2. Materials and Methods
2.1. Model description: optimal control
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 (SH), short term exposed (EsH), long term exposed (ElH), and infectious (IH). Mosquito population M(t) is also divided into two classes: susceptible (SM), and infectious (IM). 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.
Table 2.The parameter values for the model
Table 2.
| Parameter |
Value |
|
| bm |
0.7949 [0.1,1.5] |
| bmh |
0.5 |
| βhm |
0.5 |
| σ |
0.3 [0.25,0.5] |
| r |
0.07 [0.01,0.5] |
| p |
0.25 |
| Tsh |
1/25.9 |
| Tlh |
1/360.3 |
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)│ui(t) is piecewise continuous on [0, T], 0 ≤ui(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.)
Figure 1.Optimal controls when B1= B2= 1000 with high mosquito population.
2.2. Numerical simulation
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).
2.3. Results
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.
Figure 2.Optimal controls when B1 = 10, B2 = 1000 with high mosquito population.
Figure 3.Optimal controls when B1= 1000; B2 = 10 with high mosquito population.
Figure 4.Optimal controls when B1= B2 = 1000 with low mosquito population.
Figure 5.Optimal controls when B1= 10, B2 = 1000 with low mosquito population.
Figure 6.Optimal controls when B1= 1000; B2 = 10 with low mosquito population.
3. Discussion and Conclusions
After 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 time-dependent 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 mosquito-reduction 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 solution
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 = [SH, EsH, ElH, IH, SM, IM]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) : = SH(t)+EsH(t) + ElH(t)+IH(t).
and
M(t) : =SM(t) + IM (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.
Appendix B. Analysis of optimal control control problem
We are to prove the existence of optimal control pairs for the system (1). Firstly, We set control space
U = {(u1 , u2) │ui is piecewise continuous on [0, T],0 ≤ ui(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 gi is the right hand side of the differential equation of the ith state variable and also
x(t) = (SH, EsH, ElH, IH, SM, IM), 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 SH, EsH, ElH, IH, SM and IM, 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).
