Research Article Transmission Dynamics and Optimal Control of Malaria in Kenya

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1 Hindawi Publising Corporation Discrete Dynamics in Nature and Society Volume 216 Article ID pages ttp://dx.doi.org/1.1155/216/ Researc Article Transmission Dynamics and Optimal Control of Malaria in Kenya Gabriel Otieno Josep K. Koske and Jon M. Mutiso DepartmentofStatisticsandComputerScienceMoiUniversityP.O.Box39Eldoret31Kenya Correspondence sould be addressed to Gabriel Otieno; Received 16 November 215; Revised 5 January 216; Accepted 6 April 216 Academic Editor: Xiaoua Ding Copyrigt 216 Gabriel Otieno et al. Tis is an open access article distributed under te Creative Commons Attribution License wic permits unrestricted use distribution and reproduction in any medium provided te original work is properly cited. Tis paper proposes and analyses a matematical model for te transmission dynamics of malaria wit four-time dependent control measures in Kenya: insecticide treated bed nets (ITNs) treatment indoor residual spray (IRS) and intermittent preventive treatment of malaria in pregnancy (IPTp). We first considered constant control parameters and calculate te basic reproduction number and investigate existence and stability of equilibria as well as stability analysis. We proved tat if R 1tedisease-free equilibrium is globally asymptotically stable in D.IfR >1 te unique endemic equilibrium exists and is globally asymptotically stable. Te model also exibits backward bifurcation at R =1.IfR >1 te model admits a unique endemic equilibrium wic is globally asymptotically stable in te interior of feasible region D. Te sensitivity results sowed tat te most sensitive parameters are mosquito deat rate and mosquito biting rates. We ten consider te time-dependent control case and use Pontryagin s Maximum Principle to derive te necessary conditions for te optimal control of te disease using te proposed model. Te existence of optimal control problem is proved. Numerical simulations of te optimal control problem using a set of reasonable parameter values suggest tat te optimal control strategy for malaria control in endemic areas is te combined use of treatment and IRS; for epidemic prone areas is te use of treatment and IRS; for seasonal areas is te use of treatment; and for low risk areas is te use of ITNs and treatment. Control programs tat follow tese strategies can effectively reduce te spread of malaria disease in different malaria transmission settings in Kenya. 1. Introduction Malaria is a leading cause of mortality and morbidity among te under-five group and te pregnant women in Sub- Saaran Africa [1]. Tese groups are at ig risk due to weakened and immature immunity respectively. Wit te recent conversion of te Millennium Development Goals (MDGs) to Sustainable Development Goals (SDGs) as part of Global Malaria Action Plan for a malaria-free world by 23 reducing malaria is critical to post-215 malaria strategies and for acieving te SDGs suc as ensuring ealty lives and promoting well-being for all at all ages. Most Kenyans are vulnerable to malaria because of poverty inadequate ealt care infrastructures and low income of te country. Prompt access to effective treatment for malaria is unacceptably low in Kenya due to te socioeconomic barriers to accessing ealt care. Tese callenges call for urgent development of effective and optimal strategies for preventing and controlling spread of malaria. Malaria transmission is igly variable across Kenya because of te different transmission intensities driven by climate and temperature. Kenya as four malaria epidemiological zones: te endemic areas te seasonal malaria transmission te malaria epidemic prone areas and te low risk malaria areas [2]. Malaria is caused by Plasmodium parasites and it is transmitted from one individual to anoter bytebiteofinfectedfemaleanopelinemosquitoes[1]. Malaria is an entirely preventable and treatable disease provided te currently recommended interventions are properly implemented. Controlling malaria transmission involves interrupting te malaria transmission for specific transmission settings and for te most at-risk groups of malaria. World Healt organization (WHO) recommended malaria intervention strategies include te use of long-lasting insecticidetreated bed nets (LLINs) indoor residual spray (IRS) cemoprevention for te most vulnerable suc as intermittent preventive treatment for pregnant women (IPTp) confirmation of malaria diagnostics troug rapid diagnostics tests (RDTs)

2 2 Discrete Dynamics in Nature and Society and microscopy for every suspected case and timely treatment wit artemisinin-based combination terapies (ACTs) [1 3]. Matematical modelling as become an important tool in understanding te dynamics of disease transmission and in decision making processes regarding intervention programs for disease control. Matematical models provide a framework for understanding te transmission dynamics for malaria and can be used for te optimal allocation of different interventions against malaria [4 5]. Optimal control is a branc of matematics developed to find optimal ways of controlling a dynamic system [6]. Application of optimal control teory can be an important tool for estimating te efficacy of various policies and control measures and te cost of implementing tem. Optimal control approaces avebeensuccessfullypreviouslyusedindecisionmaking for te infectious diseases suc as tuberculosis [7 8] HIV [9] and influenza [1]. Optimal control teory as also been applied in malaria control to assess te impact of antimalarial control measures by formulating te model as an optimal control problem. Most malaria models for analyzing effect of interventions in optimal control used te standard Susceptible-Exposed-Infectious-Recovered (SEIR) model for umans and Susceptible-Exposed-Infectious (SEI) model for mosquitoes. Okosun et al. [11] considered tree control variables in assessing te optimal control and cost effectiveness of te interventions but not for different settings. Mwamtobe et al. [12] used tree control variables and for only one region in Malawi. Kim et al. [13] used two control efforts in te optimal control model for malaria transmission in Sout Korea. Agusto et al. [14] used tree system control variables. Silva and Torres [15] used one control variable. IPTp is one of te WHO recommended prevention terapies for te pregnant women. IPTp as been sown to be effective in reducing maternal and infant mortality tat are related to malaria for te most at-risk group for malaria [16 19]. Very few studies ave been carried out in applying optimal control teory to malaria transmission models for different transmission settings. Te combined effect of ITNs IRS and natural deat on reducing te mosquito population as not been demonstrated in optimal control teory in malaria control. Te effect of IPTp wic is WHO recommended preventive terapy for te most at-risk group for malaria (pregnant women) as not been studied in optimal control teory. No optimal control model for malaria interventions for different transmission settings exits for Kenya. No optimal controlmodelforfourcontrolvariablesincorporatingte IPTp malaria intervention studies exits for Kenya. No optimal control model as been stratified by te age group (under five) and specific categories (pregnant women). In tis paper a model for malaria transmission dynamics wit four control strategies is formulated and analyzed. We ten formulate an optimal control problem and derive expressions for te optimal control for te malaria transmissionmodelwitfourcontrolvariablesanenusete optimal control teory to study te effectiveness of all possible combinations of four malaria preventive measures among te pregnant women and cildren under five years of age. 2. Model Formulation Te model is formulated by considering te uman and mosquito subgroups. Te considered model consists of population of susceptible S ExposedumansE infected umans I recovered umans R susceptiblemosquitoess m exposed mosquitoes E m andinfectedmosquitoesi m.te total population sizes at time t for umans and mosquitoes are denoted by N (t) and N m (t) respectively. We employ te SEIRS type model for umans to describe a disease wit temporaryimmunityonrecoveryfrominfection.mosquitoes are assumed not to recover from te parasites so te mosquito population can be described by te SEI model. In te model we incorporate four time-dependent control measures simultaneously: (i) te use of treated bed nets u 1 (t) (ii) treatment of infective umans u 2 (t) (iii) spray of insecticides u 3 (t) and (iv) treatment to protect pregnant women and teir newborn cildren: intermittent preventive treatment for pregnant women (IPTp) u 4 (t). S (t) represents te number of individuals not yet infected wit te malaria parasite at time t E (t) represents individuals wo are infected but not yet infectious I (t) is te class representing tose wo are infected wit malaria and are capable of transmitting te disease to susceptible mosquitoes and R (t) represents te class of individuals wo ave temporarily recovered from te disease. Figure 1 describes te dynamics of malaria in uman and mosquito populations togeter wit interventions. Te susceptible umans (pregnant women and cildren under te age of five) (S ) are recruited at te rate Λ.Tey eiter die from natural causes (at a rate μ )ormovetote exposed class (E ) by acquiring malaria troug contact wit infectious mosquitoes at a rate (1 u 1 )(βεφi m /N )S or (1 u 4 )(βεφi m /N w )S wereβ is te transmission probability per bite ε is te per capita biting rate of mosquitoes φ is te contact rate of vector per uman per unit time u 1 (t) [ 1] is te preventive measure using ITNs u 4 (t) [ 1] is te preventive measure using IPTp I m (t) is te infectious mosquitoes at time t N (t) is te total number of individuals (pregnant women and cildren under te age of five) and N w (t) is te total number of pregnant women. Susceptible class S is divided into wole population (cildren under te age of five and pregnant women) being exposed and te population for te pregnant women being exposed. Exposed individuals move to te infectious class after te development of clinical symptoms at te rate α.infectiousindividualsare assumed to recover at a rate b+τu 2 wereb isterateof spontaneous recovery u 2 (t) [ 1] is te control on treatment of infected individuals and τ [1]is te efficacy of treatment. Infectious individuals wo do not recover die at arateδ +μ. Individuals infected wit malaria suffer a disease induced deat at rate of δ and natural deat μ. Infected individuals ten progress to partially immune group were upon recovery te partially immune individual loses immunity at te rate ψ and becomes susceptible again. Susceptible mosquitoes (S m ) are recruited at te rate Λ m and acquire malaria infection (following contact wit umans infected wit malaria) at te rate λ m. Tey eiter die from natural causes (at a rate μ m ) or move to te exposed class

