 119868lowast

Vyields

120597119867

1205971199061

= 119860119906lowast

1+ 120573Vℎ1205991205821119868

lowast

V 119878lowast

ℎminus 120573Vℎ1205991205822119868

lowast

V 119878lowast

ℎ= 0

120597119867

1205971199062

= 11986021199062 + (1 minus 120588) 1205781205821119868lowast

ℎminus 1205781205823119868

lowast

ℎ+ 1205781205881205824119868

lowast

ℎ= 0

120597119867

1205971199063

= 1198603119906lowast

3minus 1205911205825119878

lowast

V minus 1205911205826119864lowast

V minus 1205911205827119868lowast

V = 0

(54)

for which

119906lowast

1=120573Vℎ120599 (1205822 minus 1205821) 119868

lowast

V 119878lowast

ℎ

1198601

119906lowast

2=120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868

lowast

ℎ

1198602

1199063 =120591 (1205825119878

lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

(55)

Then

119906lowast

1= max0min(1

120573Vℎ120599 (1205822 minus 1205821) 119868lowast

V 119878lowast

ℎ

1198601

)

119906lowast

2= max0min(1


ℎ

1198602

)

119906lowast

3= max0min(1

120591 (1205825119878lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

)

(56)




3



119889119878lowast

ℎ

119889119905= Λ ℎ + (1 minus 1205811) 120579119873

lowast

ℎ

+ (120601 + 1205781199062) (1 minus 120588) 119868lowast

ℎminus 120573Vℎ120599119868

lowast

V 119878lowast

ℎ


lowast

V 119878lowast

ℎ

1198601

))


ℎ+ 120595119877lowast

ℎ

119889119864lowast

ℎ

119889119905= 120573Vℎ120599119868

lowast

V 119878lowast

ℎ


lowast

V 119878lowast

ℎ

1198601

))

+ 1205811120579119873lowast

ℎminus 120572ℎ119864

lowast

ℎminus 120583ℎ119864

lowast

ℎ


119889119868lowast

ℎ

119889119905

= 120572ℎ119864lowast

ℎ

minus (120601 + 120578


lowast

ℎ

1198602

))

+120583ℎ + 120575ℎ) 119868lowast

ℎ

119889119877lowast

ℎ


times 119868lowast

ℎtimes (1198602)

minus1))) 120588119868ℎ

minus (120583ℎ + 120595)119877ℎ

119889119878lowast

V

119889119905

= Λ V minus (120583V + 120591


V + 1205826119864lowast

V + 1205827119868lowast

V )

times(1198603)minus1))) 119878

lowast


ℎ119878lowast

V

119889119864lowast

V

119889119905= 120573ℎV120599119868

lowast

ℎ119878lowast

V


V + 1205826119864lowast

V + 1205827119868lowast

V )

times(1198603)minus1))) 119864

lowast


V

119889119868lowast

V

119889119905= 120572V119864V


V + 1205826119864lowast

V +1205827119868lowast

V )

times(1198603)minus1))) 119868

lowast

V

(57)


ℎ 119868lowast

ℎ 119877lowast

ℎ 119906lowast

1 119906lowast

2 119906lowast

3 1205821 1205822 1205827)

119867lowast

= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V

+1

2(1198601(max0min(1

120573Vℎ120599 (1205822 minus 1205821) 119868lowast

V 119878lowast

ℎ

1198601

))

2


ℎ


+1198603(max0min(1120591(1205825119878

lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

))

2

)

+ 1205821

119889119878lowast

ℎ

119889119905+ 1205822

119889119864lowast

ℎ

119889119905+ 1205823

119889119868lowast

ℎ

119889119905+ 1205824

119889119877lowast

ℎ

119889119905+ 1205825

119889119878lowast

V

119889119905

+ 1205826

119889119864lowast

V

119889119905+ 1205827

119889119868lowast

V

119889119905







u1 ne 0 u3 ne 0 u2 = 0


0

10

20

30

40

50

60

70

times102In

fect

ed h

uman

pop

ulat

ion

0 10 20 30 40 50 60 70 80 90 100

Time (years)

(a)



fect

ed h

uman

pop

ulat

ion

0

10

20

30

40

50

60

70

0 10 20 30 40 50 60 70 80 90 100

times102

Time (days)

(b)

u2 ne 0 u3 ne 0 u1 = 0


0 10 20 30 40 50 600

5

10

15

20

25

30

35

40

times102

Infe

cted

hum

an p

opul

atio

n

Time (years)

(c)

u1 ne 0 u2 ne 0 u3 0ne ==u2 ne 0 u1 0 u3 0

=u1 ne 0 u2 0 u3 = 0 = =u3 ne 0 u1 0 u2 0

0

05

1

15

2

25

3

times104

0 10 20 30 40 50 60 70

Infe

cted

hum

an p

opul

atio

n

Time (years)

(d)








of total 22 30 53








409

93 53 47 7 4484

400440 446

486449

9 9 9 9 9 90

100

200

300

400

500

600

Bite

from

anin

fect

edm

osqu

ito

Fem

ale

mos

quito

Stag

nant

wat

er

Envi

ronm

ent

man

agem

ent

Oth

er

YesNo


wat

erso

aked

in ra

inRa

ined

on

Anopheles

Do not know


Seek treatment


47

418

493

462

409

454

83

8

38

92

0 100 200 300 400 500 600

Other

YesNo


House sprayed(IRS)

Use of ITNor LLN










5 Numerical Results






R0 = 18894

0 05 1 15 2 25 3 35 4

Time (years)

0

20

40

60

80

100

120

Susc

eptib

le in

divi

dual

s


(a)

R0 = 18894

0 05 1 15 2 25 3 35 4

Time (years)

0

20

40

60

80

100

120

140

times102

Expo

sed

indi

vidu

als

Exposed individuals

(b)

R0 = 18894

0 05 1 15 2 25 3 35 49

10

11

12

13

14

15

16

17

Time (years)

Infe

cted

indi

vidu

als


(c)

0 100 200 300 400 500 600 7000

1

2

3

4

5

6

7

8

Time (days)

Reco

vere

d in

divi

dual

s


R0 = 18894

times102

(d)








No Yes


of total 36 57 92



Time (days)

R0 = 18894

Prev

alen

ce

0 20 40 60 80 100 120 140 160 180 2000

01

02

03

04

05

06

07



6 Conclusions












of total 425 384 809



Hum

an p

opul

atio

n


times104

Time (years)0 05 1 15 2 25 3 35 4

0

05

1

15

2

25

(a)

times105

Hum

an p

opul

atio

n


Time (years)0 10 20 30 40 50 60

0

02

04

06

08

1

12

14

16

18

2

(b)


















Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119873ℎ

minus 120575ℎ119894 + 120572ℎ + 120579] 119890

120572ℎ119890 = [Λ ℎ

119873ℎ

+ 120601 + 1205781199062 + 120575ℎ + 120579 minus 120575ℎ119894] 119894

(120601 + 1205781199062) 120588119894 = [Λ ℎ

119873ℎ

+ 120595 + 120579 minus 120575ℎ119894] 119903

Λ ℎ

119873ℎ

= 120575ℎ119894 + 120583ℎ

(1 minus 119910 minus 119911) 120573ℎV120599119894 = [Λ V

119873V+ 120572V + 1205911199063]119910

120572V119910 = [Λ V

119873V+ 1205781199063] 119911

Λ V

119873V= 120583V

(24)



(120583ℎ + 120572ℎ) 119890 + (1 minus 1199061) 120573Vℎ120599119911119890


(120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ119911) 119890



120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911

(25)


120572ℎ119890 = (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894

120572ℎ119890 = (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894

119890 =(120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894

120572ℎ

(26)



=(120583ℎ + 120601 + 1205781199062 + 120575ℎ) 119894

120572ℎ

(1 minus 1199061) 120573Vℎ120599120572ℎ119911119894

+ [120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911 (120583ℎ + 120601 + 1205781199062 + 120575ℎ)] 119894

= (1 minus 1199061) (1 minus 119903) 120573Vℎ120599120572ℎ119911 + 1205811120579120572ℎ

+ 120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911

(27)


119894 = ((1 minus 1199061) (1 minus 119903) 120573Vℎ120599120572ℎ119911 + 1205811120579120572ℎ

+120583ℎ + 120572ℎ + (1 minus 1199061) 120573Vℎ120599119911)

times ((1 minus 1199061) 120573Vℎ120599120572ℎ119911119894


(28)


(120601 + 1205781199062) 120588119894 = (120583ℎ + 120595) 119903

119903 =(120601 + 1205781199062) 119894

120583ℎ + 120595

(29)


119894 =1198871 (1 minus 1198861119894) 119888119911 + 119889 + 119892 + 119887ℎ119911

1198871119888119911 + 119892 + 1198871ℎ119896119911

(1198871119888119911 + 119892 + 1198871ℎ119896119911 + 11988711198881198861119911

1198871119888119911 + 119892 + 1198871ℎ119896119911) 119894 =

1198871119888119911 + 119889 + 119892 + 1198871ℎ119911

1198871119888119911 + 119892 + 1198871ℎ119896119911

119894 =1198871119888119911 + 119889 + 119892 + 1198871ℎ119911

1198871119888119911 + 119892 + 1198871ℎ119896119911 + 11988711198881198861119911

(30)


(120583V + 120572V + 1205911199063) 119910 + 120573ℎV119894119910 = 120573ℎV120599 (1 minus 119911) 119894

119910 =(1 minus 119911) 120573ℎV120599119894

120583V + 120572V + 1205911199063 + 120573ℎV120599119894

(31)


120572V119910 = (120583V + 1205911199063) 119911

119911 =120572V119910

120583V + 1205911199063

(32)


119911 =(1 minus 119911) 120572V120573ℎV120599119894

(120583V + 1205911199063) (120583V + 120572V + 1205911199063 + 120573ℎV120599119894)

=1198982119894

1198992 (1199011 + 1199021119894)minus

1198982119894119911

1198992 (1199011 + 1199021119894)

=1198982119894

11989921199011 + 11989921199021119894 + 1198982119894

(33)


119894 =119889 + 119892 + (1198871119888 + 1198871ℎ) (1198982119894 (11989921199011 + 11989921199021119894 + 1198982119894))

119892 + (1198871119888 + 1198871ℎ119896 + 11988711198881198861) (1198982119894 (11989921199011 + 11989921199021119894 + 1198982119894))

(34)



1198601198942+ 119861119894 + 119862 = 0 (35)

where119860 = 11989211989921199021 + 1198921198982 + 11988711198881198982 + 1198871ℎ1198961198982 + 119887111988811988611198982

= (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 120572V120573ℎV120599

+ (1 minus 1199061) 120573Vℎ120573ℎV120572ℎ120572V1205992

+ (1 minus 1199061) (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 120573Vℎ120573ℎV120572V1205992

+ (1 minus 1199061) (120601 + 1205781199062

120583ℎ + 120595)120573Vℎ120573ℎV120572ℎ120572V120599

2

gt 0

119862 = minus11988911989921199011 minus 11989921199011

= minus (120583V + 1205911199063) (120583V + 120572V + 1205911199063) [(1 minus 1199061)]

lt 0

119861 = 11988911989921199021 + 1198891198982 + 11989211989921199021 + 1198921198982 + 1198871ℎ1198982 + 11989211989921199011 minus 11988711198881198982

