APPLICATION OF OPTIMAL CONTROL TO THE EPIDEMIOLOGY OF MALARIA

Transcription

1 Electronic Journal of Differential Equations, Vol. 212 (212, No. 81, pp ISSN: URL: ttp://ejde.mat.txstate.edu or ttp://ejde.mat.unt.edu ftp ejde.mat.txstate.edu APPLICATION OF OPTIMAL CONTROL TO THE EPIDEMIOLOGY OF MALARIA FOLASHADE B. AGUSTO, NIZAR MARCUS, KAZEEM O. OKOSUN Abstract. Malaria is a deadly disease transmitted to umans troug te bite of infected female mosquitoes. In tis paper a deterministic system of differential equations is presented and studied for te transmission of malaria. Ten optimal control teory is applied to investigate optimal strategies for controlling te spread of malaria disease using treatment, insecticide treated bed nets and spray of mosquito insecticide as te system control variables. Te possible impact of using combinations of te tree controls eiter one at a time or two at a time on te spread of te disease is also examined. 1. Introduction Malaria is a common and serious disease. It is reported tat te incidence of malaria in te world may be in te order of 3 million clinical cases eac year. Malaria mortality is estimated at almost 2 million deats worldwide per year. Te vast numbers of malaria deats occur among young cildren in Africa, especially in remote rural areas. In addition, an estimated over 2 billion people are at risk of infection, no vaccines are available for te disease [25, 43]. Malaria is transmitted to umans troug te bite of an infected female Anopeles mosquito, following te successful sporozoite inoculation, plasmodium falciparum is usually first detected 7-11 days. Tis is followed after few days of te bites, by clinical symptoms suc as sweats, sills, pains, and fever. Mosquitoes on te oter and acquire infection from infected uman after a blood meal. Altoug malaria is life-treatening it is still preventable and curable if te infected individual seek treatment early. Prevention is usually by te use of insecticide treated bed nets and spraying of insecticide but according to te World Healt Organization position statement on insecticide treated mosquito nets [44], te insecticide treated bed nets(itns, long-lasting insecticide nets (LLINs, indoor residual spraying (IRS, and te oter main metod of malaria vector control, may not be sufficiently effective alone to acieve and maintain interruption of transmission of malaria, particularly in olo-endemic areas of Africa. 2 Matematics Subject Classification. 92B5, 93A3, 93C15. Key words and prases. Malaria; optimal control; insecticide treated bed nets; mosquito insecticide. c 212 Texas State University - San Marcos. Submitted July 2, 21. Publised May 22,

2 2 F. B. AGUSTO, N. MARCUS, K. O. OKOSUN EJDE-212/81 Many studies ave been carried out to quantify te impact of malaria infection in umans [6, 14, 19, 24, 33, 36]. Many of tese studies focuses only on te transmission of te disease in uman and te vector populations but recently, Ciyaka et.al [1] formulated a deterministic system of differential equations wit two latent periods in te non-constant ost and vector populations in order to teoretically assess te potential impact of personal protection, treatment and possible vaccination strategies on te transmission dynamics of malaria. Blayne et al [5], used a time dependent model to study te effects of prevention and treatment on malaria, similarly Okosun [29] used a time dependent model to study te impact of a possible vaccination wit treatment strategies in controlling te spread of malaria in a model tat includes treatment and vaccination wit waning immunity. Tus, following te WHO position statement [44] it is instructive to carry out modeling studies to determine te impact of various combinations of control strategies on te transmission dynamics of malaria. In tis paper, we use treatment of symptomatic individuals, personal protection and te straying of insecticide as control measures and ten consider tis time dependent control measures using optimal control teory. Time dependent control strategies ave been applied for te studies of HIV/AIDS disease, Tuberculosis, Influenza and SARS [1, 2, 7, 17, 2, 39, 42, 46]. Optimal control teory as been applied to models wit vector-borne diseases [5, 31, 4, 45]. Our goal is to develop matematical model for uman-vector interactions wit control strategies, wit te aim of investigating te role of personal protection, treatment and spraying of insecticides in malaria transmission, in line wit concerns raised WHO [44]; in order to determine optimal control strategies wit various combinations of te control measures for controlling te spread of malaria transmission. Te paper is organized as follows: in Section 2, we give te description of te uman-vector model, stating te assumptions and definitions of te various parameters of te model. Te analysis of te equilibrium points are discussed in Sections 2.2 and 3. In Section 4, we state te control problem as well as te objective functional to be minimized, we ten apply te Pontryagin s Maximum Principle to find te necessary conditions for te optimal control. In Sections 5, we sow te simulation results to illustrate te population dynamics wit preventative measures and treatment as controls. 2. Model formulation Te model sub-divides te total uman population at time t, denoted by N (t, into te following sub-populations of susceptible individuals (S (t, tose exposed to malaria parasite (E (t, individuals wit malaria symptoms (I (t, partially immune uman (R (t. So tat N (t = S (t + E (t + I (t + R (t. Te total vector (mosquito population at time t, denoted by N v (t, is subdivided into susceptible mosquitoes (S v (t, mosquitoes exposed to te malaria parasite (E v (t and infectious mosquitoes(i v (t. Tus, N v (t = S v (t + E v (t + I v (t. It is assumed tat susceptible umans are recruited into te population at a constant rate Λ. Susceptible individuals acquire malaria infection following contact wit infectious mosquitoes (at a rate βε φ, were β is te transmission probability per bite and ε is te biting rate of mosquitoes, φ is contact rate of vector per

