1. IntroductionMalaria is a mosquitoborne infectious disease caused by a eukaryotic protist of the genus Plasmodium. Malaria is naturally transmitted by the bite of a female Anopheles mosquito. The primary vector in Korea is reported to be A sinensis. Since the reemergence of Plasmodium vivax malaria in 1993[1,2], it has been endemic and continues to cause extensive morbidity in Korea, despite the huge efforts invested to control it.The first mathematical malaria model proposed by Ross [3], was subsequently modified by MacDonald, which has influenced both the modeling and the application of control strategies to malaria [4]. Recently, the optimal control theory has been applied to malaria Okosun et al [5], and to vectorborne disease Lashari
Table 1.
2.1. Model description: optimal controlTo construct a deterministic model for P vivax malaria transmission with control terms, the model of Nah et al [11] was modified and optimal control terms inspired by the model of Blayneh et al [7] were added as follows:In the model, human population H(t) is divided into four classes: susceptible (S_{H}), short term exposed (E^{s}_{H}), long term exposed (E^{l}_{H}), and infectious (I_{H}). Mosquito population M(t) is also divided into two classes: susceptible (S_{M}), and infectious (I_{M}). Note that the mosquito population M(t) is not constant while human population H(t) is constant.The factor of 1 – u_{1}(t) reduces the reproduction rate of the mosquito population. It is assumed that the mortality rate of mosquitoes (susceptible and infected) increases at a rate proportional to u_{1}(t), where ρ > 0 is a rate constant. In the human population, the associated force of infection is reduced by a factor of 1 – u_{2}(t), where u_{2}(t) measures the level of successful prevention efforts. In fact, the control u_{2}(t) represents the use of prevention measures to minimize mosquitohuman contacts. Table 1 lists detailed descriptions of the parameters. The system (1) has a unique solution set. (See Appendix A for detail.)An optimal control problem can now be formulated for the transmission dynamics of P vivax malaria transmission in Korea. The goal is to show that it is possible to implement time dependent antimalaria control techniques while minimizing the cost of implementation of such control measures.
Table 2.
Parameter  Value 



b_{m}  0.7949 [0.1,1.5] 
b_{mh}  0.5 
β_{hm}  0.5 
σ  0.3 [0.25,0.5] 
r  0.07 [0.01,0.5] 
p  0.25 
T^{s}_{h}  1/25.9 
T^{l}_{h}  1/360.3 
2.2. Numerical simulationUsing the forwardbackward sweep method, the optimality system was solved numerically. This consists of 12 ordinary differential equations from the state and adjoint equations, coupled with the two controls. In choosing upper bounds for the controls, it was supposed that the two controls would not be 100% effective, so the upper bounds of u_{1} and u_{2} were chosen to be 0.8. The weight in the objective functional is A_{1} = 1000. The parameters in Table 2 were adopted from other articles [11] and used for our simulation.We simulate the model in different scenarios. Figure 1 depicts scenarios for the state variables of the model for the case when the cost is the same for the two controls. Figure 2 depicts scenarios for the state variables of the model for the case when the cost of prevention measures to minimize mosquitohuman contact is more expensive than the cost of reducing the reproduction rate of the mosquito population. Figure 3 depicts scenarios for the state variables of the model for the case when the cost of reducing the reproduction rate of the mosquito population are more expensive than the cost of prevention measures to minimize mosquitohuman contacts.It is also worth noting that different initial mosquitoes populations do not have effect on the optimal strategies (Figures 4  6).
2.3. ResultsIf the cost of reducing the reproduction rate of the mosquito population is more than that of prevention measures to minimize mosquitohuman contacts, the u_{2} control needs to be taken for a longer time, comparing the other situations (Figures 1 to 3). In that situation, full effort for u_{2} is needed after the high peak of infected human population.On the other hand, Figures 4 to 6 suggest that even though the mosquito population is not so high in initial point, full efforts for u_{1} and u_{2} are needed for at least some of the time.
3. Discussion and ConclusionsAfter 1993’s reemergence of malaria, the endemicity of P vivax malaria is becoming a growing concern in South Korea. Public health advisories were subsequently issued to apply community mosquito control and personal protection.The purpose of this work is to suggest optimal control strategies of P vivax malaria in different scenarios. In all cases, optimal control programs lead effectively reduce the number of infectious individuals. We have used a deterministic model with timedependent parameters to develop the transmission dynamics of P vivax malaria in Korea. For numerical simulations, most parameters were adopted from other articles [11].Mathematical model and numerical simulations suggest that the use of mosquitoreduction strategies is more effective than personal protection in some cases but not always. Public health authorities should choose the proper control strategy where their situation lies in the scenarios discussed in the Results section.