= (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599

+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 120572V120573ℎV120599

+ (1 minus 1199061) 120573Vℎ120573ℎV120572V1205992

+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) (120583V + 120572V + 1205911199063)


= (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599

+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599


+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) (120583V + 120572V + 1205911199063) (1 minusR2

119890)

(36)


119861 = (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599

+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 1205812120579120572V120573ℎV120599

+ (1 minus 1199061) 120573Vℎ120573ℎV120572V1205992+ (120583ℎ + 120572ℎ) (120583V + 1205911199063)

times (120583V + 120572V + 1205911199063) (1 minusR2

119890)

lt 0

(37)








= 119864V = 119868V = 0 with R1198900 = R119890 | 120575ℎ = 0

(38)




119889119864ℎ

119889119905= (1 minus 1199061) 120582ℎ (

Λ ℎ

120583ℎ


120583ℎ

minus (120572ℎ + 120583ℎ) 119864ℎ

119889119868ℎ

119889119905= 120572ℎ119864ℎ minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ

119889119864V

119889119905= 120582V (

Λ V


119889119868V

119889119905= 120572V119864V minus (120583V + 1205911199063) 119868V

(39)


120597

120597119864ℎ

[(1 minus 1199061) 120573Vℎ120599

119864ℎ119868ℎ119864VΛ ℎ120583ℎ(Λ ℎ

120583ℎ


+1205811120579Λ ℎ

119864ℎ119868ℎ119864V119868V120583ℎminus(120572ℎ + 120583ℎ)

119868ℎ119864V119868V]

+120597

120597119868ℎ

[120572ℎ

119868ℎ119864V119868Vminus(120601 + 1205781199062 + 120583ℎ + 120575ℎ)

119864ℎ119864V119868V]


+120597

120597119864V[

120573ℎV120599

119864ℎ119864V119868VΛ ℎ120583ℎ(Λ V

120583Vminus 119864V minus 119868V)

minus(120572V + 120583V + 1205911199063)

119864ℎ119868ℎ119864V]

+120597

120597119868V[

120572V

119864ℎ119868ℎ119868Vminus(120583V + 1205911199063)

119864ℎ119868ℎ119864V]

= minus120573Vℎ120599

1198642

ℎ119868ℎ119864V

minus120573Vℎ120599120583ℎ

1198642

ℎ119864V

(1 minus119877ℎ

Λ ℎ120583ℎ

) minus1205811120579Λ ℎ

1198642

ℎ119868ℎ119864V119868V120583ℎ

minus120572ℎ

1198682

ℎ119864V119868V

minus120573ℎV120599

119864ℎ1198642V119868V

[1 minus120583ℎ

Λ ℎ

] minus120572V

119864ℎ119868ℎ1198682V

lt 0

(40)








Γ (1199061 1199062 1199063) = int

119879119891

0

(1198621119864ℎ + 1198622119868ℎ + 1198623119873V

+1

2(11986011199062

1+ 1198602119906

2

2+ 1198603119906

2

3)) 119889119905

(41)



2




2

3 A lin-


2

1 1198602119906

2

2 and 1198603119906

2

3 such that the

terms (12)11986011199062

1 (12)1198602119906

2

2 and (12)1198603119906

2

3describe the cost




1(119905) 119906lowast

2(119905) and 119906

lowast

3(119905) such

that

Γ (119906lowast

1 119906lowast

2 119906lowast

3) = min(119906111990621199063)isinΦ2

Γ (1199061 1199062 1199063) (42)

where



(43)



119871 = 1198621119864ℎ + 1198622119868ℎ + 1198623119873V +1

2(11986011199062

1+ 1198602119906

2

2+ 1198603119906

2

3)

(44)



119867 = 119871 + 1205821

119889119878ℎ

119889119905+ 1205822

119889119864ℎ

119889119905+ 1205823

119889119868ℎ

119889119905+ 1205824

119889119877ℎ

119889119905

+ 1205825

119889119878V

119889119905+ 1205826

119889119864V

119889119905+ 1205827

119889119868V

119889119905

= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V +1

2(11986011199062

1+ 1198602119906

2

2+ 1198603119906

2

3)

+ 1205821 [Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119868ℎ



+ 1205823 [120572ℎ119864ℎ minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ]

+ 1205824 [(120601 + 1205781199062) 120588119868ℎ minus (120583ℎ + 120595)119877ℎ]


+ 1205826 [120573ℎV120599119868ℎ119878V minus (120572V + 120583V + 1205911199063) 119864V]

+ 1205827 [120572V119864V minus (120583V + 1205911199063) 119868V]

(45)



1 119906lowast

2 119906lowast


min(119906111990621199063)isinΦ2

Γ (1199061 1199062 1199063) = Γ (119906lowast

1 119906lowast

2 119906lowast

3) (46)


Γ (1199061 1199062 1199063) ge 1205851(100381610038161003816100381611990611003816100381610038161003816

2+100381610038161003816100381611990621003816100381610038161003816

2100381610038161003816100381611990631003816100381610038161003816

2)1205982

minus 1205852(47)





0 =120597119867 (119905 120594 119906 120582)

120597119906

1205821015840=120597119867 (119905 120594 119906 120582)

120597120594

119889120594

119889119905= minus

120597119867 (119905 120594 119906 120582)

120597120582

(48)



2 119906lowast

3) with


ℎ 119868lowast

ℎ 119877lowast

ℎ 119878lowast

V 119864lowast

V 119868lowast


minus1205821015840


minus 120583ℎ1205821 + 12057912058111205822

minus1205821015840


minus1205821015840

3= (1 minus 1205811) 1205791205821 + (120601 + 1205781199062) [(1 minus 120588) 1205821 + 1205881205824]

+ 12057912058111205822 minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 1205823

+ 120573ℎV120599 (1205826 minus 1205825) 119878V + 1198622

minus1205821015840

4= (1 minus 1205811) 1205791205821 + 1205951205821 + 12057912058111205822 minus (120583ℎ + 120595) 1205824

minus1205821015840

5= 120573ℎV120599 (1205826 minus 1205825) 119868ℎ minus (120583V + 1205911199063) 1205825 + 1198623

minus1205821015840

6= 120572V (1205827 minus 1205826) minus (120583V + 1205911199063) 1205826 + 1198623

minus1205821015840


(49)


1205821 (119879119891) = 1205822 (119879119891) = 1205823 (119879119891) = 1205824 (119879119891)

= 1205825 (119879119891) = 1205826 (119879119891) = 1205827 (119879119891) = 0

(50)


2 119906lowast

3) that mini-


119906lowast

1= max0min(1

120573Vℎ120599 (1205822 minus 1205821) 119868lowast

V 119878lowast

ℎ

1198601

)

119906lowast

2= max0min(1


ℎ

1198602

)

119906lowast

3= max0min(1

120591 (1205825119878lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

)

(51)




minus1198891205821

119889119905=120597119867

120597119878ℎ


minus 120583ℎ1205821 + 12057912058111205822

minus1198891205822

119889119905=120597119867

120597119864ℎ

= (1 minus 1205811) 1205791205821 + 12057912058111205822

+ 120572ℎ (1205823 minus 1205822) minus 120583ℎ1205822 + 1198621

minus1198891205823

119889119905=120597119867

120597119868ℎ

= (1 minus 1205811) 1205791205821

+ (120601 + 1205781199062) [(1 minus 120588) 1205821 + 1205881205824] + 12057912058111205822


minus1198891205824

119889119905=120597119867

120597119877ℎ

= (1 minus 1205811) 1205791205821 + 1205951205821 + 12057912058111205822 minus (120583ℎ + 120595) 1205824

minus1198891205825

119889119905=120597119867


minus1198891205826

119889119905=120597119867

120597119864V= 120572V (1205827 minus 1205826) minus (120583V + 1205911199063) 1205826 + 1198623

minus1198891205827

119889119905=120597119867

120597119868V= (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119878ℎ

minus (120583V + 1205911199063) 1205827 + 1198623

(52)


1205821 (119879119891) = 1205822 (119879119891) = 1205823 (119879119891) = 1205824 (119879119891)

= 1205825 (119879119891) = 1205826 (119879119891) = 1205827 (119879119891) = 0

(53)


lowast

ℎ

119864ℎ = 119864lowast

ℎ 119868ℎ = 119868

lowast

ℎ 119877ℎ = 119877

lowast

ℎ 119878V = 119878

lowast

V 119864V = 119864lowast

V and 119868V = 119868lowast

Vyields

120597119867

1205971199061

= 119860119906lowast

1+ 120573Vℎ1205991205821119868

lowast

V 119878lowast

ℎminus 120573Vℎ1205991205822119868

lowast

V 119878lowast

ℎ= 0

120597119867

1205971199062

= 11986021199062 + (1 minus 120588) 1205781205821119868lowast

ℎminus 1205781205823119868

lowast

ℎ+ 1205781205881205824119868

lowast

ℎ= 0

120597119867

1205971199063

= 1198603119906lowast

3minus 1205911205825119878

lowast

V minus 1205911205826119864lowast

V minus 1205911205827119868lowast

V = 0

(54)

for which

119906lowast

1=120573Vℎ120599 (1205822 minus 1205821) 119868

lowast

V 119878lowast

ℎ

1198601

119906lowast

2=120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868

lowast

ℎ

1198602

1199063 =120591 (1205825119878

lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

(55)

Then

119906lowast

1= max0min(1

120573Vℎ120599 (1205822 minus 1205821) 119868lowast

V 119878lowast

ℎ

1198601

)

119906lowast

2= max0min(1


ℎ

1198602

)

119906lowast

3= max0min(1

120591 (1205825119878lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

)

(56)




3



119889119878lowast

ℎ

119889119905= Λ ℎ + (1 minus 1205811) 120579119873

lowast

ℎ

+ (120601 + 1205781199062) (1 minus 120588) 119868lowast

ℎminus 120573Vℎ120599119868

lowast

V 119878lowast

ℎ


lowast

V 119878lowast

ℎ

1198601

))


ℎ+ 120595119877lowast

ℎ

119889119864lowast

ℎ

119889119905= 120573Vℎ120599119868

lowast

V 119878lowast

ℎ


lowast

V 119878lowast

ℎ

1198601

))

+ 1205811120579119873lowast

ℎminus 120572ℎ119864

lowast

ℎminus 120583ℎ119864

lowast

ℎ


119889119868lowast

ℎ

119889119905

= 120572ℎ119864lowast

ℎ

minus (120601 + 120578


lowast

ℎ

1198602

))