3 Discrete Dynamics in Nature and Society 3 ψr (1 u 1 )λ S Λ S E α E I (b + τu 2 )I R (1 u 4 )λ w S μ S β μ (δ +μ )I λ E μ R I m α m E m E m (1 u 1 )λ m S m S m Λ m au 1 I m +pu 3 I m +μ m I m au 1 E m +pu 3 E m +μ m E m au 1 S m +pu 3 S m +μ m S m Figure 1: Malaria model wit interventions. by acquiring malaria troug contacts wit infected umans at a rate (1 u 1 )(λεφi /N )S m wereλ is te probability for a vector to get infected after biting an infectious uman and I (t) are individuals infected wit malaria at time t.te mosquito population is reduced due to te use of insecticides spray at a rate pu 3 wereu 3 (t) [ 1] represents te control due to IRS and p represents te efficacy of IRS. Mosquito population is also reduced as a result of natural deat (μ m ) and at te rate au 1 wereu 1 (t) represents te control due to ITNs and a is te efficacy due to ITNs. Newly infected mosquitoes are moved into te exposed class (E m )atarateα m andprogressestoteclassofsymptomaticmosquitoes(i m ). Te state variables of te model are represented and described in Table 1. Table 2 describes and sows parameters of te model. Table 3 describes and represents malaria prevention and control strategies practiced in Kenya. Tefollowingassumptionsavebeenusedinteformulation of te model: (i) Population for uman and mosquito being constant (no immigrants). (ii) No recovery for infected mosquitoes. (iii) Mosquitoes not dying due to disease infection. (iv) All parameters in te model being nonnegative. Putting te above formulations and assumptions togeter gives te following system of nonlinear differential equations describing te dynamics of malaria in uman and mosquito populations togeter wit interventions: ds =Λ +ψr (1 u 1 )λ S (1 u 4 )λ w S μ S Variable S (t) E (t) I (t) R (t) S m (t) E m (t) I m (t) N (t) N w (t) N m (t) Table 1: State variables of te malaria model. Description Number of susceptible individuals (pregnant and under 5) at time t Number of exposed individuals (pregnant and under 5) at time t Number of infectious umans (pregnant and under 5) at time t Number of recovered umans (pregnant and under 5) at time t Number of susceptible mosquitoes at time t Number of exposed mosquitoes at time t Number of infectious mosquitoes at time t Total number of individuals (pregnant and under 5) at time t Total number of pregnant individuals at time t Total mosquito population at time t de =(1 u 1 )λ S +(1 u 4 )λ w S (α +μ )E di =α E (δ +μ )I (b+τu 2 )I dr =(b+τu 2 )I (ψ+μ )R ds m =Λ m (1 u 1 )λ m S m (μ m +au 1 +pu 3 )S m de m =(1 u 1 )λ m S m α m E m

4 4 Discrete Dynamics in Nature and Society Table 2: Description of parameter variables of te malaria model. Parameter φ ε β λ μ μ m ψ α α m Λ Λ m δ b λ λ w λ m Variable u 1 (t) u 2 (t) u 3 (t) u 4 (t) p τ a Description Mosquito contact rate wit uman Mosquito biting rate Probability of uman getting infected (te probability of transmission of infection from an infectious mosquito to a susceptible uman provided tere is a bite) Probability of a mosquito getting infected (te probability of transmission of infection from an infectious uman to a susceptible mosquito provided tere is a bite) Per capita natural deat rate of umans Per capita natural deat rate of mosquitoes Per capita rate of loss of immunity by recovered individuals Humans progression rate from exposed to infected Mosquitoes progression rate from exposed to infected Recruitment rate of uman by birt and by getting pregnant Recruitment of mosquitoes by birt Per capita disease induced deat rate for umans (pregnant and under 5) Proportion of spontaneous individual recovery Force of infection for susceptible umans (pregnant and under 5) to exposed individuals Force of infection for susceptible pregnant umans to exposed individuals Force of infection for susceptible mosquitoes to exposed mosquitoes Table 3: Prevention and control variables in te model. Description Preventive measure using insecticide-treated bed nets (ITNs) Te control effort on treatment of infectious individuals Preventing measure using indoor residual spray (IRS) Preventive measure using intermittent preventive treatment for pregnant women (IPTp) Rate constant due to use of indoor residual spray Rate constant due to use of treatment effort Rate constant due to use of insecticide-treated bed nets (μ m +au 1 +pu 3 )E m Wit initial conditions S () E () I () R () S m () E m () I m () λ m =λεφi /N is te per capita incidence rate among mosquitoes (force of infection for susceptible mosquitoes) λ = βεφi m /N is te force of infection for susceptible umans (pregnant and under 5) and λ w =βεφi m /N w is te force of infection for susceptible pregnant umans. Te total population size for te uman is N =S +E + I +R and for mosquito is N m =S m +E m +I m and teir differential equations are given by dn / = Λ μ N δ I and dn m / = Λ m (μ m +au 1 +pu 3 )N m respectively Matematical Analysis of te Malaria Model wit Intervention Strategies. We will assume tat te control parameters are constant so as to determine te basic reproduction number steady states and teir stability as well as te bifurcation analysis. We describe te basic properties and analysis of te formulated malaria model wit control strategies troug matematical analysis of te formulated model Basic Properties of te Model: Positivity and Invariant Regions. All te state variables and parameters for model (1) are nonnegative for all t. Te feasible solutions set for model (1) given by D={(S E I R S m E m I m ) R 7 + :(S S m ) > (E I R E m I m ) ; N Λ μ ; N m Λ m μ m +au 1 +pu 3 } is positively invariant and ence model (1) is biologically epidemiologically meaningful and matematically well posed in te domain D. System (1) as always a disease-free equilibrium given by (2) (3) E =(S E I R S m E m I m ) =( Λ Λ (4) m μ (μ m +au 1 +pu 3 ) ). di m =α m E m (μ m +au 1 +pu 3 )I m. (1) Basic Reproduction Number. Te matrices F and V for te new infection terms and te remaining transfer terms at disease-free equilibrium [26] respectively are given by

5 Discrete Dynamics in Nature and Society 5 (1 u 1 )βεφ+(1 u 4 )βεφ F= (1 u [ 1 )λεφμ Λ m α (μ m +au 1 +pu 3 ) ] [ ] (μ +α ) α V= (δ +μ +b+τu 2 ) [ α m +μ m +au 1 +pu 3. ] [ α m μ m +au 1 +pu 3 ] (5) It follows tat te basic reproduction of model (1) denoted by R isgivenby R =ρ(fv 1 ) α R = (1 u 1 )λεφλ m μ (1 u 1 )βεφα m +α (1 u 1 )λεφλ m μ (1 u 4 )βεφα m (μ m +au 1 +pu 3 )(μ +α )(δ +μ +b+τu 2 )Λ (α m +μ m +au 1 +pu 3 )(μ m +au 1 +pu 3 ). (6) Stability Analysis of Disease-Free Equilibrium Point (1) Local Stability of Disease-Free Equilibrium Point Proof. Te Jacobian matrix (J) of te malaria model (1) at te disease-free equilibrium point is given by Teorem 1. Te disease-free equilibrium point for system (1) is locally asymptotically stable if R <1. (α +μ ) (1 u 1 )βεφ+(1 u 4 )βεφ α (δ +μ +b+τu 2 ) (b+τu 2 ) (ψ+μ ). (7) (1 u [ 1 )λεφλ m μ (α (μ m +au 1 +pu 3 )Λ m +μ m +au 1 +pu 3 ) ] [ α m (μ m +au 1 +pu 3 ) ] Te eigenvalues of te Jacobian matrix are te solutions of te caracteristic equation J λi = (α +μ +λ) (1 u 1 )βεφ+(1 u 4 )βεφ α (δ +μ +b+τu 2 +λ) (b+τu 2 ) (ψ+μ +λ) (1 u [ 1 )λεφλ m μ (α (μ m +au 1 +pu 3 )Λ m +μ m +au 1 +pu 3 +λ) ] [ α m (μ m +au 1 +pu 3 +λ) ] =. (8)

6 6 Discrete Dynamics in Nature and Society Expanding te determinant into a caracteristic equation we ave (δ +μ )(α +μ +λ)(δ +μ +b+τu 2 +λ)(α m +μ m +au 1 +pu 3 +λ)(μ m +au 1 +pu 3 +λ) (1 u 1) 2 λε 2 φ 2 Λ m μ α m βα +(1 u 4 )(1 u 1 )λε 2 φ 2 Λ m μ α m βα (μ m +au 1 +pu 3 )Λ m =. (9) Henceweave were A 1 =(μ m +au 1 +pu 3 ) A 2 =(α m +μ m +au 1 +pu 3 ) A 3 =(δ +μ +b+τu 2 ) A 4 =(α +μ ) Q=((1 u 1 ) 2 λε 2 φ 2 Λ m μ α m βα +(1 u 4 )(1 u 1 )λε 2 φ 2 Λ m μ α m βα ) ((μ m +au 1 +pu 3 )Λ m ) 1 (λ + A 1 )(λ+a 2 )(λ+a 3 )(λ+a 4 ) Q= λ 4 +B 1 λ 3 +B 2 λ 2 +B 3 λ+b 4 = B 1 =A 4 +A 3 +A 2 +A 1 B 2 =A 4 (A 3 +A 2 +A 1 )+A 3 (A 2 +A 1 )+A 1 A 2 B 3 =A 4 A 3 A 2 +A 4 A 3 A 1 +A 4 A 2 A 1 +A 3 A 2 A 1 B 4 =A 4 A 3 A 2 A 1 Q. (1) (11) Tus applying te Rout-Hurwitz criteria [27] to polynomial (1) we ave tat all determinants of te Hurwitz matrices arepositive.henceallteeigenvaluesoftejacobianave negative real part implying tat te DFE point is (at least) locally asymptotically stable (R <1). Next we study te global beavior of te disease-free equilibrium for system (1). (2) Global Stability of Disease-Free Equilibrium Point Teorem 2. Te DFE E ofsystemof(1)isgloballyasymptotically stable if R <1. Proof. We consider te following Lyapunov function: were c 1 = c 2 = c 3 = L=c 1 E +c 2 I +c 3 E m +c 4 I m (12) α (μ m +au 1 +pu 3 )(α +μ )(b+τu 2 +δ +μ ) 1 (μ m +au 1 +pu 3 )(b+τu 2 +δ +μ ) 1 (1 u 1 )εφλλ m c 4 = α m +μ m +au 1 +pu 3 (1 u 1 )εφλλ m α m. (13) Computing te derivative of L along te solution of te system of differential equation (1) L= α (μ m +au 1 +pu 3 )(α +μ )(b+τu 2 +δ +μ ) [(1 u 1)λ S +(1 u 4 )λ w S (α +μ )E ] + 1 (μ m +au 1 +pu 3 )(b+τu 2 +δ +μ ) [α 1 E (δ +μ )I (b+τu 2 )I ]+ [(1 u (1 u 1 )εφλλ 1 )λ m S m α m E m m (μ m +au 1 +pu 3 )E m ]+ α m +μ m +au 1 +pu 3 (1 u 1 )εφλλ m α m [α m E m (μ m +au 1 +pu 3 )I m ] L=[ α [(1 u 1 )λ S +(1 u 4 )λ w S (α +μ )] (μ m +au 1 +pu 3 )(α +μ )(b+τu 2 +δ +μ ) ]E +[ [α E (δ +μ )I (b+τu 2 )] (μ m +au 1 +pu 3 )(b+τu 2 +δ +μ ) ]I +[ [(1 u 1)λ m S m α m E m (μ m +au 1 +pu 3 )] (1 u 1 )εφλλ m ]E m +[ (α m +μ m +au 1 +pu 3 )(μ m +au 1 +pu 3 ) (1 u 1 )εφλλ m α m ]I m (14) L= (α m +μ m +au 1 +pu 3 )(μ m +au 1 +pu 3 +δ ) α [ (1 u 1 )λεφλ m μ (1 u 1 )βεφα m +α (1 u 1 )λεφλ m μ (1 u 4 )βεφα m (1 u 1 )εφλλ m α m (μ m +au 1 +pu 3 )(μ +α )(δ +μ +b+τu 2 )Λ (α m +μ m +au 1 +pu 3 )(μ m +au 1 +pu 3 ) 1]I L= (α m +μ m +au 1 +pu 3 )(μ m +au 1 +pu 3 +δ ) [R (1 u 1 )εφλλ m α 1]I iff R 1. m