3 EJDE-212/81 APPLICATION OF OPTIMAL CONTROL 3 uman per unit time. Susceptible individuals infected wit malaria are moved to te exposed class (E at te rate βε φ and ten progress to te infectious class, following te development of clinical symptoms (at a rate α. Individuals wit malaria symptoms are effectively treated (at a rate τ were ( τ 1. Human spontaneous recovery rate is given by b, were b < τ. And individuals infected wit malaria suffer a disease-induced deat (at a rate ψ. Infected individual ten progress to te partially immuned group. Upon recovery, te partially immuned individual losses immunity (at te rate κ and becomes susceptible again. Susceptible mosquitoes (S v are generated at te rate Λ v and acquire malaria infection (following effective contacts wit umans infected wit malaria at a rate λφε v (I + ηr, were λ is te probability of a vector getting infected troug te infectious uman and ε v is te biting rate of mosquitoes. We assume tat umans in te R (t class can still transmit te disease, tus, te modification parameter η [, 1 gives te reduced infectivity of te recovered individuals [11, 32]. Mosquitoes are assumed to suffer natural deat at a rate µ v, regardless of teir infection status. Newly-infected mosquitoes are moved into te exposed class (E v, and progress to te class of symptomatic mosquitoes (I v following te development of symptoms (at a rate α v. Tus, putting te above formulations and assumptions togeter gives te following uman-vector model, given by system of ordinary differential equations below as ds = Λ + κr βε φi v S µ S, de = βε φi v S (α + µ E, di = α E (b + τi (ψ + µ I, dr = (b + τi (κ + µ R, (2.1 ds v = Λ v λφε v (I + ηr S v µ v S v, de v = λφε v (I + ηr S v (α v + µ v E v, di v = α ve v µ v I v, Te associated model variables and parameters are described in Table Basic properties of te malaria model Positivity and boundedness of solutions. For te malaria transmission model (2.1 to be epidemiologically meaningful, it is important to prove tat all its state variables are non-negative for all time. In oter words, solutions of te model system (2.1 wit non-negative initial data will remain non-negative for all time t >. Teorem 2.1. Let te initial data S (, E (, I (, R (, S v (, E v (, I v (. Ten te solutions (S, E, I, R, S v, E v, I v of te malaria model (2.1 are non-negative for all t >. Furtermore lim sup N (t Λ t µ, lim sup t N v (t Λ v µ v,

4 4 F. B. AGUSTO, N. MARCUS, K. O. OKOSUN EJDE-212/81 wit N = S + E + I + R and N v = S v + E v + I v. Proof. Let t 1 = sup{t > : S (t >, E (t >, I (t >, R (t >, S v (t >, I v (t >, E v (t > }. Since S ( >, E ( >, I ( >, R ( >, S v ( >, E v ( >, I v ( >, ten, t 1 >. If t 1 <, ten S, E, I, R, S v, E v or I v is equal to zero at t 1. It follows from te first equation of te system (2.1, tat Tus, Hence, ds = Λ βε φi v S µ S + κr d { S (t exp[(βε φi v + µ t] } = (Λ + κr exp[(βε φi v + µ t] S (t 1 exp[(βε φi v + µ t] S ( = so tat t1 (Λ + κr exp[(βε φi v + µ p]dp and S (t 1 = S ( exp[ (βε φi v + µ t 1 ] + exp[ (βε φi v + µ t 1 ] t1 (Λ + κr exp[(βε φi v + µ p]dp >. R (t 1 = R ( exp[ (µ + κt 1 ] + exp[(µ + κt 1 ] >. t1 (b + τi exp[(µ + κp]dp It can similarly be sown tat E >, I >, S v >, E v > and I v > for all t >. For te second part of te proof, it sould be noted tat < I (t N (t and < I v (t N v (t. Adding te first four equations and te last tree equations of te model (2.1 gives dn (t = Λ µ N (t ψi (t, Tus, dn v (t = Λ v µ v N v (t. Λ (µ + ψn (t dn (t Λ µ N (t, Λ v µ v N v (t dn v(t Λ v µ v N v (t. Hence, respectively, (2.2 and as required. Λ µ + ψ Λ v lim inf t lim inf µ v t N (t lim sup N (t Λ, t µ N v(t lim sup N v (t Λ v, t µ v

5 EJDE-212/81 APPLICATION OF OPTIMAL CONTROL Invariant regions. Te malaria model (2.1 will be analyzed in a biologicallyfeasible region as follows. Te system (2.1 is split into two parts, namely te uman population (N ; wit N = S + E + I + R and te vector population (N v ; wit N v = S v + E v + I v. Consider te feasible region wit D = D D v R 4 + R 3 +, D = {(S, E, I, R R 4 + : S + E + I + R Λ µ }, D v = {(S v, E v, I v R 3 + : S v + E v + I v Λ v µ v } Te following steps are done to establis te positive invariance of D (i.e., solutions in D remain in D for all t >. Te rate of cange of te umans and mosquitoes populations is given in equation (2.2, it follows tat dn (t Λ µ N (t, dn v (t Λ v µ v N v (t. (2.3 A standard comparison teorem [21] can ten be used to sow tat N (t N (e µ t + Λ µ (1 e µ t and N v (t N v (e µvt + Λv µ v (1 e µvt. In particular, N (t Λ µ and N v (t Λv µ v if N ( Λ µ and N v ( Λv µ v respectively. Tus, te region D is positively-invariant. Hence, it is sufficient to consider te dynamics of te flow generated by (2.1 in D. In tis region, te model can be considered as been epidemiologically and matematically well-posed [15]. Tus, every solution of te basic model (2.1 wit initial conditions in D remains in D for all t >. Terefore, te ω-limit sets of te system (2.1 are contained in D. Tis result is summarized below. Lemma 2.2. Te region D = D D v R 4 + R 3 + is positively-invariant for te basic model (2.1 wit non-negative initial conditions in R Stability of te disease-free equilibrium (DFE. Te malaria model (2.1 as a DFE, obtained by setting te rigt-and sides of te equations in te model to zero, given by ( E = (S, E, I, R, Sv, Ev, Iv Λ =,,,, Λ v,,. µ µ v Te linear stability of E can be establised using te next generation operator metod [42] on te system (2.1, te matrices F and V, for te new infection terms and te remaining transfer terms, are, respectively, given by βε φs F = λε v φsv λε v φηsv,