+120583ℎ + 120575ℎ) 119868lowast

ℎ

119889119877lowast

ℎ


times 119868lowast

ℎtimes (1198602)

minus1))) 120588119868ℎ

minus (120583ℎ + 120595)119877ℎ

119889119878lowast

V

119889119905

= Λ V minus (120583V + 120591


V + 1205826119864lowast

V + 1205827119868lowast

V )

times(1198603)minus1))) 119878

lowast


ℎ119878lowast

V

119889119864lowast

V

119889119905= 120573ℎV120599119868

lowast

ℎ119878lowast

V


V + 1205826119864lowast

V + 1205827119868lowast

V )

times(1198603)minus1))) 119864

lowast


V

119889119868lowast

V

119889119905= 120572V119864V


V + 1205826119864lowast

V +1205827119868lowast

V )

times(1198603)minus1))) 119868

lowast

V

(57)


ℎ 119868lowast

ℎ 119877lowast

ℎ 119906lowast

1 119906lowast

2 119906lowast

3 1205821 1205822 1205827)

119867lowast

= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V

+1

2(1198601(max0min(1

120573Vℎ120599 (1205822 minus 1205821) 119868lowast

V 119878lowast

ℎ

1198601

))

2


ℎ


+1198603(max0min(1120591(1205825119878

lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

))

2

)

+ 1205821

119889119878lowast

ℎ

119889119905+ 1205822

119889119864lowast

ℎ

119889119905+ 1205823

119889119868lowast

ℎ

119889119905+ 1205824

119889119877lowast

ℎ

119889119905+ 1205825

119889119878lowast

V

119889119905

+ 1205826

119889119864lowast

V

119889119905+ 1205827

119889119868lowast

V

119889119905







u1 ne 0 u3 ne 0 u2 = 0


0

10

20

30

40

50

60

70

times102In

fect

ed h

uman

pop

ulat

ion

0 10 20 30 40 50 60 70 80 90 100

Time (years)

(a)



fect

ed h

uman

pop

ulat

ion

0

10

20

30

40

50

60

70

0 10 20 30 40 50 60 70 80 90 100

times102

Time (days)

(b)

u2 ne 0 u3 ne 0 u1 = 0


0 10 20 30 40 50 600

5

10

15

20

25

30

35

40

times102

Infe

cted

hum

an p

opul

atio

n

Time (years)

(c)

u1 ne 0 u2 ne 0 u3 0ne ==u2 ne 0 u1 0 u3 0

=u1 ne 0 u2 0 u3 = 0 = =u3 ne 0 u1 0 u2 0

0

05

1

15

2

25

3

times104

0 10 20 30 40 50 60 70

Infe

cted

hum

an p

opul

atio

n

Time (years)

(d)








of total 22 30 53








409

93 53 47 7 4484

400440 446

486449

9 9 9 9 9 90

100

200

300

400

500

600

Bite

from

anin

fect

edm

osqu

ito

Fem

ale

mos

quito

Stag

nant

wat

er

Envi

ronm

ent

man

agem

ent

Oth

er

YesNo


wat

erso

aked

in ra

inRa

ined

on

Anopheles

Do not know


Seek treatment


47

418

493

462

409

454

83

8

38

92

0 100 200 300 400 500 600

Other

YesNo


House sprayed(IRS)

Use of ITNor LLN










5 Numerical Results






R0 = 18894

0 05 1 15 2 25 3 35 4

Time (years)

0

20

40

60

80

100

120

Susc

eptib

le in

divi

dual

s


(a)

R0 = 18894

0 05 1 15 2 25 3 35 4

Time (years)

0

20

40

60

80

100

120

140

times102

Expo

sed

indi

vidu

als

Exposed individuals

(b)

R0 = 18894

0 05 1 15 2 25 3 35 49

10

11

12

13

14

15

16

17

Time (years)

Infe

cted

indi

vidu

als


(c)

0 100 200 300 400 500 600 7000

1

2

3

4

5

6

7

8

Time (days)

Reco

vere

d in

divi

dual

s


R0 = 18894

times102

(d)








No Yes


of total 36 57 92



Time (days)

R0 = 18894

Prev

alen

ce

0 20 40 60 80 100 120 140 160 180 2000

01

02

03

04

05

06

07



6 Conclusions












of total 425 384 809



Hum

an p

opul

atio

n


times104

Time (years)0 05 1 15 2 25 3 35 4

0

05

1

15

2

25

(a)

times105

Hum

an p

opul

atio

n


Time (years)0 10 20 30 40 50 60

0

02

04

06

08

1

12

14

16

18

2

(b)


















Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198601198942+ 119861119894 + 119862 = 0 (35)

where119860 = 11989211989921199021 + 1198921198982 + 11988711198881198982 + 1198871ℎ1198961198982 + 119887111988811988611198982

= (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 120572V120573ℎV120599

+ (1 minus 1199061) 120573Vℎ120573ℎV120572ℎ120572V1205992

+ (1 minus 1199061) (120583ℎ + 120601 + 1205781199062 + 120575ℎ) 120573Vℎ120573ℎV120572V1205992

+ (1 minus 1199061) (120601 + 1205781199062

120583ℎ + 120595)120573Vℎ120573ℎV120572ℎ120572V120599

2

gt 0

119862 = minus11988911989921199011 minus 11989921199011

= minus (120583V + 1205911199063) (120583V + 120572V + 1205911199063) [(1 minus 1199061)]

lt 0

119861 = 11988911989921199021 + 1198891198982 + 11989211989921199021 + 1198921198982 + 1198871ℎ1198982 + 11989211989921199011 minus 11988711198881198982

= (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599

+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 120572V120573ℎV120599

+ (1 minus 1199061) 120573Vℎ120573ℎV120572V1205992

+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) (120583V + 120572V + 1205911199063)


= (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599

+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599


+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) (120583V + 120572V + 1205911199063) (1 minusR2

119890)

(36)


119861 = (120583V + 1205911199063) 1205811120579120572ℎ120573ℎV120599 + 1205811120579120572ℎ120572V120573ℎV120599

+ (120583ℎ + 120572ℎ) (120583V + 1205911199063) 120573ℎV120599 + (120583ℎ + 120572ℎ) 1205812120579120572V120573ℎV120599

+ (1 minus 1199061) 120573Vℎ120573ℎV120572V1205992+ (120583ℎ + 120572ℎ) (120583V + 1205911199063)

times (120583V + 120572V + 1205911199063) (1 minusR2

119890)

lt 0

(37)








= 119864V = 119868V = 0 with R1198900 = R119890 | 120575ℎ = 0

(38)




119889119864ℎ

119889119905= (1 minus 1199061) 120582ℎ (

Λ ℎ

120583ℎ


120583ℎ

minus (120572ℎ + 120583ℎ) 119864ℎ

119889119868ℎ

119889119905= 120572ℎ119864ℎ minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ

119889119864V

119889119905= 120582V (

Λ V


119889119868V

119889119905= 120572V119864V minus (120583V + 1205911199063) 119868V

(39)


120597

120597119864ℎ

[(1 minus 1199061) 120573Vℎ120599

119864ℎ119868ℎ119864VΛ ℎ120583ℎ(Λ ℎ

120583ℎ


+1205811120579Λ ℎ

119864ℎ119868ℎ119864V119868V120583ℎminus(120572ℎ + 120583ℎ)

119868ℎ119864V119868V]

+120597

120597119868ℎ

[120572ℎ

119868ℎ119864V119868Vminus(120601 + 1205781199062 + 120583ℎ + 120575ℎ)

119864ℎ119864V119868V]


+120597

120597119864V[

120573ℎV120599

119864ℎ119864V119868VΛ ℎ120583ℎ(Λ V

120583Vminus 119864V minus 119868V)

minus(120572V + 120583V + 1205911199063)

119864ℎ119868ℎ119864V]

+120597

120597119868V[

120572V

119864ℎ119868ℎ119868Vminus(120583V + 1205911199063)

119864ℎ119868ℎ119864V]

= minus120573Vℎ120599

1198642

ℎ119868ℎ119864V

minus120573Vℎ120599120583ℎ

1198642

ℎ119864V

(1 minus119877ℎ

Λ ℎ120583ℎ

) minus1205811120579Λ ℎ

1198642

ℎ119868ℎ119864V119868V120583ℎ

minus120572ℎ

1198682

ℎ119864V119868V

minus120573ℎV120599

119864ℎ1198642V119868V

[1 minus120583ℎ

Λ ℎ

] minus120572V

119864ℎ119868ℎ1198682V

lt 0

(40)








Γ (1199061 1199062 1199063) = int

119879119891

0

(1198621119864ℎ + 1198622119868ℎ + 1198623119873V

+1

2(11986011199062

1+ 1198602119906

2

2+ 1198603119906

2

3)) 119889119905

(41)



2




2

3 A lin-


2

1 1198602119906

2

2 and 1198603119906

2

3 such that the

terms (12)11986011199062

1 (12)1198602119906

2

2 and (12)1198603119906

2

3describe the cost




1(119905) 119906lowast

2(119905) and 119906

lowast

3(119905) such

that

Γ (119906lowast

1 119906lowast

2 119906lowast

3) = min(119906111990621199063)isinΦ2

Γ (1199061 1199062 1199063) (42)

where



(43)



119871 = 1198621119864ℎ + 1198622119868ℎ + 1198623119873V +1

2(11986011199062

1+ 1198602119906

2

2+ 1198603119906

2

3)

(44)



119867 = 119871 + 1205821

119889119878ℎ

119889119905+ 1205822

119889119864ℎ

119889119905+ 1205823

119889119868ℎ

119889119905+ 1205824

119889119877ℎ

119889119905

+ 1205825

119889119878V

119889119905+ 1205826

119889119864V

119889119905+ 1205827

119889119868V

119889119905

= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V +1

2(11986011199062

1+ 1198602119906

2

2+ 1198603119906

2

3)

+ 1205821 [Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119868ℎ



+ 1205823 [120572ℎ119864ℎ minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ]

+ 1205824 [(120601 + 1205781199062) 120588119868ℎ minus (120583ℎ + 120595)119877ℎ]


+ 1205826 [120573ℎV120599119868ℎ119878V minus (120572V + 120583V + 1205911199063) 119864V]

+ 1205827 [120572V119864V minus (120583V + 1205911199063) 119868V]

(45)



1 119906lowast

2 119906lowast


min(119906111990621199063)isinΦ2

Γ (1199061 1199062 1199063) = Γ (119906lowast

1 119906lowast

2 119906lowast

3) (46)


Γ (1199061 1199062 1199063) ge 1205851(100381610038161003816100381611990611003816100381610038161003816

2+100381610038161003816100381611990621003816100381610038161003816

2100381610038161003816100381611990631003816100381610038161003816

2)1205982

minus 1205852(47)





0 =120597119867 (119905 120594 119906 120582)

120597119906

1205821015840=120597119867 (119905 120594 119906 120582)

120597120594

119889120594

119889119905= minus

120597119867 (119905 120594 119906 120582)

120597120582

(48)



2 119906lowast

3) with


ℎ 119868lowast

ℎ 119877lowast

ℎ 119878lowast

V 119864lowast

V 119868lowast


minus1205821015840


minus 120583ℎ1205821 + 12057912058111205822

minus1205821015840


minus1205821015840

3= (1 minus 1205811) 1205791205821 + (120601 + 1205781199062) [(1 minus 120588) 1205821 + 1205881205824]

+ 12057912058111205822 minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 1205823