7 Discrete Dynamics in Nature and Society 7 Tus we ave establised tat L if R <1and te equality L=olds if and only if R =1and E =I =E m =I m =. If R >1ten L>wen S (t) and S m (t) are sufficiently close to Λ /μ and Λ m /(μ m +au 1 +pu 3 ) respectively except wen E =I =E m =I m =. On te boundary wen E =I =E m =I m = N (t) = Λ μ N and N m (t) = Λ m (μ m +au 1 +pu 3 ) and N (t) Λ /μ N m (t) Λ m /(μ m +au 1 +pu 3 ) as t. Terefore te largest compact invariant D = {(S E I R S m E m I m ) R 7 + : L=}wen R <1is te singleton {E } in D. By LaSalle s invariant principle [28] E is globally asymptotically stable for R <1in D. Next we investigate te endemic equilibrium and its stability of system (1) Stability Analysis of Endemic Equilibrium Point. First we determine te existence of te endemic equilibrium points. Te endemic equilibrium (E 1 ) of te model is given by E 1 = (S E I R S m E m I m ). (15) To derive E 1 we solve model (1) by equating it to zero. Substituting and solving for I (as an expression of parameters only) troug some algebraic manipulation give Λ +ψ (b + τu 2)I (ψ + μ ) +[ (1 u 1)βεφ ][ N (1 u 1 )(μ m +au 1 +pu 3 )R m I (1 u 1 )λεφi +N (μ m +au 1 +pu 3 ) ] N [ w μ (1 u 1 )λεφi +Λ N w (μ m +au 1 +pu 3 ) μ N w R 2 (1 u 1) 2 ] (μ m +au 1 +pu 3 )(α +μ )(δ +μ +b+τu 2 )+α (1 u 4 )(1 u 1 )(μ m +au 1 +pu 3 )βεφr m μ +[ (1 u 4)βεφ ][ N w (1 u 4 )(μ m +au 1 +pu 3 )R m I (1 u 4 )λεφi +N w (μ m +au 1 +pu 3 ) ] N [ w μ (1 u 1 )λεφi +Λ N w (μ m +au 1 +pu 3 ) μ N w R 2 (1 u 1) 2 ] (μ m +au 1 +pu 3 )(α +μ )(δ +μ +b+τu 2 )+α (1 u 4 )(1 u 1 )(μ m +au 1 +pu 3 )βεφr m μ (16) N μ [ w μ (1 u 1 )λεφi +Λ N w (μ m +au 1 +pu 3 ) μ N w R 2 (1 u 1) 2 ] (μ m +au 1 +pu 3 )(α +μ )(δ +μ +b+τu 2 )+α (1 u 4 )(1 u 1 )(μ m +au 1 +pu 3 )βεφr m μ = or were A (I )2 +BI +C= (17) A=(ψ+μ )Λ [μ N w R 2 (1 u 1) 2 (μ m +au 1 +pu 3 )(α +μ )(δ +μ +b+τu 2 ) +α (1 u 4 )(1 u 1 )(μ m +au 1 +pu 3 )βεφr m μ ] [(1 u 1 )λεφi +N (μ m +au 1 +pu 3 )] N w N +ψ(b+τu 2 )I [μ N w R 2 (1 u 1) 2 (μ m +au 1 +pu 3 )(α +μ )(δ +μ +b+τu 2 ) λεφi +Λ N w (μ m +au 1 +pu 3 )] N w +(ψ +μ )[(1 u 4 )βεφ][(1 u 4 )(μ m +au 1 +pu 3 ) R m I ][N wμ (1 u 1 )λεφi +Λ N w (μ m +au 1 +pu 3 )] N C=(ψ+μ )μ [N w μ (1 u 1 )λεφi +Λ N w (μ m +au 1 +pu 3 )] [(1 u 1 )λεφi +N (μ m +au 1 +pu 3 )] N w N. (18) We use te quadratic formula to find te roots of (17); tat is +α (1 u 4 )(1 u 1 )(μ m +au 1 +pu 3 )βεφr m μ ] [(1 u 1 )λεφi +N (μ m +au 1 +pu 3 )] N w N B=(ψ+μ )[(1 u 1 )βεφ][(1 u 1 ) (μ m +au 1 +pu 3 )R m I ][N wμ (1 u 1 ) I = B ± B 2 4AC 2A = B + B 2 4AC 2A I = B + B 2 4AC 2A or B B 2 4AC 2A = B B 2 4AC 2A =Φ. (19)

8 8 Discrete Dynamics in Nature and Society Hence S N = w μ (1 u 1 )λεφφ+λ N w (μ m +au 1 +pu 3 ) μ N w R 2 (1 u 1) 2 (μ m +au 1 +pu 3 )(α +μ )(δ +μ +b+τu 2 )+α (1 u 4 )(1 u 1 )(μ m +au 1 +pu 3 )βεφr m μ E I R =[ ((1 u 1)βεφ+(1 u 4 )βεφ)(1 u 1 )(μ m +au 1 +pu 3 )R m Φ ] N w N (μ +α )((1 u 1 )λεφφ+n (μ m +au 1 +pu 3 )) N [ w μ (1 u 1 )λεφφ+λ N w (μ m +au 1 +pu 3 ) μ N w R 2 (1 u 1) 2 ] (μ m +au 1 +pu 3 )(α +μ )(δ +μ +b+τu 2 )+α (1 u 4 )(1 u 1 )(μ m +au 1 +pu 3 )βεφr m μ α =Φ=[ ((1 u 1 )βεφ+(1 u 4 )βεφ)(1 u 1 )(μ m +au 1 +pu 3 )R m Φ (δ +μ +b+τu 2 )N w N (μ +α ) ((1 u 1 )λεφφ+n (μ m +au 1 +pu 3 )) ] N [ w μ (1 u 1 )λεφφ+λ N w (μ m +au 1 +pu 3 ) μ N w R 2 (1 u 1) 2 ] (μ m +au 1 +pu 3 )(α +μ )(δ +μ +b+τu 2 )+α (1 u 4 )(1 u 1 )(μ m +au 1 +pu 3 )βεφr m μ (b + τu =[ 2 )α ((1 u 1 )βεφ+(1 u 4 )βεφ)(1 u 1 )(μ m +au 1 +pu 3 )R m Φ (μ +ψ)(δ +μ +b+τu 2 )N w N (μ +α ) ((1 u 1 )λεφφ+n (μ m +au 1 +pu 3 )) ] N [ w μ (1 u 1 )λεφφ+λ N w (μ m +au 1 +pu 3 ) μ N w R 2 (1 u 1) 2 ] (μ m +au 1 +pu 3 )(α +μ )(δ +μ +b+τu 2 )+α (1 u 4 )(1 u 1 )(μ m +au 1 +pu 3 )βεφr m μ (2) S m = Λ m N (1 u 1 )λεφφ+(μ m +au 1 +pu 3 )N E m =[ (1 u 1 )α λεφ ((1 u 1 )(μ m +au 1 +pu 3 )R m Φ) Λ m N N (α m +μ m +au 1 +pu 3 )((1 u 1 )λεφφ+n (μ m +au 1 +pu 3 )) ((1 u 1 )λεφφ+(μ m +au 1 +pu 3 )N ) ] I m = (1 u 1 )(μ m +au 1 +pu 3 )R m Φ (1 u 1 )λεφφ+n (μ m +au 1 +pu 3 ). From te quadratic equation (17) we analyze te possibility of multiple endemic equilibria. It is important to note tat te coefficient A is always positive wit B and C aving different signs. All te negative terms in C areinteformof(1 u) were u is te control tat is bounded by 1. It follows tat (i) tere is a unique endemic equilibrium if B<and C=or B 2 4AC= (ii) tere is a unique endemic equilibrium if C<(i.e. if R >1) (iii) tere are two endemic equilibria if C> B<and B 2 4AC> (iv) tere are no endemic equilibria oterwise. Note tat te ypotesis C>is equivalent to R <1. Tusteresultsoftissectioncanbesummarizedinte following teorem. Teorem 3. If R < 1 E is an equilibrium of system (1) and it is locally asymptotically stable. Furtermore tere exists an endemic equilibrium if conditions in (i) are satisfied or two endemic equilibria if conditions in (iii) are satisfied. If R >1tenE is unstable and tere exists a unique endemic equilibrium. Item (iii) indicates te possibility of backward bifurcation in model (1) wen R <1. In te next section we will prove teoccurrenceofmultipleequilibriaforr <1comes from te backward bifurcation and tis will give information on te local stability of te endemic equilibrium. (1) Local Stability Analysis of Endemic Equilibrium Point. We usecentremanifoleory[29]toinvestigatetestability of te endemic equilibria for model (1) were we carry out bifurcation analysis of system (1) at R = 1. We consider a transmission rate β as bifurcation parameter Ψ so tat R =1. To apply te teory we introduce dimensionless state variables into system (1). Te system of (1) can be written as ds =Λ +ψr μ S (1 u 1)βεφS I m N (1 u 4)βεφS I m N w de = (1 u 1)βεφS I m + (1 u 4)βεφS I m N N w μ E α E di =α E (δ +μ )I (b+τu 2 )I

