To 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 – u₁(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 u₁(t), where ρ > 0 is a rate constant. In the human population, the associated force of infection is reduced by a factor of 1 – u₂(t), where u₂(t) measures the level of successful prevention efforts. In fact, the control u₂(t) represents the use of prevention measures to minimize mosquito-human 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 anti-malaria control techniques while minimizing the cost of implementation of such control measures.

An optimal control problem with the objective cost functional can be given by

subject to the state system given by (1).

The goal is to minimize the infected human populations and the cost of implementing the control. In the objective cost functional, the quantities A , B₁ and B₂ represent the weight constants of infected human, for mosquito control and prevention of mosquito-human contacts, respectively. The costs associated with mosquito control and prevention of mosquito-human contacts are described in the terms

and

respectively.

Optimal control functions

need to be found such that

subject to the system of equations given by (1), where U = [(u₁,u₂)│u_i(t) is piecewise continuous on [0, T], 0 ≤u_i(t) ≤ 1, i = 1, 2} is the control set:

Such optimal control functions

exist, and the optimality system can be derived. (See Appendix B for detail.)

Using the forward-backward 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 u₁ and u₂ were chosen to be 0.8. The weight in the objective functional is A₁ = 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 mosquito-human 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 mosquito-human contacts.

It is also worth noting that different initial mosquitoes populations do not have effect on the optimal strategies (Figures 4 - 6).

If the cost of reducing the reproduction rate of the mosquito population is more than that of prevention measures to minimize mosquito-human contacts, the u₂ control needs to be taken for a longer time, comparing the other situations (Figures 1 to 3). In that situation, full effort for u₂ 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 u₁ and u₂ are needed for at least some of the time.

We 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 = [u₁, u₂]^T and

So 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 follows

respectively. To show the existence of solution of the system (1), we have to prove that F (X ,U). satisfy a Lipschitz condition. Let

H(t) : = S_H(t)+E^s_H(t) + E^l_H(t)+I_H(t).

and

M(t) : =S_M(t) + I_M (t)

But

For any given pairs (X₁, U), (X₂, V)

U = (u₁ , u₂)^T , V = (v₁, v₂)^T, we obtain,

We estimate the 4 terms in the right side of (i):

and

where

Hence, the system (1) satisfy all conditions of the Picard-Lindelof Theorem ([12,13]) and also the function F(X, U) is continuously differentiable. Therefore, the system (1) have a unique solution.

We are to prove the existence of optimal control pairs for the system (1). Firstly, We set control space

U = {(u₁ , u₂) │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 control

and

such that

subject 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 u₁ = u₂ = 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 u₁ and u₂ have to satisfy. The integrand in the functional,

is convex function on the control u₁ and u₂. Also we can easily check that there exist a constant ρ > 1, a numbers ω₁ ≥ 0 and ω₂ > 0 such that

which completes the existence of an optimal control.To find the optimal solution we apply Pontryagin’s Maximum Principle ([15-17]) 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 u₁ and u₂. 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 also

x(t) = (S_H, E^s_H, E^l_H, I_H, S_M, I_M), u(t) = (u₁(t), u₂(t)) and λ(t) = (λ₁(t), λ₂(t), (λ₃(t), λ₄(t), λ₅(t), λ₆(t)).

By applying Pontryagin’s Maximum Principle([18]) if (x^*(t), u^*(t)) is an optimal control, then there exists a non-trivial vector function λ(t) satisfying the following equalities:

If follows from the derivation above

Now, we apply the necessary conditions to the Hamiltonian

Theorem 3. Let

and

be optimal state solutions with associated optimal control variables

for the optimal control problem (1) and (2). Then, there exist adjoint variables λ₁(t), λ₂(t), λ₃(t), λ₄(t), λ₅(t) and λ₆(t). and λ₆(t) that satisfy

with transversality conditions(or boundary conditions)

Furthermore, the optimal control

and

are given by

Proof. To determine the adjoint equations and the transversality conditions, we use the Hamiltonian (4). From setting

and also differentiating the Hamiltonian (4) with respect to S_H, E^s_H, E^l_H, I_H, S_M and I_M, we obtain

By the optimality conditions, we have

Using the property of the control space, we obtain the characterizations of

and

in (6). From the fixed of start time, we have transvesality conditions (5).