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).