+ 120573ℎV120599 (1205826 minus 1205825) 119878V + 1198622

minus1205821015840

4= (1 minus 1205811) 1205791205821 + 1205951205821 + 12057912058111205822 minus (120583ℎ + 120595) 1205824

minus1205821015840

5= 120573ℎV120599 (1205826 minus 1205825) 119868ℎ minus (120583V + 1205911199063) 1205825 + 1198623

minus1205821015840

6= 120572V (1205827 minus 1205826) minus (120583V + 1205911199063) 1205826 + 1198623

minus1205821015840


(49)


1205821 (119879119891) = 1205822 (119879119891) = 1205823 (119879119891) = 1205824 (119879119891)

= 1205825 (119879119891) = 1205826 (119879119891) = 1205827 (119879119891) = 0

(50)


2 119906lowast

3) that mini-


119906lowast

1= max0min(1

120573Vℎ120599 (1205822 minus 1205821) 119868lowast

V 119878lowast

ℎ

1198601

)

119906lowast

2= max0min(1


ℎ

1198602

)

119906lowast

3= max0min(1

120591 (1205825119878lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

)

(51)




minus1198891205821

119889119905=120597119867

120597119878ℎ


minus 120583ℎ1205821 + 12057912058111205822

minus1198891205822

119889119905=120597119867

120597119864ℎ

= (1 minus 1205811) 1205791205821 + 12057912058111205822

+ 120572ℎ (1205823 minus 1205822) minus 120583ℎ1205822 + 1198621

minus1198891205823

119889119905=120597119867

120597119868ℎ

= (1 minus 1205811) 1205791205821

+ (120601 + 1205781199062) [(1 minus 120588) 1205821 + 1205881205824] + 12057912058111205822


minus1198891205824

119889119905=120597119867

120597119877ℎ

= (1 minus 1205811) 1205791205821 + 1205951205821 + 12057912058111205822 minus (120583ℎ + 120595) 1205824

minus1198891205825

119889119905=120597119867


minus1198891205826

119889119905=120597119867

120597119864V= 120572V (1205827 minus 1205826) minus (120583V + 1205911199063) 1205826 + 1198623

minus1198891205827

119889119905=120597119867

120597119868V= (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119878ℎ

minus (120583V + 1205911199063) 1205827 + 1198623

(52)


1205821 (119879119891) = 1205822 (119879119891) = 1205823 (119879119891) = 1205824 (119879119891)

= 1205825 (119879119891) = 1205826 (119879119891) = 1205827 (119879119891) = 0

(53)


lowast

ℎ

119864ℎ = 119864lowast

ℎ 119868ℎ = 119868

lowast

ℎ 119877ℎ = 119877

lowast

ℎ 119878V = 119878

lowast

V 119864V = 119864lowast

V and 119868V = 119868lowast

Vyields

120597119867

1205971199061

= 119860119906lowast

1+ 120573Vℎ1205991205821119868

lowast

V 119878lowast

ℎminus 120573Vℎ1205991205822119868

lowast

V 119878lowast

ℎ= 0

120597119867

1205971199062

= 11986021199062 + (1 minus 120588) 1205781205821119868lowast

ℎminus 1205781205823119868

lowast

ℎ+ 1205781205881205824119868

lowast

ℎ= 0

120597119867

1205971199063

= 1198603119906lowast

3minus 1205911205825119878

lowast

V minus 1205911205826119864lowast

V minus 1205911205827119868lowast

V = 0

(54)

for which

119906lowast

1=120573Vℎ120599 (1205822 minus 1205821) 119868

lowast

V 119878lowast

ℎ

1198601

119906lowast

2=120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868

lowast

ℎ

1198602

1199063 =120591 (1205825119878

lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

(55)

Then

119906lowast

1= max0min(1

120573Vℎ120599 (1205822 minus 1205821) 119868lowast

V 119878lowast

ℎ

1198601

)

119906lowast

2= max0min(1


ℎ

1198602

)

119906lowast

3= max0min(1

120591 (1205825119878lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

)

(56)




3



119889119878lowast

ℎ

119889119905= Λ ℎ + (1 minus 1205811) 120579119873

lowast

ℎ

+ (120601 + 1205781199062) (1 minus 120588) 119868lowast

ℎminus 120573Vℎ120599119868

lowast

V 119878lowast

ℎ


lowast

V 119878lowast

ℎ

1198601

))


ℎ+ 120595119877lowast

ℎ

119889119864lowast

ℎ

119889119905= 120573Vℎ120599119868

lowast

V 119878lowast

ℎ


lowast

V 119878lowast

ℎ

1198601

))

+ 1205811120579119873lowast

ℎminus 120572ℎ119864

lowast

ℎminus 120583ℎ119864

lowast

ℎ


119889119868lowast

ℎ

119889119905

= 120572ℎ119864lowast

ℎ

minus (120601 + 120578


lowast

ℎ

1198602

))

+120583ℎ + 120575ℎ) 119868lowast

ℎ

119889119877lowast

ℎ


times 119868lowast

ℎtimes (1198602)

minus1))) 120588119868ℎ

minus (120583ℎ + 120595)119877ℎ

119889119878lowast

V

119889119905

= Λ V minus (120583V + 120591


V + 1205826119864lowast

V + 1205827119868lowast

V )

times(1198603)minus1))) 119878

lowast


ℎ119878lowast

V

119889119864lowast

V

119889119905= 120573ℎV120599119868

lowast

ℎ119878lowast

V


V + 1205826119864lowast

V + 1205827119868lowast

V )

times(1198603)minus1))) 119864

lowast


V

119889119868lowast

V

119889119905= 120572V119864V


V + 1205826119864lowast

V +1205827119868lowast

V )

times(1198603)minus1))) 119868

lowast

V

(57)


ℎ 119868lowast

ℎ 119877lowast

ℎ 119906lowast

1 119906lowast

2 119906lowast

3 1205821 1205822 1205827)

119867lowast

= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V

+1

2(1198601(max0min(1

120573Vℎ120599 (1205822 minus 1205821) 119868lowast

V 119878lowast

ℎ

1198601

))

2


ℎ


+1198603(max0min(1120591(1205825119878

lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

))

2

)

+ 1205821

119889119878lowast

ℎ

119889119905+ 1205822

119889119864lowast

ℎ

119889119905+ 1205823

119889119868lowast

ℎ

119889119905+ 1205824

119889119877lowast

ℎ

119889119905+ 1205825

119889119878lowast

V

119889119905

+ 1205826

119889119864lowast

V

119889119905+ 1205827

119889119868lowast

V

119889119905







u1 ne 0 u3 ne 0 u2 = 0


0

10

20

30

40

50

60

70

times102In

fect

ed h

uman

pop

ulat

ion

0 10 20 30 40 50 60 70 80 90 100

Time (years)

(a)



fect

ed h

uman

pop

ulat

ion

0

10

20

30

40

50

60

70

0 10 20 30 40 50 60 70 80 90 100

times102

Time (days)

(b)

u2 ne 0 u3 ne 0 u1 = 0


0 10 20 30 40 50 600

5

10

15

20

25

30

35

40

times102

Infe

cted

hum

an p

opul

atio

n

Time (years)

(c)

u1 ne 0 u2 ne 0 u3 0ne ==u2 ne 0 u1 0 u3 0

=u1 ne 0 u2 0 u3 = 0 = =u3 ne 0 u1 0 u2 0

0

05

1

15

2

25

3

times104

0 10 20 30 40 50 60 70

Infe

cted

hum

an p

opul

atio

n

Time (years)

(d)








of total 22 30 53








409

93 53 47 7 4484

400440 446

486449

9 9 9 9 9 90

100

200

300

400

500

600

Bite

from

anin

fect

edm

osqu

ito

Fem

ale

mos

quito

Stag

nant

wat

er

Envi

ronm

ent

man

agem

ent

Oth

er

YesNo


wat

erso

aked

in ra

inRa

ined

on

Anopheles

Do not know


Seek treatment


47

418

493

462

409

454

83

8

38

92

0 100 200 300 400 500 600

Other

YesNo


House sprayed(IRS)

Use of ITNor LLN










5 Numerical Results






R0 = 18894

0 05 1 15 2 25 3 35 4

Time (years)

0

20

40

60

80

100

120

Susc

eptib

le in

divi

dual

s


(a)

R0 = 18894

0 05 1 15 2 25 3 35 4

Time (years)

0

20

40

60

80

100

120

140

times102

Expo

sed

indi

vidu

als

Exposed individuals

(b)

R0 = 18894

0 05 1 15 2 25 3 35 49

10

11

12

13

14

15

16

17

Time (years)

Infe

cted

indi

vidu

als


(c)

0 100 200 300 400 500 600 7000

1

2

3

4

5

6

7

8

Time (days)

Reco

vere

d in

divi

dual

s


R0 = 18894

times102

(d)








No Yes


of total 36 57 92



Time (days)

R0 = 18894

Prev

alen

ce

0 20 40 60 80 100 120 140 160 180 2000

01

02

03

04

05

06

07



6 Conclusions












of total 425 384 809



Hum

an p

opul

atio

n


times104

Time (years)0 05 1 15 2 25 3 35 4

0

05

1

15

2

25

(a)

times105

Hum

an p

opul

atio

n


Time (years)0 10 20 30 40 50 60

0

02

04

06

08

1

12

14

16

18

2

(b)


















Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+120597

120597119864V[

120573ℎV120599

119864ℎ119864V119868VΛ ℎ120583ℎ(Λ V

120583Vminus 119864V minus 119868V)

minus(120572V + 120583V + 1205911199063)

119864ℎ119868ℎ119864V]

+120597

120597119868V[

120572V

119864ℎ119868ℎ119868Vminus(120583V + 1205911199063)

119864ℎ119868ℎ119864V]

= minus120573Vℎ120599

1198642

ℎ119868ℎ119864V

minus120573Vℎ120599120583ℎ

1198642

ℎ119864V

(1 minus119877ℎ

Λ ℎ120583ℎ

) minus1205811120579Λ ℎ

1198642

ℎ119868ℎ119864V119868V120583ℎ

minus120572ℎ

1198682

ℎ119864V119868V

minus120573ℎV120599

119864ℎ1198642V119868V

[1 minus120583ℎ

Λ ℎ

] minus120572V

119864ℎ119868ℎ1198682V

lt 0

(40)








Γ (1199061 1199062 1199063) = int

119879119891

0

(1198621119864ℎ + 1198622119868ℎ + 1198623119873V

+1

2(11986011199062

1+ 1198602119906

2

2+ 1198603119906

2

3)) 119889119905

(41)



2




2

3 A lin-


2

1 1198602119906

2

2 and 1198603119906

2

3 such that the

terms (12)11986011199062

1 (12)1198602119906

2

2 and (12)1198603119906

2

3describe the cost




1(119905) 119906lowast

2(119905) and 119906

lowast

3(119905) such

that

Γ (119906lowast

1 119906lowast

2 119906lowast

3) = min(119906111990621199063)isinΦ2

Γ (1199061 1199062 1199063) (42)

where



(43)