9 Discrete Dynamics in Nature and Society 9 dr =(b+τu 2 )I μ R ψr (1 u 4)Ψ εφx 7 x 1 μ Λ = 1 ds m de m di m =Λ m (1 u 1)λεφI S m N (μ m +au 1 +pu 3 )S m = (1 u 1)λεφI S m N α m E m (μ m +au 1 +pu 3 )E m =α m E m (μ m +au 1 +pu 3 )I m. (21) We let x 1 =S x 2 =E x 3 =I x 4 =R x 5 =S m x 6 =E m and x 7 =I m. Terefore system (1) in vector form can be written as dx i =H(X i ) (22) were X i =(x 1 x 2...x 7 ) T and H=( ) T are transposed matrices and N =Λ /μ wit Ψ =β dx 1 =Λ +ψx 4 μ S (1 u 1)Ψ εφx 7 x 1 μ Λ dx 2 = (1 u 1)Ψ εφx 7 x 1 μ Λ + (1 u 4)Ψ εφx 7 x 1 μ (μ Λ +α )x 2 = 2 dx 3 =α x 2 (δ +μ +b+τu 2 )x 3 = 3 dx 4 =(b+τu 2 )x 3 (μ +ψ)x 4 = 4 dx 5 =Λ m (1 u 1)λεφx 3 x 5 μ Λ (μ m +au 1 +pu 3 )x 5 = 5 dx 6 dx 7 = (1 u 1)λεφx 3 x 5 μ Λ (α m +μ m +au 1 +pu 3 )x 6 = 6 =α m x 6 (μ m +au 1 +pu 3 )x 7 = 7. (23) Let Ψ be te bifurcation parameter; te system is linearized at disease-free equilibrium point wen β=ψ wit R =1. Tat is α R = α m Λ m μ (1 u 1 ) 2 φ 2 εβλ + α α m Λ m μ (1 u 1 )(1 u 4 )φ 2 ε 2 β Λ (pu 3 +au 1 +μ m ) 2 (μ +α 1 )(pu 3 +μ m +au 1 +α m )(μ +δ +b+τu 2 ) 1 2 = α α m Λ m μ (1 u 1 ) 2 φ 2 εβλ + α α m Λ m μ (1 u 1 )(1 u 4 )φ 2 ε 2 β Λ (pu 3 +au 1 +μ m ) 2 (μ +α 1 )(pu 3 +μ m +au 1 +α m )(μ +δ +b+τu 2 ) (24) Ψ = Λ (pu 3 +au 1 +μ m ) 2 (μ +α 1 )(pu 3 +μ m +au 1 +α m )(μ +δ +b+τu 2 ) α α m Λ m μ (1 u 1 ) 2 φ 2 ελ + α α m Λ m μ (1 u 1 )(1 u 4 )φ 2 ε 2. Te Jacobian matrix of (1) calculated at Ψ is given by μ ψ Ψ εφ α μ Ψ εφ α δ μ b τu 2 b+τu 2 μ ψ (1 u 1 )λεφλ m μ (μ Λ (μ m +au 1 +pu 3 ) m +au 1 +pu 3 ). (25) ( (1 u 1 )λεφλ m μ ) α Λ (μ m +au 1 +pu 3 ) m μ m au 1 pu 3 ( α m μ m au 1 pu 3 ) Centre manifold approac [29] is ten applied.

10 1 Discrete Dynamics in Nature and Society A rigt eigenvector associated wit te eigenvalue zero is w=(w 1 w 2...w 7 ).Solvingtesystemweavetefollowing rigt eigenvector: w 1 = ψw 4 Ψ εφw 7 μ w 2 = Ψ εφw 7 α +μ w 3 = α w 2 b+τu 2 +μ +δ w 4 = (b + τu 2)w 3 μ +ψ w 5 = (1 u 1)λεφΛ m μ w 3 Λ (μ m +au 1 +pu 3 ) 2 w 6 = w 7 = (1 u 1 )λεφλ m μ w 3 Λ (μ m +au 1 +pu 3 )(α m +μ m +au 1 +pu 3 ) α m w 6 μ m +au 1 +pu 3 >. (26) TelefteigenvectorssatisfyingV w = 1 are V =(V 1 V 2...V 7 ). Solving te system te left eigenvector will be given by V 1 = V 2 = α V 3 α +μ V V 3 = 6 (1 u 1 )λεφλ m μ Λ (μ m +au 1 +pu 3 )( δ u b τu 2 ) V 4 = V 5 = V 6 = V 7 = α m V 7 μ m α m au 1 pu 3 Ψ εφv 2 μ m +au 1 +pu 3. (27) Computing for te sign of a and b as indicated in te teorem gives a=v 2 w 7 [ (1 u 1)Ψ εφμ (1 u 4 )Ψ εφμ Λ ] ([ Ψ εw 7 (α +μ ) ]+[ α w 2 (b + τu 2 +μ +δ ) ]) + V 6 w 3 [ (1 u 1)λεφμ Λ ] (1 u ([ 1 )λεφμ Λ m w 3 Λ (μ m +au 1 +pu 3 )(α m +μ m +au 1 +pu 3 ) ] α +[ m w 3 (μ m +au 1 +pu 3 ) ]) b=v 2 w 7 εφ > so tat b is always positive. Terefore te following result is establised. (28) Teorem 4. Model (1) exibits backward bifurcation at R =1 wenever a>and b>and R <1.Wenevera<and b> ten model (1) exibits a forward bifurcation at R =1. Finally we will investigate te global stability of te endemic equilibrium in te feasible region. (2) Global Stability Analysis of Endemic Equilibrium Point. Global stability results for te endemic equilibrium can be obtained wen it is unique and wenever it exists. We ave establised in Teorem 4 tat if R > 1 tis implies te existence and uniqueness of te endemic equilibrium. Te global beavior of te endemic equilibrium of model (1) wen it exists is explored by proving tat suc an equilibrium is globally asymptotic stable in te interior of te feasible region D. We will use te geometric approac to global stability as described by Li and Muldowney [3]. Te following conditions are required for te global stability of te endemic equilibrium E 1 : (i) te uniqueness of E 1 in te interior of D (condition H 1 ); (ii) te existence of an absorbing compact set in te interior of D (condition H 2 ); and (iii) te fulfillment of a Bendixson criterion (i.e. inequality (A.6)). Teorem 5. If a<and b>and R >1 ten te endemic equilibrium of te malaria model (1) is globally asymptotically stable in te interior of D. Proof. Following Li and Muldowney [3] for system (1) under te assumption of R > 1 satisfies conditions (H 1 )-(H 2 ) te existence of te endemic equilibrium as also been sown and te instability of DFE implies te uniform persistence [31]; tat is tere exists a constant c>suc tat any solutions (S (t) I (t) I m (t)) wit (S () I () I m ()) in te interior of D satisfy min { lim inf S t (t) lim inf I t (t) lim inf I t m (t)}. (29) Te uniform persistence togeter wit boundedness of D is equivalent to te existence of a compact set in te interior of D wic is absorbing for (4) as described by Hutson and Scmitt [32]. Tus (H 1 ) is verified. Moreover E 1 is te only equilibrium in te interior of Dsotat(H 2 ) is also verified. Wat remains is to find conditions for wic te Bendixson criterion given by (A.6) is verified. To tis aim let us begin by observing tat from te Jacobian matrix

11 Discrete Dynamics in Nature and Society 11 associated wit a general solution (S I I m ) of reduced system (1) we get te second additive compound matrix J 2 : J 2 (S I I m ) a 11 a 12 (1 u 1 )εφβs +(1 u 4 )εφβs =( (1 u 1 )εφλ(n m E m I m ) a 22 δ ) (1 u 1 )εφβi m +(1 u 4 )εφβi m a 33 (3) were +(1 u 1 )εφβi m +(1 u 4 )εφβi m (31) a 11 =μ +α +(1 u 1 )εφβi m +(1 u 4 )εφβi m +μ +δ +b+τu 2 a 12 =(1 u 1 )εφβs +(1 u 4 )εφβs a 22 =μ +δ +(1 u 1 )εφβi m +(1 u 4 )εφβi m +(1 u 1 )εφλi +μ m +au 1 +pu 3 a 33 =m+μ m +au 1 +pu 3 +(1 u 1 )εφλi were m=(λ m /μ m +au 1 +pu 3 )/(Λ /μ ). Coose now matrix P(x) = P(S I I m )=diag(1 I /I m I /I m ).TenP f P 1 = diag( I /I I m /I m I /I I m /I m ) and te matrix B=P f P 1 +PJ 2 P 1 can be written in block form as were B=[ B 11 B 12 B 21 B 22 ] (32) B 11 = (μ +α +(1 u 1 )εφβi m +(1 u 4 )εφβi m +μ +b+τu 2 +δ ) B 12 = [((1 u 1 )εφβs +(1 u 4 )εφβs ) I m I (1 u 1 )εφβs I m I +(1 u 4 )εφβs I m I ] B 21 =[(1 u 1 ) ελφi m (N m E m +I m )( I T )] I m I I B 22 = [ m a I I 22 m [ I (1 u 1 )εφβi m (1 u 4 )εφβi m (1 u 1 )εφλi I ]. m a [ I I 33 m ] (33) Te vector norm in R 3 + is ere cosen to be (x y z) = max { x y z }. (34) Let σ( ) denote te Lozinskii measure wit respect to tis norm. Using te metod of estimating σ( ) in [29] we ave σ (B) sup {g 1 g 2 } (35) = sup {σ 1 (B 11 )+ B 12 σ 1 (B 22 )+ B 21 } were B 12 and B 21 arematrixnormswitrespecttote L vector norm and σ 1 denotes te Lozinskii measure wit respect to L norm. Since B 11 is a scalar its Lozinskii measure wit respect to any norm in R 1 is equal to B 11. Terefore σ 1 (B 11 )= (μ +α +(1 u 1 )εφβi m +(1 u 4 ) εφβi m +μ +b+τu 2 +δ ) I σ 1 (B 22 )=max { I m (μ I I +α m +(1 u 1 )εφβi m +(1 u 4 )εφβi m +(1 u 1 )εφλi +μ m +au 1 +pu 3 ) I I m (1 I I m u 1 )εφβi m (1 u 4 )εφβi m (1 u 1 )εφλi μ δ b τu 2 μ m au 1 pu 3 }