6 6 F. B. AGUSTO, N. MARCUS, K. O. OKOSUN EJDE-212/81 k 1 α 1 k 2 V = (b + τ k 3 k 4, α 2 µ v were k 1 = α + µ, k 2 = b + τ + ψ + µ, k 3 = κ + µ, k 4 = α v + µ v. It follows tat te reproduction number of te malaria system (2.1, denoted by R, is α 1 α 2 λβ[k 3 + η(b + τ]φ R = 2 ɛ ε v S S v, (2.4 k 3 k 4 k 2 k 1 µ v Furter, using [42, Teorem 2], te following result is establised. Teorem 2.3. Te DFE of te model (2.1, given by R, is locally asymptotically stable (LAS if R < 1, and unstable if R > Existence of endemic equilibrium point (EEP Next conditions for te existence of endemic equilibria for te model (2.1 is explored. Let E 1 = ( S, E, I, R, Sv, Ev, Iv, be te arbitrary endemic equilibrium of model (2.1, in wic at least one of te infected components of te model is non-zero. Let λ = βφε I v, (3.1 λ v = λφε v (I + ηr (3.2 be te force of infection in umans and in te vector. Setting te rigt-and sides of te equations in (2.1 to zero gives te following expressions (in terms of λ and λ v S Λ = k 1k 2 k 3 (λ + µ k 1 k 2 k 3 κλ α (b + τ, E k 2 λ = Λ k 3 (λ + µ k 1 k 2 k 3 κλ α (b + τ, I λ = Λ k 3 α 1 (λ + µ k 1 k 2 k 3 κλ α (b + τ, R = (b + τλ Λ α 1 (λ + µ k 1 k 2 k 3 κλ Sv Λ v = (λ v + µ v, E v = Substituting (3.3 and (3.2 into (3.1, gives a λ α (b + τ, λ v Λ v k 4 (λ v + µ v, I v = α v λ v Λ v k 4 µ v (λ v + µ v + b, were a = k 4 µ v {λφε v Λ α [k 3 + η(b + τ] + µ v [k 3 k 1 k 2 κα (b + τ]} b = µ µ 2 vk 4 k 3 k 1 k 2 (1 R 2. (3.3 Te coefficient a is always positive, te coefficient b is positive (negative if R is less tan (greater tan unity. Furtermore, tere is no positive endemic equilibrium if b. If b <, ten tere is a unique endemic equilibrium (given by λ = b /a. Tis result is summarized below.

7 EJDE-212/81 APPLICATION OF OPTIMAL CONTROL 7 Lemma 3.1. Te model (2.1 as a unique positive endemic equilibrium wenever R > 1, and no positive endemic equilibrium oterwise Global stability of endemic equilibrium for a special case. In tis section, we investigate te global stability of te endemic equilibrium of model (2.1, for te special case wen κ, tat tere is no lost of immunity. Using te approac in te proof of Lemma 2.2, it can be sown tat te region D = D D v R 4 + R 3 +, were D = { (S, E, I, R D : S S }, D v = { (S v, E v, I v D v : S v S v}. is positively-invariant for te special case of te model (2.1 described above. It is convenient to define D = { (S, E, I, R, S v, E v, I v D : E = I = R = E v = I v }. Teorem 3.2. Te unique endemic equilibrium, Ẽ1, of te model (2.1 is GAS in D\ D wenever R κ= > 1. Proof. Let R > 1, so tat te unique endemic equilibrium (Ẽ1 exists. Consider te non-linear Lyapunov function ( F = S S S ln S ( S + E E E ln E E + k ( 1 I I α I ln I I + k ( 2k 1 R α γ R ln R ( R + Sv Sv Sv ln S ( v Sv + Ev Ev Ev ln E v Ev ( Iv R + k 4 Iv α v Iv ln I v Iv were γ = b + τ and te Lyapunov derivative is F = (1 S + S + S (1 S v S v + S v, (1 E Ė + k ( 1 E α (1 E v Ėv + k ( 4 E v α v 1 I I + k 2k 1 I α γ 1 I v I v. I v (1 R R R Substituting te expressions for te derivatives in F (from (2.1 wit κ gives ( F = Λ λ S µ S S Λ λ S µ S S ( + λ S k 1 E E λ S k 1 E E + k 1 α ( α E k 2 I k 1 + k 2k 1 α γ (γi k 3 R k 2k 1 α γ I ( α E k 2 I α I R ( γi k 3 R R ( + Λ v λ v S v µ v S v S v Λ v λ v S v µ v S v S v + λ v S v k 4 E v + E v E v ( λ v S v k 4 E v