119871 = 1198621119864ℎ + 1198622119868ℎ + 1198623119873V +1

2(11986011199062

1+ 1198602119906

2

2+ 1198603119906

2

3)

(44)



119867 = 119871 + 1205821

119889119878ℎ

119889119905+ 1205822

119889119864ℎ

119889119905+ 1205823

119889119868ℎ

119889119905+ 1205824

119889119877ℎ

119889119905

+ 1205825

119889119878V

119889119905+ 1205826

119889119864V

119889119905+ 1205827

119889119868V

119889119905

= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V +1

2(11986011199062

1+ 1198602119906

2

2+ 1198603119906

2

3)

+ 1205821 [Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119868ℎ



+ 1205823 [120572ℎ119864ℎ minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ]

+ 1205824 [(120601 + 1205781199062) 120588119868ℎ minus (120583ℎ + 120595)119877ℎ]


+ 1205826 [120573ℎV120599119868ℎ119878V minus (120572V + 120583V + 1205911199063) 119864V]

+ 1205827 [120572V119864V minus (120583V + 1205911199063) 119868V]

(45)



1 119906lowast

2 119906lowast


min(119906111990621199063)isinΦ2

Γ (1199061 1199062 1199063) = Γ (119906lowast

1 119906lowast

2 119906lowast

3) (46)


Γ (1199061 1199062 1199063) ge 1205851(100381610038161003816100381611990611003816100381610038161003816

2+100381610038161003816100381611990621003816100381610038161003816

2100381610038161003816100381611990631003816100381610038161003816

2)1205982

minus 1205852(47)





0 =120597119867 (119905 120594 119906 120582)

120597119906

1205821015840=120597119867 (119905 120594 119906 120582)

120597120594

119889120594

119889119905= minus

120597119867 (119905 120594 119906 120582)

120597120582

(48)



2 119906lowast

3) with


ℎ 119868lowast

ℎ 119877lowast

ℎ 119878lowast

V 119864lowast

V 119868lowast


minus1205821015840


minus 120583ℎ1205821 + 12057912058111205822

minus1205821015840


minus1205821015840

3= (1 minus 1205811) 1205791205821 + (120601 + 1205781199062) [(1 minus 120588) 1205821 + 1205881205824]

+ 12057912058111205822 minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 1205823

+ 120573ℎV120599 (1205826 minus 1205825) 119878V + 1198622

minus1205821015840

4= (1 minus 1205811) 1205791205821 + 1205951205821 + 12057912058111205822 minus (120583ℎ + 120595) 1205824

minus1205821015840

5= 120573ℎV120599 (1205826 minus 1205825) 119868ℎ minus (120583V + 1205911199063) 1205825 + 1198623

minus1205821015840

6= 120572V (1205827 minus 1205826) minus (120583V + 1205911199063) 1205826 + 1198623

minus1205821015840


(49)


1205821 (119879119891) = 1205822 (119879119891) = 1205823 (119879119891) = 1205824 (119879119891)

= 1205825 (119879119891) = 1205826 (119879119891) = 1205827 (119879119891) = 0

(50)


2 119906lowast

3) that mini-


119906lowast

1= max0min(1

120573Vℎ120599 (1205822 minus 1205821) 119868lowast

V 119878lowast

ℎ

1198601

)

119906lowast

2= max0min(1


ℎ

1198602

)

119906lowast

3= max0min(1

120591 (1205825119878lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

)

(51)




minus1198891205821

119889119905=120597119867

120597119878ℎ


minus 120583ℎ1205821 + 12057912058111205822

minus1198891205822

119889119905=120597119867

120597119864ℎ

= (1 minus 1205811) 1205791205821 + 12057912058111205822

+ 120572ℎ (1205823 minus 1205822) minus 120583ℎ1205822 + 1198621

minus1198891205823

119889119905=120597119867

120597119868ℎ

= (1 minus 1205811) 1205791205821

+ (120601 + 1205781199062) [(1 minus 120588) 1205821 + 1205881205824] + 12057912058111205822


minus1198891205824

119889119905=120597119867

120597119877ℎ

= (1 minus 1205811) 1205791205821 + 1205951205821 + 12057912058111205822 minus (120583ℎ + 120595) 1205824

minus1198891205825

119889119905=120597119867


minus1198891205826

119889119905=120597119867

120597119864V= 120572V (1205827 minus 1205826) minus (120583V + 1205911199063) 1205826 + 1198623

minus1198891205827

119889119905=120597119867

120597119868V= (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119878ℎ

minus (120583V + 1205911199063) 1205827 + 1198623

(52)


1205821 (119879119891) = 1205822 (119879119891) = 1205823 (119879119891) = 1205824 (119879119891)

= 1205825 (119879119891) = 1205826 (119879119891) = 1205827 (119879119891) = 0

(53)


lowast

ℎ

119864ℎ = 119864lowast

ℎ 119868ℎ = 119868

lowast

ℎ 119877ℎ = 119877

lowast

ℎ 119878V = 119878

lowast

V 119864V = 119864lowast

V and 119868V = 119868lowast

Vyields

120597119867

1205971199061

= 119860119906lowast

1+ 120573Vℎ1205991205821119868

lowast

V 119878lowast

ℎminus 120573Vℎ1205991205822119868

lowast

V 119878lowast

ℎ= 0

120597119867

1205971199062

= 11986021199062 + (1 minus 120588) 1205781205821119868lowast

ℎminus 1205781205823119868

lowast

ℎ+ 1205781205881205824119868

lowast

ℎ= 0

120597119867

1205971199063

= 1198603119906lowast

3minus 1205911205825119878

lowast

V minus 1205911205826119864lowast

V minus 1205911205827119868lowast

V = 0

(54)

for which

119906lowast

1=120573Vℎ120599 (1205822 minus 1205821) 119868

lowast

V 119878lowast

ℎ

1198601

119906lowast

2=120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868

lowast

ℎ

1198602

1199063 =120591 (1205825119878

lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

(55)

Then

119906lowast

1= max0min(1

120573Vℎ120599 (1205822 minus 1205821) 119868lowast

V 119878lowast

ℎ

1198601

)

119906lowast

2= max0min(1


ℎ

1198602

)

119906lowast

3= max0min(1

120591 (1205825119878lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

)

(56)




3



119889119878lowast

ℎ

119889119905= Λ ℎ + (1 minus 1205811) 120579119873

lowast

ℎ

+ (120601 + 1205781199062) (1 minus 120588) 119868lowast

ℎminus 120573Vℎ120599119868

lowast

V 119878lowast

ℎ


lowast

V 119878lowast

ℎ

1198601

))


ℎ+ 120595119877lowast

ℎ

119889119864lowast

ℎ

119889119905= 120573Vℎ120599119868

lowast

V 119878lowast

ℎ


lowast

V 119878lowast

ℎ

1198601

))

+ 1205811120579119873lowast

ℎminus 120572ℎ119864

lowast

ℎminus 120583ℎ119864

lowast

ℎ


119889119868lowast

ℎ

119889119905

= 120572ℎ119864lowast

ℎ

minus (120601 + 120578


lowast

ℎ

1198602

))

+120583ℎ + 120575ℎ) 119868lowast

ℎ

119889119877lowast

ℎ


times 119868lowast

ℎtimes (1198602)

minus1))) 120588119868ℎ

minus (120583ℎ + 120595)119877ℎ

119889119878lowast

V

119889119905

= Λ V minus (120583V + 120591


V + 1205826119864lowast

V + 1205827119868lowast

V )

times(1198603)minus1))) 119878

lowast


ℎ119878lowast

V

119889119864lowast

V

119889119905= 120573ℎV120599119868

lowast

ℎ119878lowast

V


V + 1205826119864lowast

V + 1205827119868lowast

V )

times(1198603)minus1))) 119864

lowast


V

119889119868lowast

V

119889119905= 120572V119864V


V + 1205826119864lowast

V +1205827119868lowast

V )

times(1198603)minus1))) 119868

lowast

V

(57)


ℎ 119868lowast

ℎ 119877lowast

ℎ 119906lowast

1 119906lowast

2 119906lowast

3 1205821 1205822 1205827)

119867lowast

= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V

+1

2(1198601(max0min(1

120573Vℎ120599 (1205822 minus 1205821) 119868lowast

V 119878lowast

ℎ

1198601

))

2


ℎ


+1198603(max0min(1120591(1205825119878

lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

))

2

)

+ 1205821

119889119878lowast

ℎ

119889119905+ 1205822

119889119864lowast

ℎ

119889119905+ 1205823

119889119868lowast

ℎ

119889119905+ 1205824

119889119877lowast

ℎ

119889119905+ 1205825

119889119878lowast

V

119889119905

+ 1205826

119889119864lowast

V

119889119905+ 1205827

119889119868lowast

V

119889119905







u1 ne 0 u3 ne 0 u2 = 0


0

10

20

30

40

50

60

70

times102In

fect

ed h

uman

pop

ulat

ion

0 10 20 30 40 50 60 70 80 90 100

Time (years)

(a)



fect

ed h

uman

pop

ulat

ion

0

10

20

30

40

50

60

70

0 10 20 30 40 50 60 70 80 90 100

times102

Time (days)

(b)

u2 ne 0 u3 ne 0 u1 = 0


0 10 20 30 40 50 600

5

10

15

20

25

30

35

40

times102

Infe

cted

hum

an p

opul

atio

n

Time (years)

(c)

u1 ne 0 u2 ne 0 u3 0ne ==u2 ne 0 u1 0 u3 0

=u1 ne 0 u2 0 u3 = 0 = =u3 ne 0 u1 0 u2 0

0

05

1

15

2

25

3

times104

0 10 20 30 40 50 60 70

Infe

cted

hum

an p

opul

atio

n

Time (years)

(d)








of total 22 30 53








409

93 53 47 7 4484

400440 446

486449

9 9 9 9 9 90

100

200

300

400

500

600

Bite

from

anin

fect

edm

osqu

ito

Fem

ale

mos

quito

Stag

nant

wat

er

Envi

ronm

ent

man

agem

ent

Oth

er

YesNo


wat

erso

aked

in ra

inRa

ined

on

Anopheles

Do not know


Seek treatment


47

418

493

462

409

454

83

8

38

92

0 100 200 300 400 500 600

Other

YesNo


House sprayed(IRS)

Use of ITNor LLN










5 Numerical Results






R0 = 18894

0 05 1 15 2 25 3 35 4

Time (years)

0

20

40

60

80

100

120

Susc

eptib

le in

divi

dual

s


(a)

R0 = 18894

0 05 1 15 2 25 3 35 4

Time (years)

0

20

40

60

80

100

120

140

times102

Expo

sed

indi

vidu

als

Exposed individuals

(b)

R0 = 18894

0 05 1 15 2 25 3 35 49

10

11

12

13

14

15

16

17

Time (years)

Infe

cted

indi

vidu

als


(c)

0 100 200 300 400 500 600 7000

1

2

3

4

5

6

7

8

Time (days)

Reco

vere

d in

divi

dual

s


R0 = 18894

times102

(d)








No Yes


of total 36 57 92



Time (days)