12 12 Discrete Dynamics in Nature and Society I σ 1 (B 22 )= I m (μ I I +α +(1 u 1 )εφβi m +(1 m u 4 )εφβi m +(1 u 1 )εφλi +μ m +au 1 +pu 3 ) B 12 = ((1 u 1)εφβS +(1 u 4 )εφβs ) I m I B 21 =(1 u 1)εφλ(N m E m I m ) I I m. Terefore g 1 = (μ +α +(1 u 1 )εφβi m +(1 u 4 )εφβi m +μ +δ +b+τu 2 )+((1 u 1 )εφβs +(1 u 4 )εφβs ) I m I g 2 =(1 u 1 )εφλ(n m E m I m ) I I m + I I I m I m (μ +α +μ m +au 1 +pu 3 +(1 u 1 )εφλi ). We rewrite te last two equations of system (1) for I and as I I = ((1 u 1 )εφβs +(1 u 4 )εφβs ) I m I (μ +δ +b+τu 2 ) I m I m =(1 u 1 )εφλ(n m E m I m ) I I m μ m au 1 pu 3. Substituting (39) into (37) and (4) into (38) we ave g 1 (t) I = I (μ +α +(1 u1)εφβim +(1 u4)εφβim) g 2 (t) = I I (μ +α +(1 u 1 )εφλi ). (36) (37) (38) I m (39) (4) (41) For te uniform persistence constant ε> tere exists a time T >independent of x() Ktecompactabsorbingset suc tat I (t) >ε I m (t) >ε; (42) Table 4: Sensitivity indices (SI) of R to parameters for te malaria model. Parameter Sensitivity indices Endemic Seasonal Epidemic Low risk μ μ m α α m λ β ε Λ Λ m b τ δ φ for t>t we ave g 1 (t) I I (43) (μ +α +(1 u 1 )εφβε+(1 u 4 )εφβε) I g 2 (t) (μ I +α +(1 u 1 )εφλε). Relations (41) imply I σ (B) μ for t>t I (44) were μ=min {μ +α +(1 u 1 )εφβε+(1 u 4 )εφβεμ (45) +α +(1 u 1 )εφλε}. Along eac solution (S (t) I (t) I m (t)) to (1) wit (S () I () I m ()) K were K is te compact absorbing set we ave for t>t t 1 t σ (B) ds 1 T t μ t T. t σ (B) ds + 1 t ln I (t) I (T ) (46) Wic implies q 2 < μ/2 <. Tis proves tat te unique endemic equilibrium is globally asymptotically stable wenever it exist tus completing te proof Sensitivity Analysis. Sensitivity analysis is to assess terelativeimpactofeacofteparametersoftebasic reproductive number. Te normalized forward sensitivity index of te reproduction number wit respect to tese parameters given in Table 4 is computed. Te index measures

13 Discrete Dynamics in Nature and Society 13 Parameter Table 5: Parameter values for te full malaria model. Estimated value Endemic Epidemic Seasonal Low risk μ KNBS (29 Census estimates) [2] μ m.4 Estimated α.7143 Estimated α m.99 Citnis et al. [21] λ.42 Estimated β.655 Estimated ε.2 Blayne et al. [22] ψ.195 Estimated Λ KNBS (estimates based on 29 Census) [2] Λ m.71 Niger and Gumel [23] b.5 Ciyaka et al. [24] τ.5 Assumed δ.5 KNBS and ICF Macro [25] p.25 Assumed a.25 Assumed φ.52 Blayne et al. [22] λ Estimated λ w Estimated λ m Estimated N KNBS (29 Census estimate) [2] N w KNBS (29 Census estimate) [2] N m Estimated Source terelativecangeinavariablewitrespecttorelative canges in parameters. Definition 6. Following Citnis et al. [33] te normalized forward sensitivity index of a variable tat depends on a parameter l is defined as ξ l = (l/) ( / l). Te sensitivity index of te model parameters is given by ξ R μ α = 2(α +μ ) ξ R pu α m = 3 +μ m +au 1 2(α m +pu 3 +au 1 +μ m ) ξ R μ μ m = m (2α m +3(μ m +au 1 +pu 3 )) 2(α m +μ m +au 1 +pu 3 )(μ m +au 1 +pu 3 ) ξ R δ δ = 2(b+τu 2 +δ +μ ) ξ R b b = 2(b+τu 2 +δ +μ ) ξ R μ 2 μ = +α δ +α b+α τu 2 2(μ +δ +b+τu 2 )(μ +α ). (47) Sensitivity indices for te control parameters are given by ξ R u 1 = μ 1 1 μ 1 ξ R τμ u 2 = 2 2(μ +ε+b+τμ 2 ) ξ R u 3 = p (3pu 3 +3μ m +au 1 +2α m )u 3 2(pu 3 +au 1 +μ m )(pu 3 +μ m +α m ) ξ R u u 4 = 4 (1 u 4 ). (48) Troug sensitivity analysis it is observed tat te most sensitive parameters to R across all te settings were te mosquito s natural deat rate μ m and mosquito biting rate ε; tis was followed by te by te mosquito contact rate wit umans φ probability of mosquito getting infected λ teprobabilityofumansgettinginfectedβ ane recruitment rate of mosquitoes and umans (see Table 4). Sensitivity Indices of R. Sensitivity indices are calculated using parameters in Table 5.

14 14 Discrete Dynamics in Nature and Society 3. Analysis of Optimal Control 3.1. Analysis of Optimal Control of te Malaria Model wit Intervention Strategies. We consider te case of timedependent control variables. Te malaria dynamics model is extended and an optimal control problem is formulated. We formulate an optimal control model for malaria disease in order to determine optimal malaria control strategies (ITNs IRS IPTp and treatment) wit minimal implementation cost. For te optimal control problem of te given system we consider te control variables u(t) = (u 1 u 2 u 3 u 4 ) U relative to te state variables S E I R S m E m I m were control variables are bounded and measured wit U = {(u 1 u 2 u 3 u 4 ) Uis Lebesgue measurable on [ 1] u i (t) 1 t [ T] i=1234}. We define te objective function as subject to (49) T J(u 1 u 2 u 3 u 4 )= (A 1 N m +A 2 I +A 3 E + 1 (5) 2 (B 1u 2 1 +B 2u 2 2 +B 3u 2 3 +B 4u 2 4 )) ds =Λ +ψr (1 u 1 )λ S (1 u 4 )λ w S de μ S =(1 u 1 )λ S +(1 u 4 )λ w S (α +μ )E di =α E (δ +μ )I (b+τu 2 )I dr =(b+τu 2 )I (ψ+μ )R ds m =Λ m (1 u 1 )λ m S m de m (μ m +au 1 +pu 3 )S m =(1 u 1 )λ m S m α m E m (μ m +au 1 +pu 3 )E m di m =α m E m (μ m +au 1 +pu 3 )I m S () E () I () R () S m () E m () I m (). (51) In te objective function T is te final time and quantities A 1 A 2 anda 3 are weigts constants of te total mosquito population infected individuals and exposed individuals respectively wile B 1 B 2 B 3 andb 4 are weigt constants for use wit ITNs treatment effort IRS and IPTp efforts respectively. Te total mosquito population (N m =S m +E m + I m ) is part of te objective function because it is affected by te use of IRS and ITNs. In addition E and I are included in te objective function because individuals in tese classes are affectedbyteuseofitnsiptpsanreatmentrespectively. A quadratic cost on te controls was cosen in line wit wat is known in te literature on epidemic optimal controls for malaria[1112].tecostofimplementingpersonalprotection using ITNs is B 1 u 2 1 treatment of infected individuals is B 2u 2 2 spraying of ouses wit IRS is B 3 u 2 3 and preventive metod of IPTp is B 4 u 2 4. A linear function as been cosen for te cost incurred by exposed individuals A 3 E infectedindividuals A 2 I anemosquitopopulationa 1 N m. A quadratic form is used for te cost on te controls B 1 u 2 1 B 2u 2 2 B 3u 2 3 and B 4 u 2 4 suctatteterms(1/2)b 1u 2 1 (1/2)B 2u 2 2 (1/2)B 3u 2 3 and (1/2)B 4 u 2 4 describe te cost associated wit te ITNs treatment mosquito control (IRS) and cemoprevention (IPTp) respectively. Our aim wit te given objective function is to minimize total number of mosquito population N m tenumberof exposed umans E (t) andinfectedumansi (t) wile minimizing te cost of control u 1 (t) u 2 (t) u 3 (t) andu 4 (t). Weselecttomodeltecontroleffortsviaalinearcombination of quadratic terms and te constants wic represents a measure of te relative cost of te interventions over [ 1]. We seek an optimal control u 1 u 2 u 3 andu 4 suc tat J(u 1 u 2 u 3 u 4 )= min u 1 u 2 u 3 u 4 U J(u 1u 2 u 3 u 4 ). (52) Pontryagin s Maximum Principle is used to solve tis optimal control problem and te derivation of necessary conditions tat an optimal control must satisfy [6]. Pontryagin s Maximum Principle converts te state system (1) and objective function (51) into a problem of minimizing pointwise te Lagrangian L and Hamiltonian H witrespecttou 1 u 2 u 3 andu Existence of Optimal Control Problem. Te existence of an optimal control can be proved by using te teorem given in Fleming and Risel [34]. It can be clearly seen tat te system of equations given by (1) is bounded from above by a linear system. According to te result in Fleming and Risel [34] te solution exists if te following ypoteses are met: (H 1 ): te set of controls and corresponding state variables is nonempty. (H 2 ): te control set U is convex and closed.