Appendix A. The existence and uniqueness of solutionWe consider system (1). We obtain the existence and uniqueness of solution. In here we are given a suitable control set.Theorem 1. The system (1) with any initial condition has a unique solution.Proof. We can rewrite (1) as :where X = [S_{H}, E^{s}_{H}, E^{l}_{H}, I_{H}, S_{M}, I_{M}]^{T},U = [u_{1}, u_{2}]^{T} andSo let G(X,U) = AX+F (X,U). Defined matrix A is a linear. So A is a bounded operator. Define a matrix norm and a vector norm as followsrespectively. To show the existence of solution of the system (1), we have to prove that F (X ,U). satisfy a Lipschitz condition. LetH(t) : = S_{H}(t)+E^{s}_{H}(t) + E^{l}_{H}(t)+I_{H}(t).andM(t) : =S_{M}(t) + I_{M} (t)ButFor any given pairs (X_{1}, U), (X_{2}, V)U = (u_{1} , u_{2})^{T} , V = (v_{1}, v_{2})^{T}, we obtain,We estimate the 4 terms in the right side of (i):andwhereHence, the system (1) satisfy all conditions of the PicardLindelof Theorem ([12,13]) and also the function F(X, U) is continuously differentiable. Therefore, the system (1) have a unique solution.
Appendix B. Analysis of optimal control control problemWe are to prove the existence of optimal control pairs for the system (1). Firstly, We set control spaceU = {(u_{1} , u_{2}) │u_{i} is piecewise continuous on [0, T],0 ≤ u_{i}(t) ≤ 1, i = 1, 2}.We consider an optimal control problem to minimize the objective functional:Theorem 2. There exist an optimal controlandsuch thatsubject to the control system (1) with initial conditions. Proof. To prove the existence of an optimal control pairs we use the result in [14]. The set of control and corresponding state variables is a nonempty. Because for each control pairs we have proved in the Theorem 1 that there exists corresponding state solutions. And also it is ok when the control u_{1} = u_{2} = 0. Note that the control and the state variables are nonnegative values. The control space U is close and convex by definition. In the minimization problem, the convexity of the objective functional in u_{1} and u_{2} have to satisfy. The integrand in the functional,is convex function on the control u_{1} and u_{2}. Also we can easily check that there exist a constant ρ > 1, a numbers ω_{1} ≥ 0 and ω_{2} > 0 such thatwhich completes the existence of an optimal control.To find the optimal solution we apply Pontryagin’s Maximum Principle ([1517]) to the constrained control problem, then the principle converts (1), (2) and (3) in to a problem of minimizing pointwise a Hamiltonian,with respect to u_{1} and u_{2}. The Hamiltonian for our problem is the integrand of the objective functional coupled with the six right hand sides of the state equations:where g_{i} is the right hand side of the differential equation of the ith state variable and alsox(t) = (S_{H}, E^{s}_{H}, E^{l}_{H}, I_{H}, S_{M}, I_{M}), u(t) = (u_{1}(t), u_{2}(t)) and λ(t) = (λ_{1}(t), λ_{2}(t), (λ_{3}(t), λ_{4}(t), λ_{5}(t), λ_{6}(t)).By applying Pontryagin’s Maximum Principle([18]) if (x^{*}(t), u^{*}(t)) is an optimal control, then there exists a nontrivial vector function λ(t) satisfying the following equalities:If follows from the derivation aboveNow, we apply the necessary conditions to the HamiltonianTheorem 3. Letandbe optimal state solutions with associated optimal control variablesfor the optimal control problem (1) and (2). Then, there exist adjoint variables λ_{1}(t), λ_{2}(t), λ_{3}(t), λ_{4}(t), λ_{5}(t) and λ_{6}(t). and λ_{6}(t) that satisfywith transversality conditions(or boundary conditions)Furthermore, the optimal controlandare given byProof. To determine the adjoint equations and the transversality conditions, we use the Hamiltonian (4). From settingand also differentiating the Hamiltonian (4) with respect to S_{H}, E^{s}_{H}, E^{l}_{H}, I_{H}, S_{M} and I_{M}, we obtainBy the optimality conditions, we haveUsing the property of the control space, we obtain the characterizations ofandin (6). From the fixed of start time, we have transvesality conditions (5).