8 8 F. B. AGUSTO, N. MARCUS, K. O. OKOSUN EJDE-212/81 so tat F = λ S + k 1 E + k ( 4 α v E v µ v I v + k 4 I ( v α v E v µ v I v. α v α v I v (1 S S I k 1 + λ v Sv (1 S v S v ( + µ S I E + k 2k 1 E v λ v S v + k 4 Ev E v α + µ v S v Iv k 4 2 S S I k 2k 1 S + λ S S α ( 2 S v S v I v E v + k 4µ v α v R R I + k 3k 2 k 1 S v + λ v Sv S v I v E λ S E α γ R k 3k 2 k 1 α γ R k 4µ v α v I v (3.4 Finally, equation (3.4 can be furter simplified to give F = µ S (2 S S + k 1 E (5 S I I R R + k 4 E v S R R (4 S v S v S + µ v S v E v E v (2 S v E v E v I v I v S v S E E S v I v I v Sv. E E Since te aritmetic mean exceeds te geometric mean, it follows tat I I (3.5 2 S S 5 S S 4 S v S v S S, E E E v E v E E E v 2 S v Ev I I I v I v S v I I S v Sv, I v Iv R R, R R Since all te model parameters are non-negative, it follows tat F for R κ= > 1. Tus, it follows from te LaSalle s Invariance Principle, tat every solution to te equations in te model (2.1 (wit initial conditions in D\ D approaces te EEP (Ẽ1 as t wenever R κ= > Analysis of optimal control We introduce into te model (2.1, time dependent preventive (, u 3 and treatment (u 2 efforts as controls to curtail te spread of malaria. Te malaria model (2.1 becomes ds = Λ + κr (1 βε φi v S µ S, de = (1 βε φi v S (α + µ E, di = α E (b + u 2 I (ψ + µ I, dr = (b + u 2 I (κ + µ R,

9 EJDE-212/81 APPLICATION OF OPTIMAL CONTROL 9 ds v = Λ v (1 λε v φ(i + ηr S v u 3 (1 ps v µ v S v, (4.1 de v = (1 λε v φ(i + ηr S v u 3 (1 pe v (α v + µ v E v, di v = α ve v u 3 (1 pi v µ v I v. Te function 1 represent te control on te use of mosquitoes treated bed nets for personal protection, and u 2 a 2, te control on treatment, were a 2 is te drug efficacy use for treatment. Te insecticides used for treating bed nets is letal to te mosquitoes and oter insects and also repels te mosquitoes, tus, reducing te number tat attempt to feed on people in te sleeping areas wit te nets [8, 44]. However, te mosquitoes can still feed on umans outside tis protective areas, and so we ave included te spraying of insecticide. Tus, eac mosquitoes group is reduced (at te rate u 3 (1 p, were (1 p is te fraction of vector population reduced and u 3 a 3, is te control function representing spray of insecticide aimed at reducing te mosquitoes sub-populations and a 3 represent te insecticide efficacy at reducing te mosquitoes population. Tis is different from wat was implemented in [5], were only two control measures of personal protection and treatment were used. Wit te given objective function J(, u 2, u 3 = tf [mi + nu cu du 2 3] (4.2 were t f is te final time and te coefficients m, n, c, d are positive weigts to balance te factors. Our goal is to minimize te number of infected umans I (t, wile minimizing te cost of control (t, u 2 (t, u 3 (t. Tus, we seek an optimal control u 1, u 2, u 3 suc tat were te control set J(u 1, u 2, u 3 = min {J(, u 2, u 3, u 2, u 3 U} (4.3,u 2,u 3 U = {(, u 2, u 3 u i : [, t f ] [, 1], Lebesgue measurable i = 1, 2, 3}. Te term mi is te cost of infection wile nu 2 1, cu 2 2 and du 2 3 are te costs of use of bed nets, treatment efforts and use of insecticides respectively. Te necessary conditions tat an optimal control must satisfy come from te Pontryagin s Maximum Principle [3]. Tis principle converts (4.1-(4.2 into a problem of minimizing pointwise a Hamiltonian H, wit respect to (, u 2, u 3 H = mi + nu cu du λ S {Λ + κr (1 βε φi v S µ S } + λ E {(1 βε φi v S (α + µ E } + λ I {α E (b + u 2 I (ψ + µ I } + λ R {(b + u 2 I (κ + µ R } + λ Sv {Λ v (1 λε v φ(i + ηr S v u 3 (1 ps v µ v S v } + λ Ev {(1 λε v φ(i + ηr S v u 3 (1 pe v (α v + µ v E v } + λ Iv {α v 2E v u 3 (1 pi v µ v I v } (4.4 were te λ S, λ E, λ I, λ R, λ Sv, λ Ev, λ Iv are te adjoint variables or co-state variables. [13, Corollary 4.1] gives te existence of optimal control due to te