15 Discrete Dynamics in Nature and Society 15 (H 3 ): rigt and side of eac equation of te state system in (1) is continuous bounded by a linear functional in te state and control and can be written as a linear function of u wit coefficients depending on time and state. (H 4 ): tere exist constants c 1 c 2 >and β>1suc tat te integrand L(y u t) of te objective functional J is convex and satisfies L (y u t) c 1 ( u u u u 4 2 ) β/2 c 2. (53) In order to confirm te above ypoteses a result given by Lukes [35] is used to establis te existence of solutions of state system (1). Since te coefficients are bounded (H 1 ) is satisfied. Te solutions are bounded and ence te control set satisfies (H 2 ) as well. Te system is bilinear in u 1 u 2 u 3 u 4 and ence te rigt and side of state system (1) satisfies condition (H 3 ) (since te solutions are bounded). Note tat te integrand of te objective functional is convex. Te last condition is also satisfied. Te state and te control variables of system (1) are nonnegative values and nonempty. Te control set U is closed and convex. Te integrand of te objective cost function J expressed by (51) is a convex function of (u 1 u 2 u 3 u 4 ) on te control set U. Te Lipscitz property of te state system wit respect to te state variables is satisfied since te state solutions are bounded. It can easily be sown tat tere exist positive numbers ξ 1 ξ 2 and a constant ε>1suc tat A 1 N m +A 2 I +A 3 E We seek to find te minimal value of te Lagrangian. To do tis we define te Hamiltonian H for te control problem wic consists of te integrand of te objective functional (Lagrangian L) and te inner product of te rigt and sides of te state equations and te costate variables or adjoint variables (λ 1 λ 2 λ 3 λ 4 λ 5 λ 6 λ 7 ) as ds H=L+λ 1 +λ de di 2 +λ 3 +λ dr 4 +λ 5 ds m +λ 6 de m +λ 7 di m. (56) Taking X=(S E I R S m E m I m ) U=(u 1 u 2 u 3 u 4 ) and λ=(λ 1 λ 2 λ 3 λ 4 λ 5 λ 6 λ 7 ) we obtain te Hamiltonian given by H (X U λ) =L(I E N m u 1 u 2 u 3 u 4 )+λ 1 [Λ +ψr (1 u 1 )λ S (1 u 4 )λ w S μ S ] +λ 2 [(1 u 1 )λ S +(1 u 4 )λ w S (α +μ )E ]+λ 3 [α E (δ +μ )I (b+τu 2 )I ]+λ 4 [(b + τu 2 )I (ψ+μ )R ] +λ 5 [Λ (1 u 1 )λ m S m (μ m +au 1 +pu 3 )S m ]+λ 6 [(1 u 1 )λ m S m α m E m (μ m +au 1 +pu 3 )E m ]+λ 7 [α m E m (57) (B 1u 2 1 +B 2u 2 2 +B 3u 2 3 +B 4u 2 4 ) (54) (μ m +au 1 +pu 3 )I m ]. ξ 1 ( u u u u 4 2 ) ε/2 ξ 2 were ξ 1 ξ 2 > A 1 A 2 B 1 B 2 B 3 B 4 >andε>1. Tis concludes existence of an optimal control since te statevariablesarebounded.henceweavetefollowing teorem. Teorem 7. Given te objective functional J(u 1 u 2 u 3 u 4 )= T (A 1N m +A 2 I +A 3 E +(1/2)(B 1 u 2 1 +B 2u 2 2 +B 3u 2 3 +B 4u 2 4 )) were U={(u 1 u 2 u 3 u 4 ) u 1 u 2 u 3 u 4 1 u i (t) 1 t [ T] i = } subject to (1) wit initial conditions ten tere exists an optimal control u =(u 1 u 2 u 3 u 4 ) suc tat J(u 1 u 2 u 3 u 4 )=min UJ(u 1 u 2 u 3 u 4 ). Lagrangian for a problem discusses ow te tecniques come and Hamiltonian elps in solving for te adjoint variable. In order to find an optimal solution first we find te Lagrangian and Hamiltonian for te optimal control problem (51). Te Lagrangian of te optimal problem is given by L(I E N m u 1 u 2 u 3 u 4 ) =A 1 N m +A 2 I +A 3 E (B 1u 2 1 +B 2u 2 2 +B 3u 2 3 +B 4u 2 4 ). (55) 3.3. Te Optimality System. In order to find te necessary conditions for tis optimal control we apply Pontryagin s Maximum Principle [6] as described by Lenart and Workman [36] as follows. If u 1 u 2 u 3 u 4 are optimal solution of an optimal control problem ten tere exists a nontrivial vector function λ(t) = (λ 1 (t) λ 2 (t)...λ n (t)) satisfying te following conditions. Te state equation is dx = H (t u 1 u 2 u 3 u 4 λ(t)). (58) λ Te optimality condition is = H (t u 1 u 2 u 3 u 4 λ(t)) u and te adjoint equation is (59) dλ = H (t u 1 u 2 u 3 u 4 λ(t)). (6) x Now we apply te necessary conditions to te Hamiltonian. Teorem 8. Given te optimal controls u 1 u 2 u 3 u 4 and solutions S E I R S m E m I m of te corresponding state

16 16 Discrete Dynamics in Nature and Society system (1) tere exist adjoint variables λ 1 λ 2 λ 3 λ 4 λ 5 λ 6 λ 7 satisfying dλ 1 dλ 2 dλ 3 = H S =(1 u 1 )λ (λ 2 λ 1 ) +(1 u 4 )λ w (λ 2 λ 1 ) μ λ 1 = H E =α (λ 3 λ 2 ) μ λ 2 +A 3 = H I =(b+τu 2 )λ 4 (b+τu 2 +μ +δ )λ 3 +A 2 ( (1 u 1)λεφS m N )λ 5 dλ 6 dλ 7 =(1 u 1 )(λ 6 λ 5 )λ m (μ m +au 1 +pu 3 )λ 5 +A 1 = H E m =α m (λ 7 λ 6 ) (μ m +au 1 +pu 3 )λ 6 +A 1 = H I m = (μ m +au 1 +pu 3 )λ 7 +A 1 +( (1 u 1)βεφS N +( (1 u 1)βεφS N (1 u 4)βεφS N w )λ 1 + (1 u 4)βεφS N w )λ 2 (61) +( (1 u 1)λεφS m N )λ 6 wit transversality conditions dλ 4 = H R =ψλ 1 (μ +ψ)λ 4 λ 1 (T) =λ 2 (T) =λ 3 (T) =λ 4 (T) =λ 5 (T) =λ 6 (T) =λ 7 (T) =. (62) dλ 5 = H S m Furtermore u 1 u 2 u 3 u 4 are represented by u 1 = max { min (1 (λ 2 λ 1 )λ S +(λ 6 λ 5 )λ m S m +as m λ 5 +ae m λ 6 +ai m λ 7 B 1 )} u 2 = max { min (1 τ(λ 3 λ 4 )I B 2 )} u 3 = max { min (1 p(λ 5S m +λ 6E m +λ 7I m ) B 3 )} (63) u 4 = max { min (1 (λ 2 λ 1 )λ w S B 4 )}. Proof. To determine te adjoint equations and te transversality conditions we use te Hamiltonian H. Te Hamiltonian function H is differentiated wit respect to S E I R S m E m andi m. Te adjoint/costate equation is given by dλ 1 dλ 2 dλ 3 = H S =(1 u 1 )λ (λ 2 λ 1 ) +(1 u 4 )λ w (λ 2 λ 1 ) μ λ 1 = H E =α (λ 3 λ 2 ) μ λ 2 +A 3 = H I dλ 4 dλ 5 =(b+τu 2 )λ 4 (b+τu 2 +μ +δ )λ 3 +A 2 ( (1 u 1)λεφS m N )λ 5 +( (1 u 1)λεφS m N )λ 6 = H R =ψλ 1 (μ +ψ)λ 4 = H S m =(1 u 1 )(λ 6 λ 5 )λ m (μ m +au 1 +pu 3 )λ 5 +A 1

17 Discrete Dynamics in Nature and Society 17 dλ 6 dλ 7 = H E m =α m (λ 7 λ 6 ) (μ m +au 1 +pu 3 )λ 6 +A 1 = H I m letting S =S E =E I =I R =R S m =S m E m =E m andi m =I m yield H u 1 =B 1 u 1 +λ 1λ S λ 2λ S +λ 5λ m S m as m λ 5 ae m λ 6 ai m λ 7 = = (μ m +au 1 +pu 3 )λ 7 +A 1 +( (1 u 1)βεφS N +( (1 u 1)βεφS N wit transversality conditions (1 u 4)βεφS N w )λ 1 + (1 u 4)βεφS N w )λ 2 (64) H =B u 2 u 2 τλ 3I +τλ 4I = 2 H =B u 3 u 3 pλ 5S m pλ 6E m pλ 7I m = 3 H =B u 4 u 4 +λ wλ 2 S λ 1λ w S = 4 for wic (66) λ 1 (T) =λ 2 (T) =λ 3 (T) =λ 4 (T) =λ 5 (T) =λ 6 (T) =λ 7 (T) =. (65) u 1 = (λ 2 λ 1 )λ S +(λ 6 λ 5 )λ m S m +as m λ 5 +ae m λ 6 +ai m λ 7 B 1 In order to minimize Hamiltonian H witrespectto te controls at te optimal controls H is differentiated wit respect to u 1 u 2 u 3 andu 4 on te set Uanesolutionfor te optimal control point is obtained after equating to zero. Tis is te optimality condition. Solving H/ u 1 = H/ u 2 = H/ u 3 = and H/ u 4 = evaluating at te optimal control on te interior of te control set were <u i <1fori = and u 2 = τ(λ 3 λ 4 )I B 2 u 3 = p(λ 5S m +λ 6E m +λ 7I m ) B 3 (67) u 4 = (λ 2 λ 1 )λ w S. B 4 By applying te boundary condition of eac control te solution of (67) becomes u 1 = max { min (1 (λ 2 λ 1 )λ S +(λ 6 λ 5 )λ m S m +as m λ 5 +ae m λ 6 +ai m λ 7 B 1 )} u 2 = max { min (1 τ(λ 3 λ 4 )I B 2 )} u 3 = max { min (1 p(λ 5S m +λ 6E m +λ 7I m ) B 3 )} (68) u 4 = max { min (1 (λ 2 λ 1 )λ w S B 4 )}. Te optimality system is comprised of state system (1) adjoint system (61) initial conditions at t= boundary conditions (62) and te caracterization of te optimal control (63). Hencetestateandoptimalcontrolcanbecalculatedusing te optimality system. Hence using te fact tat te second derivatives of te Lagrangian wit respect to u 1 u 2 u 3 andu 4 respectively are positive indicates tat te optimal problem is a minimum at controls u 1 u 2 u 3 andu 4. Te optimality system is solved using te forwardbackward fourt-order Runge-Kutta sceme in R statistical computing platform [37]. Te optimal strategy is obtained by solving te state and adjoint systems and te transversality conditions. First we start to solve state (1) using te Runge- Kutta fourt-order forward in time wit a guess for te controls u 1 u 2 u 3 andu 4 over te simulated time. Ten using te current iteration of te state equations wit te initial guess for te controls te adjoint equations in system (57) are solved by a backward metod wit te transversality conditions (62). Ten te controls are updated by using a convex combination of te previous controls and te value from caracterizations (63). Tis process is repeated and iterations stopped if te values of te unknowns at te