R0 = 18894

Prev

alen

ce

0 20 40 60 80 100 120 140 160 180 2000

01

02

03

04

05

06

07



6 Conclusions












of total 425 384 809



Hum

an p

opul

atio

n


times104

Time (years)0 05 1 15 2 25 3 35 4

0

05

1

15

2

25

(a)

times105

Hum

an p

opul

atio

n


Time (years)0 10 20 30 40 50 60

0

02

04

06

08

1

12

14

16

18

2

(b)


















Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119867 = 119871 + 1205821

119889119878ℎ

119889119905+ 1205822

119889119864ℎ

119889119905+ 1205823

119889119868ℎ

119889119905+ 1205824

119889119877ℎ

119889119905

+ 1205825

119889119878V

119889119905+ 1205826

119889119864V

119889119905+ 1205827

119889119868V

119889119905

= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V +1

2(11986011199062

1+ 1198602119906

2

2+ 1198603119906

2

3)

+ 1205821 [Λ ℎ + (1 minus 1205811) 120579119873ℎ + (120601 + 1205781199062) (1 minus 120588) 119868ℎ



+ 1205823 [120572ℎ119864ℎ minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 119868ℎ]

+ 1205824 [(120601 + 1205781199062) 120588119868ℎ minus (120583ℎ + 120595)119877ℎ]


+ 1205826 [120573ℎV120599119868ℎ119878V minus (120572V + 120583V + 1205911199063) 119864V]

+ 1205827 [120572V119864V minus (120583V + 1205911199063) 119868V]

(45)



1 119906lowast

2 119906lowast


min(119906111990621199063)isinΦ2

Γ (1199061 1199062 1199063) = Γ (119906lowast

1 119906lowast

2 119906lowast

3) (46)


Γ (1199061 1199062 1199063) ge 1205851(100381610038161003816100381611990611003816100381610038161003816

2+100381610038161003816100381611990621003816100381610038161003816

2100381610038161003816100381611990631003816100381610038161003816

2)1205982

minus 1205852(47)





0 =120597119867 (119905 120594 119906 120582)

120597119906

1205821015840=120597119867 (119905 120594 119906 120582)

120597120594

119889120594

119889119905= minus

120597119867 (119905 120594 119906 120582)

120597120582

(48)



2 119906lowast

3) with


ℎ 119868lowast

ℎ 119877lowast

ℎ 119878lowast

V 119864lowast

V 119868lowast


minus1205821015840


minus 120583ℎ1205821 + 12057912058111205822

minus1205821015840


minus1205821015840

3= (1 minus 1205811) 1205791205821 + (120601 + 1205781199062) [(1 minus 120588) 1205821 + 1205881205824]

+ 12057912058111205822 minus (120601 + 1205781199062 + 120583ℎ + 120575ℎ) 1205823

+ 120573ℎV120599 (1205826 minus 1205825) 119878V + 1198622

minus1205821015840

4= (1 minus 1205811) 1205791205821 + 1205951205821 + 12057912058111205822 minus (120583ℎ + 120595) 1205824

minus1205821015840

5= 120573ℎV120599 (1205826 minus 1205825) 119868ℎ minus (120583V + 1205911199063) 1205825 + 1198623

minus1205821015840

6= 120572V (1205827 minus 1205826) minus (120583V + 1205911199063) 1205826 + 1198623

minus1205821015840


(49)


1205821 (119879119891) = 1205822 (119879119891) = 1205823 (119879119891) = 1205824 (119879119891)

= 1205825 (119879119891) = 1205826 (119879119891) = 1205827 (119879119891) = 0

(50)


2 119906lowast

3) that mini-


119906lowast

1= max0min(1

120573Vℎ120599 (1205822 minus 1205821) 119868lowast

V 119878lowast

ℎ

1198601

)

119906lowast

2= max0min(1


ℎ

1198602

)

119906lowast

3= max0min(1

120591 (1205825119878lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

)

(51)




minus1198891205821

119889119905=120597119867

120597119878ℎ


minus 120583ℎ1205821 + 12057912058111205822

minus1198891205822

119889119905=120597119867

120597119864ℎ

= (1 minus 1205811) 1205791205821 + 12057912058111205822

+ 120572ℎ (1205823 minus 1205822) minus 120583ℎ1205822 + 1198621

minus1198891205823

119889119905=120597119867

120597119868ℎ

= (1 minus 1205811) 1205791205821

+ (120601 + 1205781199062) [(1 minus 120588) 1205821 + 1205881205824] + 12057912058111205822


minus1198891205824

119889119905=120597119867

120597119877ℎ

= (1 minus 1205811) 1205791205821 + 1205951205821 + 12057912058111205822 minus (120583ℎ + 120595) 1205824

minus1198891205825

119889119905=120597119867


minus1198891205826

119889119905=120597119867

120597119864V= 120572V (1205827 minus 1205826) minus (120583V + 1205911199063) 1205826 + 1198623

minus1198891205827

119889119905=120597119867

120597119868V= (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119878ℎ

minus (120583V + 1205911199063) 1205827 + 1198623

(52)


1205821 (119879119891) = 1205822 (119879119891) = 1205823 (119879119891) = 1205824 (119879119891)

= 1205825 (119879119891) = 1205826 (119879119891) = 1205827 (119879119891) = 0

(53)


lowast

ℎ

119864ℎ = 119864lowast

ℎ 119868ℎ = 119868

lowast

ℎ 119877ℎ = 119877

lowast

ℎ 119878V = 119878

lowast

V 119864V = 119864lowast

V and 119868V = 119868lowast

Vyields

120597119867

1205971199061

= 119860119906lowast

1+ 120573Vℎ1205991205821119868

lowast

V 119878lowast

ℎminus 120573Vℎ1205991205822119868

lowast

V 119878lowast

ℎ= 0

120597119867

1205971199062

= 11986021199062 + (1 minus 120588) 1205781205821119868lowast

ℎminus 1205781205823119868

lowast

ℎ+ 1205781205881205824119868

lowast

ℎ= 0

120597119867

1205971199063

= 1198603119906lowast

3minus 1205911205825119878

lowast

V minus 1205911205826119864lowast

V minus 1205911205827119868lowast

V = 0

(54)

for which

119906lowast

1=120573Vℎ120599 (1205822 minus 1205821) 119868

lowast

V 119878lowast

ℎ

1198601

119906lowast

2=120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868

lowast

ℎ

1198602

1199063 =120591 (1205825119878

lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

(55)

Then

119906lowast

1= max0min(1

120573Vℎ120599 (1205822 minus 1205821) 119868lowast

V 119878lowast

ℎ

1198601

)

119906lowast

2= max0min(1


ℎ

1198602

)

119906lowast

3= max0min(1

120591 (1205825119878lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

)

(56)




3



119889119878lowast

ℎ

119889119905= Λ ℎ + (1 minus 1205811) 120579119873

lowast

ℎ

+ (120601 + 1205781199062) (1 minus 120588) 119868lowast

ℎminus 120573Vℎ120599119868

lowast

V 119878lowast

ℎ


lowast

V 119878lowast

ℎ

1198601

))


ℎ+ 120595119877lowast

ℎ

119889119864lowast

ℎ

119889119905= 120573Vℎ120599119868

lowast

V 119878lowast

ℎ


lowast

V 119878lowast

ℎ

1198601

))

+ 1205811120579119873lowast

ℎminus 120572ℎ119864

lowast

ℎminus 120583ℎ119864

lowast

ℎ


119889119868lowast

ℎ

119889119905

= 120572ℎ119864lowast

ℎ

minus (120601 + 120578


lowast

ℎ

1198602

))

+120583ℎ + 120575ℎ) 119868lowast

ℎ

119889119877lowast

ℎ


times 119868lowast

ℎtimes (1198602)

minus1))) 120588119868ℎ

minus (120583ℎ + 120595)119877ℎ

119889119878lowast

V

119889119905

= Λ V minus (120583V + 120591


V + 1205826119864lowast

V + 1205827119868lowast

V )

times(1198603)minus1))) 119878

lowast


ℎ119878lowast

V

119889119864lowast

V

119889119905= 120573ℎV120599119868

lowast

ℎ119878lowast

V


V + 1205826119864lowast

V + 1205827119868lowast

V )

times(1198603)minus1))) 119864

lowast


V

119889119868lowast

V

119889119905= 120572V119864V


V + 1205826119864lowast

V +1205827119868lowast

V )

times(1198603)minus1))) 119868

lowast

V

(57)


ℎ 119868lowast

ℎ 119877lowast

ℎ 119906lowast

1 119906lowast

2 119906lowast

3 1205821 1205822 1205827)

119867lowast

= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V

+1

2(1198601(max0min(1

120573Vℎ120599 (1205822 minus 1205821) 119868lowast

V 119878lowast

ℎ

1198601

))

2


ℎ


+1198603(max0min(1120591(1205825119878

lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

))

2

)

+ 1205821

119889119878lowast

ℎ

119889119905+ 1205822

119889119864lowast

ℎ

119889119905+ 1205823

119889119868lowast

ℎ

119889119905+ 1205824

119889119877lowast

ℎ

119889119905+ 1205825

119889119878lowast

V

119889119905

+ 1205826

119889119864lowast

V

119889119905+ 1205827

119889119868lowast

V

119889119905







u1 ne 0 u3 ne 0 u2 = 0


0

10

20

30

40

50

60

70

times102In

fect

ed h

uman

pop

ulat

ion

0 10 20 30 40 50 60 70 80 90 100

Time (years)

(a)



fect

ed h

uman

pop

ulat

ion

0

10

20

30

40

50

60

70

0 10 20 30 40 50 60 70 80 90 100

times102

Time (days)

(b)

u2 ne 0 u3 ne 0 u1 = 0


0 10 20 30 40 50 600

5

10

15

20

25

30

35

40

times102

Infe

cted

hum

an p

opul

atio

n

Time (years)

(c)

u1 ne 0 u2 ne 0 u3 0ne ==u2 ne 0 u1 0 u3 0

=u1 ne 0 u2 0 u3 = 0 = =u3 ne 0 u1 0 u2 0

0

05

1

15

2

25

3

times104

0 10 20 30 40 50 60 70

Infe

cted

hum

an p

opul

atio

n

Time (years)

(d)








of total 22 30 53








409

93 53 47 7 4484

400440 446

486449

9 9 9 9 9 90

100

200

300

400

500

600

Bite

from

anin

fect

edm

osqu

ito

Fem

ale

mos

quito

Stag

nant

wat

er

Envi

ronm

ent

man

agem

ent

Oth

er

YesNo


wat

erso

aked

in ra

inRa

ined

on

Anopheles

Do not know


Seek treatment


47

418

493

462

409

454

83

8

38

92

0 100 200 300 400 500 600

Other

YesNo


House sprayed(IRS)

Use of ITNor LLN










5 Numerical Results






R0 = 18894

0 05 1 15 2 25 3 35 4

Time (years)

0

20

40

60

80

100

120

Susc

eptib

le in

divi

dual

s


(a)

R0 = 18894

0 05 1 15 2 25 3 35 4

Time (years)

0

20

40

60

80

100

120

140

times102

Expo

sed

indi

vidu

als

Exposed individuals

(b)