10 1 F. B. AGUSTO, N. MARCUS, K. O. OKOSUN EJDE-212/81 convexity of te integrand of J wit respect to, u 2 and u 3, a priori boundedness of te state solutions, and te Lipscitz property of te state system wit respect to te state variables. Applying Pontryagin s Maximum Principle [3] and te existence result for te optimal control from [13], we obtain te following teorem. Teorem 4.1. Given an optimal control u 1, u 2, u 3 and solutions S, E, I, R, S v, E v, I v of te corresponding state system (4.1 tat minimizes J(, u 2, u 3 over U. Ten tere exists adjoint variables λ S, λ E, λ I, λ R, λ Sv, λ Ev, λ Iv satisfying dλ S = [(1 βε φi v + µ ]λ S + (1 βε φi v λ E dλ E = (µ + α λ E + α λ I dλ I = m [(b + u 2 + (µ + ψ]λ I + (b + u 2 λ R + (1 λε v φs v (λ Ev λ Sv dλ R = κλ S (µ + κλ R + (1 λε v φηs v (λ Sv λ Ev dλ S v = [(1 λε v φ(i + ηr + u 3 (1 p + µ v ]λ Sv + (1 λε v φ(i + ηr λ Ev dλ E v = [u 3 (1 p + α v + µ v ]λ Ev + α v λ Iv dλ I v = (1 βε φs λ S + (1 βε φs λ E [u 3 (1 p + µ v ]λ Iv (4.5 and wit transversality conditions λ S (t f = λ E (t f = λ I (t f = λ R (t f = λ Sv (t f = λ Ev (t f = λ Iv (t f (4.6 and te controls u 1, u 2 and u 3 satisfy te optimality condition { ( u 1 = max, min 1, βε φiv (λ E λ S S + λε vφ(i + ηr (λ E v λ Sv Sv 2n { ( } u 2 = max, min { ( u 3 = max, min 1, (λ I λ R I 2c 1, (1 p(s vλ Sv + E vλ Ev + I v λ Iv 2d } }, (4.7 Proof. Te differential equations governing te adjoint variables are obtained by differentiation of te Hamiltonian function, evaluated at te optimal control. Ten te adjoint system can be written as dλ S dλ I = H S = [(1 βε φi v + µ ]λ S + (1 βε φi v λ E dλ E = H E = (µ + α λ E + α λ I = H I = m [(b + u 2 + (µ + ψ]λ I + (b + u 2 λ R + (1 λε v φs v (λ Ev λ Sv

11 EJDE-212/81 APPLICATION OF OPTIMAL CONTROL 11 dλ R dλ S v dλ I v = H R = κλ S (µ + κλ R + (1 λε v φηs v (λ Sv λ Ev = H S v = [(1 λε v φ(i + ηr + u 3 (1 p + µ v ]λ Sv + (1 λε v φ(i + ηr λ Ev dλ E v wit transversality conditions = H E v = [u 3 (1 p + α v + µ v ]λ Ev + α v λ Iv = H I v = (1 βε φs λ S + (1 βε φs λ E [u 3 (1 p + µ v ]λ Iv λ S (t f = λ E (t f = λ I (t f = λ R (t f = λ Sv (t f = λ Ev (t f = λ Iv (t f (4.8 On te interior of te control set, were < u i < 1, for i = 1, 2, 3, we ave = H = 2nu 1 + βε φi v (λ S λ E S + λε v φ(i + ηr (λ Sv λ Ev S v, = H u 2 = 2cu 2 (λ I λ R I, = H u 3 = 2du 3 (1 p(s vλ Sv + E vλ Ev + I v λ Iv. Hence, we obtain (see [23] and u 1 = max u 1 = βε φiv (λ E λ S S + λε vφ(i + ηr (λ E v λ Sv Sv, 2n { (, min u 2 = (λ I λ R I, 2c u 3 = (1 p(s vλ Sv + Evλ Ev + Iv λ Iv. 2d 1, βε φiv (λ E λ S S + λε vφ(i + ηr (λ E v λ Sv Sv 2n { ( u 2 = max, min 1, (λ I λ R I } 2c (1 p(svλ Sv + Evλ Ev + Iv λ Iv }. 2d { ( u 3 = max, min 1, (4.9 Due to te a priori boundedness of te state and adjoint functions and te resulting Lipscitz structure of te ODE s, we can obtain te uniqueness of te optimal control for small t f, following tecniques from [3]. Te uniqueness of te optimal control follows from te uniqueness of te optimality system, wic consists of (4.1 and (4.5, (4.6 wit caracterization (4.7. Tere is a restriction on te lengt of time interval in order to guarantee te uniqueness of te optimality system. Tis smallness restriction of te lengt on te time is due to te opposite time orientations of te optimality system; te state problem as initial values and te adjoint problem as final values. Tis restriction is very common in control problems (see [16, 2, 22]. },

12 12 F. B. AGUSTO, N. MARCUS, K. O. OKOSUN EJDE-212/81 Next we discuss te numerical solutions of te optimality system and te corresponding optimal control pairs, te parameter coices, and te interpretations from various cases. 5. Numerical results In tis section, we study numerically an optimal transmission parameter control for te malaria model. Te optimal control is obtained by solving te optimality system, consisting of 7 ODE s from te state and adjoint equations. An iterative sceme is used for solving te optimality system. We start to solve te state equations wit a guess for te controls over te simulated time using fourt order Runge-Kutta sceme. Because of te transversality conditions (4.6, te adjoint equations are solved by a backward fourt order Runge-Kutta sceme using te current iterations solutions of te state equation. Ten te controls are updated by using a convex combination of te previous controls and te value from te caracterizations (4.7. Tis process is repeated and iterations are stopped if te values of te unknowns at te previous iterations are very close to te ones at te present iterations [23]. We explore a simple model wit preventive and treatment as control measures to study te effects of control practices and te transmission of malaria. Using various combinations of te tree controls, one control at a time and two controls at a time, we investigate and compare numerical results from simulations wit te following scenarios i. using personal protection ( witout insecticide spraying (u 3 and no treatment of te symptomatic umans (u 2 ii. treating te symptomatic umans (u 2 witout using insecticide spraying (u 3 and no personal protection (, iii. using insecticide spraying (u 3 witout personal protection ( and no treatment of te symptomatic umans (u 2, iv. treating te symptomatic umans (u 2 and using insecticide spraying (u 3 wit no personal protection (, v. using personal protection ( and insecticide spraying (u 3 wit no treatment of te symptomatic umans (u 2, vi using treatment (u 2 and personal protection ( wit no insecticide spraying (u 3, finally vii. using all tree control measures (, u 2 and u 3. For te figures presented ere, we assume tat te weigt factor c associated wit control u 2 is greater tan n and d wic are associated wit controls and u 3. Tis assumption is based on te facts tat te cost associated wit and u 3 will include te cost of insecticide and insecticide treated bed nets, and te cost associated wit u 2 will include te cost of antimalarial drugs, medical examinations and ospitalization. For te numerical simulation we ave used te following weigt factors, m = 92, n = 2, c = 65, and d = 1, initial state variables S ( = 7, E ( = 1, I (, R (, S v ( = 5, E v ( = 5, I v ( = 3 and parameter values Λ v.71, Λ.11, β.3, ε.1, ε.1, λ.5, µ.457, µ v.667, κ.14, α 1.58, α 2.556, σ.25, b.5, φ.52, ψ.2, τ.5, p.85, for wic te reproduction number R = , to illustrate te effect of different