18 18 Discrete Dynamics in Nature and Society previous iterations are very close to te ones at te present iterations [36]. 4. Numerical Results on Optimal Control Analysis and Discussion Te parameters in model (1) were estimated using clinical malaria data and demograpics statistics of Kenya. Tose tat were not available were obtained from literature publised by researcers in malaria endemic countries wic ave similar environmental conditions compared to Kenya. Table 5 provides a summary of te estimated values of all parameters of te malaria model. Data was collected from te literature Division of Malaria Control (DOMC) Kenya National Bureau of Statistics Malaria Indicator Survey for Kenya Demograpic Healt Survey (DHS) for Kenya WHO websites and ospital records (from Kisumu Kisii Cuka (Tarake-Niti) and Nyeri counties representing te four different transmission settings/epidemiological zones in Kenya). Te rates are per day. In addition te effect of te different intervention strategies is estimated as u 1 =.94 u 2 =.165 u 3 =.76and u 4 =.35 and te cost of intervention is for u 1 = $2.5 5 u 2 = $2.5 u 3 = $1.5 andu 4 = $2.5 (Wite et al. [38] and Hansen et al. [39]). Te initial state variables are constant across all te epidemiological zones and are cosen as S () = 7 E () = 25 I () = 3 R () = 3 S m () = 5 E m () = 5 andi m () = 1. Te following weigt factors were also fixed for te different epidemiological scenarios as B 1 =2 B 2 =65 B 3 =1 B 4 =1 A 3 = 1 A 2 =92and A 1 = Dynamics of Human Population wit Intervention Strategies. We simulated malaria model wit intervention strategies to find te dynamics of uman and mosquito populations. It is observed tat te control strategy leads to decrease in te number of infected uman (I )assowninfigure2. Te figure sows steady decrease in susceptible uman population at te initial period as te exposure of umans to disease increases. Tereafter grap of susceptible umans increases as te exposed and infected uman population decreases due to positive effect of te intervention strategies being implemented Numerical Results on Optimal Control Analysis. In tis section we investigate numerically te effect of te several optimal control strategies on te spread of malaria. We compare te numerical results from te simulation using one control and various combinations of two tree and four control strategies. Tis was done by comparing wen tere were not any intervention strategies and wen tere were intervention strategies. Tere are 15 different control strategies for eac of te four different epidemiological zones in Kenya tat are explored. We use te case of endemic zone wit te case of one control variable two control variables treecontrolvariablesandalltefourcontrolvariablesbeing in use for te illustration purpose. TeresultsinFigures3 6sowasignificantdifference in I and I m wit te control strategy compared to I and I m witout te control strategy. It is observed tat te control strategy leads to decrease in te number of infected umans (I ). Te uncontrollable case leads to a decrease in tenumberofinfectedmosquitoes(i m ) wile te control strategy leads to decrease in te infected number. Te control profilessowteupperbounimeforeacstrategyforeac setting before dropping to te lower bound. Tere are several combinations for te different settings as described below. Results of only one intervention strategy for te endemic epidemiological zones is as follows. (a) Optimal Protection Using ITN. Onlytecontrol(u 1 )on ITNs is used to optimize te objective function J wile te control on treatment (u 2 ) te control on IRS (u 3 ) and control on IPTp (u 4 )aresettozero. (b) Optimal Treatment.Onlytecontrol(u 2 )ontreatmentis used to optimize te objective function J wile te control on ITNs (u 1 ) te control on IRS (u 3 ) and control on IPTp (u 4 )aresettozero. (c) Optimal IRS. Onlytecontrol(u 3 ) on IRS is used to optimize te objective function J wile te control on treatment (u 2 ) te control on ITNs (u 1 ) and control on IPTp (u 4 )aresettozero. (d) Optimal IPTp. Onlytecontrol(u 4 )oniptpisused to optimize te objective function J wile te control on treatment (u 2 ) te control on IRS (u 3 ) and control on ITNs (u 1 )aresettozero. Resultsofcombining2interventionstrategiesforte endemic epidemiological zones are as follows. (a) Optimal ITNs and Treatment. Wittisstrategyte control on ITNs (u 1 ) and te treatment (u 2 )areuseo optimize te objective function J wilesettingtecontrolon IRS (u 3 ) and control on IPTp (u 4 )tozero. (b) Optimal ITN and IRS. Wittisstrategytecontrolon ITNs (u 1 ) and te IRS (u 3 ) are used to optimize te objective function J wile setting te control on treatment (u 2 )and control on IPTp (u 4 )tozero. (c) Optimal ITN and IPTp. Wit tis strategy te control on ITNs (u 1 )andiptp(u 4 ) are used to optimize te objective function J wile setting te control on treatment (u 2 )and control on IRS (u 3 )tozero. (d) Optimal Treatment and IRS. Wit tis strategy te control on treatment (u 2 ) and te IRS (u 3 )areuseooptimizete objective function J wile setting te control on ITNs (u 1 ) and control on IPTp (u 4 )tozero. (e) Optimal Treatment and IPTp. Wittisstrategyte control on treatment (u 2 )aneiptp(u 4 )areuseo optimize te objective function J wilesettingtecontrolon IRS (u 3 )andcontrolonitns(u 1 )tozero. (f) Optimal IRS and IPTp. Wit tis strategy te control on IRS (u 3 )aneiptp(u 4 ) are used to optimize te objective

19 Discrete Dynamics in Nature and Society Susceptible umans 72 Exposed umans Time 5 Time 1 (a) (b) Infectious umans 1 5 Recovered umans Time 1 5 Time 1 (c) (d) Figure 2: Simulations sowing te dynamics of uman population wit intervention strategies for te endemic setting. function J wile setting te control on treatment (u 2 )and control on ITNs (u 1 )tozero. Results of combining tree intervention strategies for te endemic epidemiological zones are as follows. (a) Optimal ITN Treatment and IRS. In tis case ITN control (u 1 ) treatment control (u 2 ) and IRS control (u 3 )areuseo optimize te objective function J wile IPTp control (u 4 )is set to zero. (b) Optimal ITN Treatment and IPTp. In tis case ITNs control (u 1 ) treatment control (u 2 ) and IPTp control (u 4 )are used to optimize te objective function J wile IRS control (u 3 )issettozero. (c) Optimal ITN IRS and IPTp.IntiscaseITNscontrol(u 1 ) IRS control (u 3 ) and IPTp control (u 4 )areuseooptimize te objective function J wile treatment control (u 2 )issetto zero. (d) Optimal Treatment IRS and IPTp.Intiscasetreatment control (u 2 ) IRS control (u 3 ) and IPTp control (u 4 )areused to optimize te objective function JwileITNscontrol(u 1 ) is set to zero.

20 2 Discrete Dynamics in Nature and Society Infected umans 1 Infected mosquitoes Time No intervention u 1 =u 2 =1u 3 =u 4 = (a) 5 1 Time No intervention u 1 =u 2 =1u 3 =u 4 = (b) Control profile Time (days) Variable u 2 =1 (c) Figure 3: Simulations of te model sowing te effect of treatment on te spread of malaria for te endemic setting. Resultsofcombiningtefourinterventionstrategiesfor te endemic epidemiological zones are as follows. Optimal ITN Treatment IRS and IPTp. Intiscaseallte control function ITNs control (u 1 ) treatment control (u 2 ) IRS control (u 3 ) and IPTp control (u 4 )areuseooptimize te objective function J. Te same procedure is repeated for oter combination for strategies and for oter epidemiological zones (epidemic seasonal and low). Based on te simulation findings for te igest number of infections being inverted at a lower cost te combined use of treatment and IRS reduces te infected uman and mosquito population faster at a lower cost for te endemic settings (15 infections at $ ). For te

21 Discrete Dynamics in Nature and Society Infected umans 1 Infected mosquitoes Time No intervention u 1 =1u 2 =1u 3 =u 4 = (a) 5 1 Time No intervention u 1 =1u 2 =1u 3 =u 4 = (b) Control profile Time (days) Variable u 1 =1 u 2 =1 (c) Figure 4: Simulations of te model sowing te effect of ITNs and treatment on te spread of malaria for te endemic setting. epidemic prone settings te use of treatment and IRS (111.3 infections at $ ) as more impact in reducing te infected uman and mosquito population. For seasonal areas muc impact will be felt wen treatment is used ( infections at $ ). For te low risk areas just te use of ITNs and treatment ( infections at ) will be sufficient to reduce infected uman and mosquito population. Tis is deduced from te intervention wic takes sorter time to start reducing te number of infected mosquitoes and umans. 5. Discussion In tis paper we formulated a matematical model for te transmission dynamics of malaria wit four time-dependent control measures in Kenya. We first consider control

22 22 Discrete Dynamics in Nature and Society Infected umans 1 Infected mosquitoes Time No intervention u 1 =1u 2 =1u 3 =1u 4 = (a) 5 1 Time No intervention u 1 =1u 2 =1u 3 =1u 4 = (b) Control profile Time (days) Variable u 1 =1 u 2 =1 u 3 =1 (c) Figure 5: Simulations of te model sowing te effect of ITNs treatment and IRS on te spread of malaria for te endemic setting. parameters to be constant and perform matematical analysis of te model. Te analysis sowed tat tere exists a domain were te model is epidemiologically and matematically well posed. Stability analysis of te model sowed tat te disease-free equilibrium is globally asymptotically stable if R 1 in D. IfR > 1 te unique endemic equilibrium exists and is globally asymptotically stable in D. Temodel exibited backward bifurcation backward bifurcation at R = 1 implying te existence of multiple endemic equilibria for R <1. Te existence of multiple endemic equilibria empasizes te fact tat R <1is not sufficient to eradicate disease from te population and te need to lower R muc below