R0 = 18894

0 05 1 15 2 25 3 35 49

10

11

12

13

14

15

16

17

Time (years)

Infe

cted

indi

vidu

als


(c)

0 100 200 300 400 500 600 7000

1

2

3

4

5

6

7

8

Time (days)

Reco

vere

d in

divi

dual

s


R0 = 18894

times102

(d)








No Yes


of total 36 57 92



Time (days)

R0 = 18894

Prev

alen

ce

0 20 40 60 80 100 120 140 160 180 2000

01

02

03

04

05

06

07



6 Conclusions












of total 425 384 809



Hum

an p

opul

atio

n


times104

Time (years)0 05 1 15 2 25 3 35 4

0

05

1

15

2

25

(a)

times105

Hum

an p

opul

atio

n


Time (years)0 10 20 30 40 50 60

0

02

04

06

08

1

12

14

16

18

2

(b)


















Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











minus1198891205821

119889119905=120597119867

120597119878ℎ


minus 120583ℎ1205821 + 12057912058111205822

minus1198891205822

119889119905=120597119867

120597119864ℎ

= (1 minus 1205811) 1205791205821 + 12057912058111205822

+ 120572ℎ (1205823 minus 1205822) minus 120583ℎ1205822 + 1198621

minus1198891205823

119889119905=120597119867

120597119868ℎ

= (1 minus 1205811) 1205791205821

+ (120601 + 1205781199062) [(1 minus 120588) 1205821 + 1205881205824] + 12057912058111205822


minus1198891205824

119889119905=120597119867

120597119877ℎ

= (1 minus 1205811) 1205791205821 + 1205951205821 + 12057912058111205822 minus (120583ℎ + 120595) 1205824

minus1198891205825

119889119905=120597119867


minus1198891205826

119889119905=120597119867

120597119864V= 120572V (1205827 minus 1205826) minus (120583V + 1205911199063) 1205826 + 1198623

minus1198891205827

119889119905=120597119867

120597119868V= (1 minus 1199061) 120573Vℎ120599 (1205822 minus 1205821) 119878ℎ

minus (120583V + 1205911199063) 1205827 + 1198623

(52)


1205821 (119879119891) = 1205822 (119879119891) = 1205823 (119879119891) = 1205824 (119879119891)

= 1205825 (119879119891) = 1205826 (119879119891) = 1205827 (119879119891) = 0

(53)


lowast

ℎ

119864ℎ = 119864lowast

ℎ 119868ℎ = 119868

lowast

ℎ 119877ℎ = 119877

lowast

ℎ 119878V = 119878

lowast

V 119864V = 119864lowast

V and 119868V = 119868lowast

Vyields

120597119867

1205971199061

= 119860119906lowast

1+ 120573Vℎ1205991205821119868

lowast

V 119878lowast

ℎminus 120573Vℎ1205991205822119868

lowast

V 119878lowast

ℎ= 0

120597119867

1205971199062

= 11986021199062 + (1 minus 120588) 1205781205821119868lowast

ℎminus 1205781205823119868

lowast

ℎ+ 1205781205881205824119868

lowast

ℎ= 0

120597119867

1205971199063

= 1198603119906lowast

3minus 1205911205825119878

lowast

V minus 1205911205826119864lowast

V minus 1205911205827119868lowast

V = 0

(54)

for which

119906lowast

1=120573Vℎ120599 (1205822 minus 1205821) 119868

lowast

V 119878lowast

ℎ

1198601

119906lowast

2=120578 (1205823 + 120588 (1205821 minus 1205824) minus 1205821) 119868

lowast

ℎ

1198602

1199063 =120591 (1205825119878

lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

(55)

Then

119906lowast

1= max0min(1

120573Vℎ120599 (1205822 minus 1205821) 119868lowast

V 119878lowast

ℎ

1198601

)

119906lowast

2= max0min(1


ℎ

1198602

)

119906lowast

3= max0min(1

120591 (1205825119878lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

)

(56)




3



119889119878lowast

ℎ

119889119905= Λ ℎ + (1 minus 1205811) 120579119873

lowast

ℎ

+ (120601 + 1205781199062) (1 minus 120588) 119868lowast

ℎminus 120573Vℎ120599119868

lowast

V 119878lowast

ℎ


lowast

V 119878lowast

ℎ

1198601

))


ℎ+ 120595119877lowast

ℎ

119889119864lowast

ℎ

119889119905= 120573Vℎ120599119868

lowast

V 119878lowast

ℎ


lowast

V 119878lowast

ℎ

1198601

))

+ 1205811120579119873lowast

ℎminus 120572ℎ119864

lowast

ℎminus 120583ℎ119864

lowast

ℎ


119889119868lowast

ℎ

119889119905

= 120572ℎ119864lowast

ℎ

minus (120601 + 120578


lowast

ℎ

1198602

))

+120583ℎ + 120575ℎ) 119868lowast

ℎ

119889119877lowast

ℎ


times 119868lowast

ℎtimes (1198602)

minus1))) 120588119868ℎ

minus (120583ℎ + 120595)119877ℎ

119889119878lowast

V

119889119905

= Λ V minus (120583V + 120591


V + 1205826119864lowast

V + 1205827119868lowast

V )

times(1198603)minus1))) 119878

lowast


ℎ119878lowast

V

119889119864lowast

V

119889119905= 120573ℎV120599119868

lowast

ℎ119878lowast

V


V + 1205826119864lowast

V + 1205827119868lowast

V )

times(1198603)minus1))) 119864

lowast


V

119889119868lowast

V

119889119905= 120572V119864V


V + 1205826119864lowast

V +1205827119868lowast

V )

times(1198603)minus1))) 119868

lowast

V

(57)


ℎ 119868lowast

ℎ 119877lowast

ℎ 119906lowast

1 119906lowast

2 119906lowast

3 1205821 1205822 1205827)

119867lowast

= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V

+1

2(1198601(max0min(1

120573Vℎ120599 (1205822 minus 1205821) 119868lowast

V 119878lowast

ℎ

1198601

))

2


ℎ


+1198603(max0min(1120591(1205825119878

lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

))

2

)

+ 1205821

119889119878lowast

ℎ

119889119905+ 1205822

119889119864lowast

ℎ

119889119905+ 1205823

119889119868lowast

ℎ

119889119905+ 1205824

119889119877lowast

ℎ

119889119905+ 1205825

119889119878lowast

V

119889119905

+ 1205826

119889119864lowast

V

119889119905+ 1205827

119889119868lowast

V

119889119905







u1 ne 0 u3 ne 0 u2 = 0


0

10

20

30

40

50

60

70

times102In

fect

ed h

uman

pop

ulat

ion

0 10 20 30 40 50 60 70 80 90 100

Time (years)

(a)



fect

ed h

uman

pop

ulat

ion

0

10

20

30

40

50

60

70

0 10 20 30 40 50 60 70 80 90 100

times102

Time (days)

(b)

u2 ne 0 u3 ne 0 u1 = 0


0 10 20 30 40 50 600

5

10

15

20

25

30

35

40

times102

Infe

cted

hum

an p

opul

atio

n

Time (years)

(c)

u1 ne 0 u2 ne 0 u3 0ne ==u2 ne 0 u1 0 u3 0

=u1 ne 0 u2 0 u3 = 0 = =u3 ne 0 u1 0 u2 0

0

05

1

15

2

25

3

times104

0 10 20 30 40 50 60 70

Infe

cted

hum

an p

opul

atio

n

Time (years)

(d)








of total 22 30 53








409

93 53 47 7 4484

400440 446

486449

9 9 9 9 9 90

100

200

300

400

500

600

Bite

from

anin

fect

edm

osqu

ito

Fem

ale

mos

quito

Stag

nant

wat

er

Envi

ronm

ent

man

agem

ent

Oth

er

YesNo


wat

erso

aked

in ra

inRa

ined

on

Anopheles

Do not know


Seek treatment


47

418

493

462

409

454

83

8

38

92

0 100 200 300 400 500 600

Other

YesNo


House sprayed(IRS)

Use of ITNor LLN










5 Numerical Results






R0 = 18894

0 05 1 15 2 25 3 35 4

Time (years)

0

20

40

60

80

100

120

Susc

eptib

le in

divi

dual

s


(a)

R0 = 18894

0 05 1 15 2 25 3 35 4

Time (years)

0

20

40

60

80

100

120

140

times102

Expo

sed

indi

vidu

als

Exposed individuals

(b)

R0 = 18894

0 05 1 15 2 25 3 35 49

10

11

12

13

14

15

16

17

Time (years)

Infe

cted

indi

vidu

als


(c)

0 100 200 300 400 500 600 7000

1

2

3

4

5

6

7

8

Time (days)

Reco

vere

d in

divi

dual

s


R0 = 18894

times102

(d)








No Yes


of total 36 57 92



Time (days)

R0 = 18894

Prev

alen

ce

0 20 40 60 80 100 120 140 160 180 2000

01

02

03

04

05

06

07



6 Conclusions












of total 425 384 809



Hum

an p

opul

atio

n


times104

Time (years)0 05 1 15 2 25 3 35 4

0

05

1

15

2

25

(a)

times105

Hum

an p

opul

atio

n


Time (years)0 10 20 30 40 50 60

0

02

04

06

08

1

12

14

16

18

2

(b)


















Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119889119868lowast

ℎ

119889119905

= 120572ℎ119864lowast

ℎ

minus (120601 + 120578


lowast

ℎ

1198602

))

+120583ℎ + 120575ℎ) 119868lowast

ℎ

119889119877lowast

ℎ


times 119868lowast

ℎtimes (1198602)

minus1))) 120588119868ℎ

minus (120583ℎ + 120595)119877ℎ

119889119878lowast

V

119889119905

= Λ V minus (120583V + 120591


V + 1205826119864lowast

V + 1205827119868lowast

V )

times(1198603)minus1))) 119878

lowast


ℎ119878lowast

V

119889119864lowast

V

119889119905= 120573ℎV120599119868

lowast

ℎ119878lowast

V


V + 1205826119864lowast

V + 1205827119868lowast

V )

times(1198603)minus1))) 119864

lowast


V

119889119868lowast

V

119889119905= 120572V119864V


V + 1205826119864lowast

V +1205827119868lowast

V )

times(1198603)minus1))) 119868

lowast

V

(57)


ℎ 119868lowast

ℎ 119877lowast

ℎ 119906lowast

1 119906lowast

2 119906lowast

3 1205821 1205822 1205827)

119867lowast

= 1198621119864ℎ + 1198622119868ℎ + 1198623119873V

+1

2(1198601(max0min(1

120573Vℎ120599 (1205822 minus 1205821) 119868lowast

V 119878lowast

ℎ

1198601

))

2


ℎ


+1198603(max0min(1120591(1205825119878

lowast

V + 1205826119864lowast

V + 1205827119868lowast

V )

1198603

))

2

)