13 EJDE-212/81 APPLICATION OF OPTIMAL CONTROL 13 Table 1. Description of Variables and Parameters of te Malaria Model (4.1 Var. S E I R S v E v I v Description Susceptible uman Exposed uman Infected uman Recovered uman Susceptible vector Exposed vector Infected vector Par. Description Est. val. References ε biting rate of umans.2-.5 [4, 18] ε v biting rate of mosquitoes.3 [18, 26, 38] β probability of uman getting infected.3 [12, 36] λ probability of a mosquito getting infected.9 [12, 36] µ Natural deat rate in umans.4 [47] µ v Natural deat rate in mosquitoes.4 [1] κ rate of loss of immunity 1/(2 365 [3, 12, 34] α 1 rate of progression from exposed to infected uman 1/17 [3, 28] α 2 rate of progression from exposed to infected mosquito 1/18 [35, 38, 27] Λ uman birt rate.11 [41] Λ v mosquitoes birt rate.71 [3, 12] ψ disease induced deat.5 [37] φ contact rate of vector per uman per unit time.6 [9] b spontaneous recovery.5 [1] η modification parameter.1 assumed optimal control strategies on te spread of malaria in a population. Tus, we ave considered te spread of malaria in an endemic population. Optimal personal protection. Only te control ( on personal protection is used to optimize te objective function J, wile te control on treatment (u 2 and te control on insecticide spray (u 3 are set to zero. In Figure 1, te results sow a significant difference in te I and I v wit optimal strategy compared to I and I v witout control. Specifically, we observed in Figure 1(a tat te control strategies lead to a decrease in te number of symptomatic uman (I as against an increases in te uncontrolled case. Similarly, in Figure 1(b, te uncontrolled case resulted in increased number of infected mosquitoes (I v, wile te control strategy lead to a decrease in te number infected. Te control profile is sown in Figure 1(c, ere we see tat te optimal personal protection control is at te upper bound till te time t f = 1 days, before dropping to te lower bound. Optimal treatment. Wit tis strategy, only te control (u 2 on treatment is used to optimize te objective function J, wile te control on personal protection ( and te control on insecticide spray (u 3 are set to zero. In Figure 2, te results sow a significant difference in te I and I v wit optimal strategy compared to I and I v witout control. But tis strategy sows tat effective treatment only as a significant impact in reducing te disease incidence among uman population. Te control profile is sown in Figure 2(c, we see tat te optimal treatment control u 2 rises to and stabilizes at te upper bound for t f = 7 days, before dropping to te lower bound.

14 14 F. B. AGUSTO, N. MARCUS, K. O. OKOSUN EJDE-212/ , u 2, u , u 2, u 3 Infected Human Infected Mosquitoes (a (b (c Figure 1. Simulations sowing te effect of personal protection only on infected uman and mosquitoes populations Optimal insecticide spraying. Wit tis strategy, only te control on insecticide spraying (u 3 is used to optimize te objective function J, wile te control on treatment (u 2 and te control on personal protection ( are set to zero. Te results in Figure 3 sow a significant difference in te I and I v wit optimal strategy compared to I and I v witout control. We see in Figure 3(a tat te control strategies resulted in a decrease in te number of symptomatic uman (I as against an increase in te uncontrolled case. Also in Figure 3(b, te uncontrolled case resulted in increased number of infected mosquitoes (I v, wile te control strategy lead to a drastic decrease in te number of infected mosquitoes. Te control profile is sown in Figure 3(c, ere we see tat te optimal insecticide spray control u 3 is at te upper bound till te time t f = 9 days, it ten reduces gradually to te lower bound. Optimal treatment and insecticide spray. Wit tis strategy, te control (u 2 on treatment and te control on (u 3 insecticide spraying are bot used to optimize te objective function J, wile te control on personal protection ( is set to zero. In Figure 4, te result sows a significant difference in te I and I v wit optimal control strategy compared to I and I v witout control. We observed in Figure 4(a tat te control strategies resulted in a decrease in te number of symptomatic umans (I as against increases in te uncontrolled case. Similarly in Figure 4(b, te uncontrolled case resulted in increased number of infected mosquitoes (I v, wile te control strategy lead to a decrease in te number of infected mosquitoes.