23 Discrete Dynamics in Nature and Society Infected umans 1 Infected mosquitoes Time No intervention u 1 =1u 2 =1u 3 =1u 4 =1 (a) 5 1 Time No intervention u 1 =1u 2 =1u 3 =1u 4 =1 (b) Control profile Time (days) Variable u1 =1 u 2 =1 u 3 =1 u 4 =1 (c) Figure 6: Simulations of te model sowing te effect of ITNs treatment IRS and IPTp on te spread of malaria for te endemic setting. onetomaketedisease-freeequilibriumstableglobally.tis beavior as important public ealt implications because it means tat bringing R below 1 is not enoug to eradicate malaria. We ten consider te case of time-dependent control variable from were we formulated an optimal control problem and derived expressions for te optimal control for te malaria model wit four control variables wit an aim of minimizing te number of malaria infections in umans (derive optimal prevention and treatment strategies) wile keeping te cost low. In te optimal control problem use of one control at a time and te different combination of interventions can be explored to investigate and compare te effects of te control strategies on malaria eradication

24 24 Discrete Dynamics in Nature and Society for different transmission settings. Te analysis of te model sowed tat te state and optimal control can be calculated using te optimality system. Te optimality system is comprised of te state system te adjoint/costate system initial conditions at t= boundary conditions (transversality) and te caracterization of te optimal control [ ]. Te results of te optimal control problem will be able to sow wic intervention or combination of te different intervention strategies as te igest impact on te control of te disease especially for different transmission settings. Our results sows tat an effective IRS use and treatment will be beneficial to te community for te control of malaria disease (infected uman and mosquito population) faster at a lower cost for te endemic settings. Tis is sligtly different from te findings of Agusto et al. [14] wo found tat te combination of te personal protection treatment and insecticides spray ad te igest impact on te control of te disease. Tis could be in endemic settings were bot preventive and treatment measures work better wic implies tat te effect of protection using IRS is better. Griffin et al. [4] found tat use of treatment long-lasting insecticides treated bed nets (LLITNs) and IRS wit ig levels coverage would result in reducing malaria transmission for ig settings toug te study did not consider te cost aspect. Te findings sow tat for te epidemic prone areas te optimal control strategy for reducing te infected uman and mosquito population was te use of treatment and IRS. Tis is sligtly different from Agusto et al. [14] findings on resource limited settings in wic te study recommended te use of personal protection and insecticides. Tis was furter different from te findings of Mwamtobe et al. [12] wo noted tat te prevention strategies (use of ITNs and IRS) lead to te reduction of bot te mosquito population and infected uman individuals. Tis is because in epidemic areas empasis is usually more placed on preventive strategies. Te results sow tat for seasonal areas muc impact will be felt wen treatment is used wic is different from Mwamtobe et al. [12] wo recommended IRS and ITNs. Tis is also comparable to Kim et al. [13] findings tat mosquitoreduction strategies are more effective tan personal protection. Tis is because in seasonal areas malaria transmission isusuallynotsoigandenceiftemosquito-reduction strategies can be implemented ten malaria transmission can be reduced. Griffin et al. [4] found tat for te ig seasonal transmission settings te use of LLITNs IRS and treatment wouldelpreducetetransmissionofmalaria. Te results sow tat for te low risk areas just te use of ITNs and treatment will be sufficient to reduce infected uman and mosquito population. Tis is comparable to Silva and Torres [15] wo found te optimal use of ITNS would prevent malaria transmission te same as Kim et al. [13]. Te findings are comparable to tose by Griffin et al. [4]. In low transmission areas prevention strategies seem to be better because te population is not infected. Tese findings support te WHO concerns on te capability of only one intervention strategy in reducing malaria transmission. Te findings are owever applicable to te designing of intervention strategies for malaria especially wen costs aspects are of concern. Tis modelling approac also addresses effectiveness of te recommended intervention for at-risk group of malaria (pregnant women) by te WHO. Te modelling approac as also been implemented in te R statistical computing platform wic is free statistical software. To te best of our knowledge tis is te first ever optimal control modelling and simulation of malaria intervention strategies in free R statistical computing platform; future testing and refinement of te model togeter wit simulation witdatafromoterrepresentativesettingssouldbedoneto improve te results and te model. Tese findings were based on te use of secondary data; a more designed study may be needed to ascertain te findings of tese studies. Regardless it does not invalidate te findings. 6. Conclusion In tis paper we derived and analyzed a deterministic model for te transmission of malaria disease wic incorporated te use of insecticide-treated bed nets (ITNs) treatment indoor residual spray (IRS) and intermittent preventive treatment for pregnant women (IPTp) and performed optimal control analysis of te model. We first consider constant control parameters from were we investigate existence and stability of equilibria as well as stability analysis. We proved tat if R 1 te disease-free equilibrium is globally asymptotically stable in D. IfR > 1 te unique endemic equilibrium exists and is globally asymptotically stable. Te model also exibits backward bifurcation at R =1. We ten consider te time-dependent control case from were we derived and analyzed te necessary conditions for te optimal control of te disease. Tere were 15 different control strategies for eac of te four different epidemiological zones in Kenya tat were explored using control plots (numerical simulations) wic compared wen tere were not any intervention strategies and wen tere were intervention strategies. Using te optimal control approac we found tat te combined use of treatment and IRS wouldreduceteigestnumberofinfectedumanand mosquito population faster at a lower cost for te endemic settings. For te epidemic prone settings te use of treatment and IRS as more impact in reducing te infected uman and mosquito population. For seasonal areas muc impact will be felt wen treatment is used. For te low risk areas just te use of ITNs and treatment will be sufficient to reduce infected uman and mosquito population. Tis was deduced from te intervention wic takes sorter time to start reducing te number of infected mosquitoes and umans. We conclude tat according to our model te optimal control strategy for malaria control in endemic areas was te combined use of treatment and IRS; in epidemic prone areas it was te use of treatment and IRS; in seasonal areas it was te useoftreatment;andinlowriskareaswasteuseofitnsand treatment. Control programs tat follow tese strategies can effectively reduce te spread of malaria disease in different malaria transmission settings in Kenya.

25 Discrete Dynamics in Nature and Society 25 Appendix Te geometric approac to global stability is as described by Li and Muldowney for proving Teorem 4. Te general metod considered is te one developed by Li & Muldowney [3]. Consider te autonomous dynamical system: x=f(x) (A.1) were f:d R n D R n is open set and simply connected and f C 1 D.Letx be an equilibrium of (A.1); tat is f(x )=.Werecalltatx is said to be globally stable in D if it is locally stable and all trajectories in D converge to x. Assume tat te following ypoteses old (H 1 ): tere exists a compact absorbing set K D; (H 2 ): equation (A.1) as a unique equilibrium x in D. Tebasicideaoftismetodistatifequilibriumx is (locally) stable ten te global stability is assured provided tat (H 1 )-(H 2 ) old and no nonconstant periodic solution of (A.1) exists. Bendixson Criterion. Li and Muldowney sowed tat if (H 1 )-(H 2 ) old and (A.1) satisfies a Bendixson criterion tat is robust under C 1 local ε-perturbations of f at all nonequilibrium nonwandering points for (A.1) ten x is globally stable in D provided it is stable. Ten a new Bendixson criterion robust under C 1 local ε-perturbation and basedonteuseoftelozinskiǐmeasureisintroduced. Function g C 1 (D R n ) is called C 1 local εperturbations of f at x D if tere exists an open neigbourood U of x in D suc tat support supp(f g) U and f g C 1 <εwere f g C 1 = sup{ f(x) g(x) + f x (x) g x (x) : x D}. Point x Dissaiobenonwondering for (A.1) for any neigborood U of x in D and tere exists arbitrary large t suc tat U x(t U) =φ. Let P(x) be a ( n 2 ) ( n 2 ) matrix-valued function tat is C 1 on D and consider were matrix P f is B=P f P 1 +PJ 2 P 1 (p ij (x)) f =( p T ij (x) x ) f(x) = p ij f(x). (A.2) (A.3) And J 2 istesecondadditivecompoundmatrixofte Jacobian matrix J; tat is J(x) = Df(x). Generally speaking for n nmatrix J=(J ij ) J 2 is ( n 2 ) ( n 2 ) matrix (for a survey on compound matrices) and teir relations to differential equations as described by Muldowney [41] and in te special case n=3oneas J 2 = [ [ J 11 +J 22 J 23 J 13 J 23 J 11 +J 33 J 12 J 31 J 21 J 22 +J 33 ] ]. (A.4) Consider Lozinskii measure L of B wit respect to vector norm in R n N=n/2[42]: 1+B 1 L (B) = lim. (A.5) + It is proved in [3] tat if (H 1 ) and (H 2 ) old condition t 1 lim supsup t x D t L (B (x (s x ))) ds < (A.6) guarantees tat tere are no orbits giving rise to a simple closed rectifiable curve in D wic is invariant for (A.1) tat is periodic orbits omoclinic orbits and eteroclinic cycles. In particular condition (A.6) is proved to be a robust Bendixson criterion for (A.1). Besides it is remarked tat under assumptions (H 1 )-(H 2 ) condition (A.6) also implies te local stability of x. As a consequence te following teorem olds [3]. Teorem A.1. Assume tat conditions (H 1 )-(H 2 ) old. Ten x is globally asymptotically stable in D provided tat function P(x) and LozinskiǐmeasureL exist suc tat condition (A.6) is satisfied. Additional Points Recommendation: in order to reduce malaria transmission in Kenya te National Malaria Control Programme sould tailor-make different intervention strategies for different epidemiological zones since some strategies work better tan oters in a resource limited setting. In addition te designedinterventionstatarepreventiveinnaturesouldbe implemented across all te different settings. Tere will be a need for te National Control Programme to create awareness on te malaria preventive measures. Competing Interests No competing interests exist. Autors Contributions All te autors ave significant contribution to tis paper and te final form of tis paper is approved by all autors. Acknowledgments Tis work was (partially) funded by an African Doctoral Dissertation Researc Fellowsip (ADDRF) award offered by African Population and Healt Researc Centre (APHRC) in partnersip wit te International Development Researc Centre (IDRC). Pase 4 of te ADDRF program is funded by IDRC (Grant no ). Tis work was also supported by National Commission for Science and Tecnology (NACOSTI/ RCD/ST&I 6t Call PD/225). References [1] WHO World Malaria Report 214 World Healt Organization Global Malaria Programme Geneva Switzerland 214 ttp://

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