+ 1205821

119889119878lowast

ℎ

119889119905+ 1205822

119889119864lowast

ℎ

119889119905+ 1205823

119889119868lowast

ℎ

119889119905+ 1205824

119889119877lowast

ℎ

119889119905+ 1205825

119889119878lowast

V

119889119905

+ 1205826

119889119864lowast

V

119889119905+ 1205827

119889119868lowast

V

119889119905







u1 ne 0 u3 ne 0 u2 = 0


0

10

20

30

40

50

60

70

times102In

fect

ed h

uman

pop

ulat

ion

0 10 20 30 40 50 60 70 80 90 100

Time (years)

(a)



fect

ed h

uman

pop

ulat

ion

0

10

20

30

40

50

60

70

0 10 20 30 40 50 60 70 80 90 100

times102

Time (days)

(b)

u2 ne 0 u3 ne 0 u1 = 0


0 10 20 30 40 50 600

5

10

15

20

25

30

35

40

times102

Infe

cted

hum

an p

opul

atio

n

Time (years)

(c)

u1 ne 0 u2 ne 0 u3 0ne ==u2 ne 0 u1 0 u3 0

=u1 ne 0 u2 0 u3 = 0 = =u3 ne 0 u1 0 u2 0

0

05

1

15

2

25

3

times104

0 10 20 30 40 50 60 70

Infe

cted

hum

an p

opul

atio

n

Time (years)

(d)








of total 22 30 53








409

93 53 47 7 4484

400440 446

486449

9 9 9 9 9 90

100

200

300

400

500

600

Bite

from

anin

fect

edm

osqu

ito

Fem

ale

mos

quito

Stag

nant

wat

er

Envi

ronm

ent

man

agem

ent

Oth

er

YesNo


wat

erso

aked

in ra

inRa

ined

on

Anopheles

Do not know


Seek treatment


47

418

493

462

409

454

83

8

38

92

0 100 200 300 400 500 600

Other

YesNo


House sprayed(IRS)

Use of ITNor LLN










5 Numerical Results






R0 = 18894

0 05 1 15 2 25 3 35 4

Time (years)

0

20

40

60

80

100

120

Susc

eptib

le in

divi

dual

s


(a)

R0 = 18894

0 05 1 15 2 25 3 35 4

Time (years)

0

20

40

60

80

100

120

140

times102

Expo

sed

indi

vidu

als

Exposed individuals

(b)

R0 = 18894

0 05 1 15 2 25 3 35 49

10

11

12

13

14

15

16

17

Time (years)

Infe

cted

indi

vidu

als


(c)

0 100 200 300 400 500 600 7000

1

2

3

4

5

6

7

8

Time (days)

Reco

vere

d in

divi

dual

s


R0 = 18894

times102

(d)








No Yes


of total 36 57 92



Time (days)

R0 = 18894

Prev

alen

ce

0 20 40 60 80 100 120 140 160 180 2000

01

02

03

04

05

06

07



6 Conclusions












of total 425 384 809



Hum

an p

opul

atio

n


times104

Time (years)0 05 1 15 2 25 3 35 4

0

05

1

15

2

25

(a)

times105

Hum

an p

opul

atio

n


Time (years)0 10 20 30 40 50 60

0

02

04

06

08

1

12

14

16

18

2

(b)


















Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










u1 ne 0 u3 ne 0 u2 = 0


0

10

20

30

40

50

60

70

times102In

fect

ed h

uman

pop

ulat

ion

0 10 20 30 40 50 60 70 80 90 100

Time (years)

(a)



fect

ed h

uman

pop

ulat

ion

0

10

20

30

40

50

60

70

0 10 20 30 40 50 60 70 80 90 100

times102

Time (days)

(b)

u2 ne 0 u3 ne 0 u1 = 0


0 10 20 30 40 50 600

5

10

15

20

25

30

35

40

times102

Infe

cted

hum

an p

opul

atio

n

Time (years)

(c)

u1 ne 0 u2 ne 0 u3 0ne ==u2 ne 0 u1 0 u3 0

=u1 ne 0 u2 0 u3 = 0 = =u3 ne 0 u1 0 u2 0

0

05

1

15

2

25

3

times104

0 10 20 30 40 50 60 70

Infe

cted

hum

an p

opul

atio

n

Time (years)

(d)








of total 22 30 53








409

93 53 47 7 4484

400440 446

486449

9 9 9 9 9 90

100

200

300

400

500

600

Bite

from

anin

fect

edm

osqu

ito

Fem

ale

mos

quito

Stag

nant

wat

er

Envi

ronm

ent

man

agem

ent

Oth

er

YesNo


wat

erso

aked

in ra

inRa

ined

on

Anopheles

Do not know


Seek treatment


47

418

493

462

409

454

83

8

38

92

0 100 200 300 400 500 600

Other

YesNo


House sprayed(IRS)

Use of ITNor LLN










5 Numerical Results






R0 = 18894

0 05 1 15 2 25 3 35 4

Time (years)

0

20

40

60

80

100

120

Susc

eptib

le in

divi

dual

s


(a)

R0 = 18894

0 05 1 15 2 25 3 35 4

Time (years)

0

20

40

60

80

100

120

140

times102

Expo

sed

indi

vidu

als

Exposed individuals

(b)

R0 = 18894

0 05 1 15 2 25 3 35 49

10

11

12

13

14

15

16

17

Time (years)

Infe

cted

indi

vidu

als


(c)

0 100 200 300 400 500 600 7000

1

2

3

4

5

6

7

8

Time (days)

Reco

vere

d in

divi

dual

s


R0 = 18894

times102

(d)








No Yes


of total 36 57 92



Time (days)

R0 = 18894

Prev

alen

ce

0 20 40 60 80 100 120 140 160 180 2000

01

02

03

04

05

06

07



6 Conclusions












of total 425 384 809



Hum

an p

opul

atio

n


times104

Time (years)0 05 1 15 2 25 3 35 4

0

05

1

15

2

25

(a)

times105

Hum

an p

opul

atio

n


Time (years)0 10 20 30 40 50 60

0

02

04

06

08

1

12

14

16

18

2

(b)


















Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













of total 22 30 53








409

93 53 47 7 4484

400440 446

486449

9 9 9 9 9 90

100

200

300

400

500

600

Bite

from

anin

fect

edm

osqu

ito

Fem

ale

mos

quito

Stag

nant

wat

er

Envi

ronm

ent

man

agem

ent

Oth

er

YesNo


wat

erso

aked

in ra

inRa

ined

on

Anopheles

Do not know


Seek treatment


47

418

493

462

409

454

83

8

38

92

0 100 200 300 400 500 600

Other

YesNo


House sprayed(IRS)

Use of ITNor LLN










5 Numerical Results






R0 = 18894

0 05 1 15 2 25 3 35 4

Time (years)

0

20

40

60

80

100

120

Susc

eptib

le in

divi

dual

s


(a)

R0 = 18894

0 05 1 15 2 25 3 35 4

Time (years)

0

20

40

60

80

100

120

140

times102

Expo

sed

indi

vidu

als

Exposed individuals

(b)

R0 = 18894

0 05 1 15 2 25 3 35 49

10

11

12

13

14

15

16

17

Time (years)

Infe

cted

indi

vidu

als


(c)

0 100 200 300 400 500 600 7000

1

2

3

4

5

6

7

8

Time (days)

Reco

vere

d in

divi

dual

s


R0 = 18894

times102

(d)








No Yes


of total 36 57 92



Time (days)

R0 = 18894

Prev

alen

ce

0 20 40 60 80 100 120 140 160 180 2000

01

02

03

04

05

06

07



6 Conclusions












of total 425 384 809



Hum

an p

opul

atio

n


times104

Time (years)0 05 1 15 2 25 3 35 4

0

05

1

15

2

25

(a)

times105

Hum

an p

opul

atio

n


Time (years)0 10 20 30 40 50 60

0

02

04

06

08

1

12

14

16

18

2

(b)


















Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















5 Numerical Results






R0 = 18894

0 05 1 15 2 25 3 35 4

Time (years)

0

20

40

60

80

100

120

Susc

eptib

le in

divi

dual

s


(a)

R0 = 18894

0 05 1 15 2 25 3 35 4

Time (years)

0

20

40

60

80

100

120

140

times102

Expo

sed

indi

vidu

als

Exposed individuals

(b)

R0 = 18894

0 05 1 15 2 25 3 35 49

10

11

12

13

14

15

16

17

Time (years)

Infe

cted

indi

vidu

als


(c)

0 100 200 300 400 500 600 7000

1

2

3

4

5

6

7

8

Time (days)

Reco

vere

d in

divi

dual

s


R0 = 18894

times102

(d)








No Yes


of total 36 57 92



Time (days)

R0 = 18894

Prev

alen

ce

0 20 40 60 80 100 120 140 160 180 2000

01

02

03

04

05

06

07



6 Conclusions












of total 425 384 809



Hum

an p

opul

atio

n


times104

Time (years)0 05 1 15 2 25 3 35 4

0

05

1

15

2

25

(a)

times105

Hum

an p

opul

atio

n


Time (years)0 10 20 30 40 50 60

0

02

04

06

08

1

12

14

16

18

2

(b)


















Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










R0 = 18894

0 05 1 15 2 25 3 35 4

Time (years)

0

20

40

60

80

100

120

Susc

eptib

le in

divi

dual

s


(a)

R0 = 18894

0 05 1 15 2 25 3 35 4

Time (years)

0

20

40

60

80

100

120

140

times102

Expo

sed

indi

vidu

als

Exposed individuals

(b)

R0 = 18894

0 05 1 15 2 25 3 35 49

10

11

12

13

14

15

16

17

Time (years)

Infe

cted

indi

vidu

als


(c)

0 100 200 300 400 500 600 7000

1

2

3

4

5

6

7

8

Time (days)

Reco

vere

d in

divi

dual

s


R0 = 18894

times102

(d)








No Yes


of total 36 57 92



Time (days)

R0 = 18894

Prev

alen

ce

0 20 40 60 80 100 120 140 160 180 2000

01

02

03

04

05

06

07



6 Conclusions












of total 425 384 809



Hum

an p

opul

atio

n


times104

Time (years)0 05 1 15 2 25 3 35 4

0

05

1

15

2

25

(a)

times105

Hum

an p

opul

atio

n


Time (years)0 10 20 30 40 50 60

0

02

04

06

08

1

12

14

16

18

2

(b)


















Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












No Yes


of total 36 57 92



Time (days)

R0 = 18894

Prev

alen

ce

0 20 40 60 80 100 120 140 160 180 2000

01

02

03

04

05

06

07



6 Conclusions












of total 425 384 809



Hum

an p

opul

atio

n


times104

Time (years)0 05 1 15 2 25 3 35 4

0

05

1

15

2

25

(a)

times105

Hum

an p

opul

atio

n


Time (years)0 10 20 30 40 50 60

0

02

04

06

08

1

12

14

16

18

2

(b)


















Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













of total 425 384 809



Hum

an p

opul

atio

n


times104

Time (years)0 05 1 15 2 25 3 35 4

0

05

1

15

2

25

(a)

times105

Hum

an p

opul

atio

n


Time (years)0 10 20 30 40 50 60

0

02

04

06

08

1

12

14

16

18

2

(b)


















Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article optimal (control of) intervention ... · research article optimal (control of)...

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