15 EJDE-212/81 APPLICATION OF OPTIMAL CONTROL , u 2, u , u 2, u 3 Infected Human Infected Mosquitoes (a (b u (c Figure 2. Simulations sowing te effect of treatment only on infected uman and mosquitoes populations Te control profile is sown in Figure 4(c, ere we see tat te optimal treatment control u 2 is at te upper bound till time t f = 5, wile te optimal insecticide spray u 3 is at te upper bound for 9 days before reducing gradually to te lower bound. Optimal personal protection and insecticide spray. Here, te control on personal protection ( and te spray of insecticide (u 3 are used to optimize te objective function J wile setting te control on treatment u 2. For tis strategy, sown in Figure 5, we observed tat te number of symptomatic uman (I and mosquitoes (I v differs considerably from te uncontrolled case. Figure 5(a, reveals tat symptomatic umans (I is lower in comparison wit te case witout control. Wile Figure 5(b, reveals a similar result of decreased number of infected mosquitoes (I v for te controlled strategy as compared wit te strategy witout control. Te control profile in Figure 5(c sows tat te control on personal protection ( is at upper bound for 6 days, wile insecticide spray (u 3 is at upper bound for t = 1 days before reducing to te lower bound. Optimal personal protection and treatment. Wit tis strategy, te control on personal protection ( and te treatment (u 2 are used to optimize te objective function J wile setting te control on spray of insecticide u 3 to zero. For tis strategy, sown in Figure 6, tere is a significant difference in te I and I v wit optimal strategy compared to I and I v witout control. We observed in Figure

16 16 F. B. AGUSTO, N. MARCUS, K. O. OKOSUN EJDE-212/ , u 2, u , u 2, u 3 Infected Human Infected Mosquitoes (a (b u (c Figure 3. Simulations sowing te effect of insecticide spraying only on infected uman and mosquitoes populations 6(a tat due to te control strategies, te number of symptomatic umans (I decreases as against te increase in te uncontrolled case. A similar decrease is observed in Figure 6(b for infected mosquitoes (I v in te control strategy, wile an increased number is observed for te uncontrolled case resulted. In Figure 6(c, te control profile, te control is at te upper bound for 118 (days and drops gradually until reacing te lower bound, wile control on treatment u 2 starts and remain at upper bound for 12 days before dropping gradually to te lower bound. Te result ere sows tat wit a personal protection coverage of 1% for 118 days and treatment coverage of 1% for 12 (days, te disease incidence will be greatly reduced. Optimal personal protection, treatment and insecticide spray. Here, all tree controls (, u 2 and u 3 are used to optimize te objective function J, wit weigt factors m = 92, n = 2, c = 65, d = 1. For tis strategy in Figure 7, we observed in Figure 7(a and 7(b tat te control strategies resulted in a decrease in te number of symptomatic umans (I and infected mosquitoes (I v as against te increased number of symptomatic umans (I and infected mosquitoes in te uncontrolled case. Te control profile sown in Figure 7(c, sows tat te control is at upper bound for t f = 6 days, wile control u 2, starts ig at about 77% and reduces to te lower bound gradually over time. Te control u 3 on te oter and is at upper bound for about 1 days before reducing to te lower bound.

17 EJDE-212/81 APPLICATION OF OPTIMAL CONTROL , u 2, u , u 2, u 3 Infected Human Infected Mosquitoes (a (b.9 u 2 u Control Profile (c Figure 4. Simulations sowing te effect of treatment and spray of insecticide on infected uman and mosquitoes populations A comparison of all four control strategies in Figures 8(a and 8(b sows tat wile all four strategies lead to a decrease in te number of infected, bot in uman and in mosquitoes. Te control strategy witout treatment resulted in a iger number of infected umans, followed by te strategy witout personal protection. Te strategy witout te spray of insecticide even toug, it gave a better result in reducing te infection in uman, gave a poorer result in reducing te mosquitoes population. Tis result sows tat wit individuals total aderence to effective use of personal protection and spray of insecticide in te population, little treatment efforts will ten be required by te community in te control of te spread of te disease. Spray of insecticide. A scenario wit reducing different fraction of vector population is simulated, te result sows tat te value of p.2 gave te lowest number of susceptible (S v vectors wile p.85 gave te least value of infected (I v vectors, tis is followed by p.6, p.85 and lastly by p = 1 (a case corresponding to no use or ineffective insecticide as expected. Tis as te resultant effect (not depicted ere on total number of vectors susceptible to malaria S v. Wen p.85, te total number of vectors susceptible to malaria, S v is 49, wen p.6, S v = 2, and lastly wen p.2, te total number of susceptible vectors to malaria, S v = 1.

18 18 F. B. AGUSTO, N. MARCUS, K. O. OKOSUN EJDE-212/ , u 2, u , u 2, u 3 Infected Human Infected Mosquitoes (a (b.9 u Control Profile (c Figure 5. Simulations sowing te effect of optimal personal protection and spray of insecticide on infected uman and mosquitoes populations 5.1. Concluding remarks. In tis paper, we presented a malaria model using a deterministic system of differential equations and establised tat te model is locally asymptotically stable wen te associated reproduction number is less tan unity. In te optimal control problem considered, we use one control at a time and te combination of two controls at a time, wile setting te oter(s to zero to investigate and compare te effects of te control strategies on malaria eradication. Tis is different from wat was investigated in [5] were only two control measures of personal protection and treatment were used wile varying te vector-ost contact rate. Our numerical results sows tat te combination of te tree (3 controls, personal protection, treatment and insecticides spray, as te igest impact on te control of te disease. Tis is followed by te combination of treatment and personal protection among te uman population; and lastly by te combination involving te use of personal protection and insecticide use. In communities were resources are scarce, we suggest tat te combination of treatment and personal protection sould be adopted, aving observed from te comparison of all four control strategies in Figure 8, tat tere is no significant difference between tis strategy and te combination of te tree (3 controls. Altoug, our recommendation agrees wit te result obtained by Blayne et al[5], our result owever sows two possible control strategies, eac wit two combinations of control measures tat are sufficient to effectively acieve and maintain interruption of transmission

19 EJDE-212/81 APPLICATION OF OPTIMAL CONTROL , u 2, u , u 2, u 3 Infected Human Infected Mosquitoes (a (b.9 u Control Profile (c Figure 6. Simulations sowing te effect of optimal personal protection and treatment on infected uman and mosquitoes populations of malaria. A result wic addresses te WHO [44] concern about te insufficiency of only one control measure to acieve and maintain interruption of transmission of malaria. Acknowledgments. K. O. Okosun acknowledges, wit tanks, te support from te Sout African Center for Epidemiological Modeling and Analysis Sout Africa (SACEMA. F.B. Agusto conducted part of tis work as a Postdoctoral Fellow at NIMBioS, National Institute for Matematical and Biological Syntesis (NIMBioS is an Institute sponsored by te National Science Foundation, te U.S. Department of Homeland Security, and te U.S. Department of Agriculture troug NSF Award #EF , wit additional support from Te University of Tennessee, Knoxville. References [1] Adams, B. M.; Banks, H. T.; Kwon, H.; Tran Hien, T.; Dynamic multidrug terapies for HIV: Optimal and STI control approaces, Mat. Bio. and Eng., 1 (2 (24, [2] Agusto, F. B.; Gumel, A. B.; Teoretical assessment of avian influenza vaccine. DCDS Series B. 13 (1 (21, [3] Anderson, R. M.; May, R. M.; Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, [4] Ariey, F.; Robert, V.; Te puzzling links between malaria transmission and drug resistance, Trends in Parasitology, 19 (23,

20 2 F. B. AGUSTO, N. MARCUS, K. O. OKOSUN EJDE-212/ , u 2, u 3, u 2, u Infected Human Infected Mosquitoes (a (b.9.8 u 2 u 3.7 Control Profile (c Figure 7. Simulations sowing te effect of optimal personal protection, treatment and spray of insecticide on infected uman and mosquitoes populations 25 Infected Human u 2, u 3,, u 3, u 2 u, u, u 1 2 3, u 2, u 3 Infected Mosquitoes u, u, u 2 3 1, u 3, u 2, u 2, u 3, u 2, u (a (b Figure 8. Simulations sowing te Comparison of te effect of te different control strategies [5] Blayne, K.; Cao, Y.; Kwon, H.; Optimal control of vector-borne diseases: Treatment and Prevention. Discrete and continuous dynamical systems series B, 11 (3 (29., [6] Brown, C. W.; Kaoui, M. E. l.; Novotni, D.; Weber, A.; Algoritmic metods for investigating equilibria in epidemic modeling, Journal of Symbolic Computation, 41 (26, [7] Castillo, C.; Optimal control of an epidemic troug educational campaigns, Electronic Journal of Differential Equations, 26, no. 125, 1 11.

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22 22 F. B. AGUSTO, N. MARCUS, K. O. OKOSUN EJDE-212/81 [34] Ruiz, D.; Poveda, G.; Vlez, I. D. etal; Modelling entomological-climatic interactions of plasmodium falciparum malaria transmission in two colombian endemic-regions: contributions to a national malaria early warning system. Malaria Journal, 5 (66 (26. [35] Serman, I. W.; Malaria: Parasite Biology, Patogenesis, and Protection, AMS Press, Wasington, DC (1998. [36] Smit, D. L.; McKenzie, F. E.; Statics and dynamics of malaria infection in Anopeles mosquitoes. Malaria Journal, 3 (24, 13 doi: / [37] Smit, R. J.; Hove-Musekwa, S. D.; Determining effective spraying periods to control malaria via indoor residual spraying in sub-saaran Africa. J. Applied Matematics and Decision Sciences. Article ID , (28. [38] Snow, R. W.; Omumbo, J.; Malaria, in Diseases and Mortality in Sub-Saaran Africa, D. T. et al Jamison, ed., Te World Bank. (26, [39] Sures, P. S.; Optimal Quarantine programmes for controlling an epidemic spread, Journal Opl. Res. Soc. 29 (3 (1978, [4] Tomé, R. C. A.; Yang, H. M.; Esteva, L.; Optimal control of Aedes aegypti mosquitoes by te sterile insect tecnique and insecticide. Matematical Biosciences, 223 (21, [41] U.S. Census Bureau, International database, 27. [42] Van den Driessce P.; Watmoug J.; Reproduction numbers and sub-tresold endemic equilibria for compartmental models of disease transmission. Matematical Biosciences, 18 (22, [43] WHO; Malaria, fact seets, ttp:// (1998. [44] WHO; Global Malaria Programme: Position Statement on ITNs, (29. [45] Wickwire, K.; A note on te optimal control of carrier - borne epidemic. J. Applied probability, 12 (1975, [46] Yan, X.; Zou, Y.; Li, J.; Optimal quarantine and isolation strategies in epidemics control. World Journal of Modelling and Simulation, 3 (3 (27, [47] Yang, H. M.; A matematical model for malaria transmission relating global warming and local socioeconomic conditions. Rev. Saude Publica, 35 (3 (21, Folasade B. Agusto Department of Matematics and Statistics, Austin Peay State University, Clarksville, Tennessee 3744, USA address: fbagusto@gmail.com Nizar Marcus Department of Matematics and Applied Matematics, University of te Western Cape, Sout Africa address: nmarcus@uwc.ac.za Kazeem O. Okosun Department of Matematics, Vaal University of Tecnology, Private Bag X21, Vanderbijlpark, 19, Sout Africa address: kazeemoare@gmail.com