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 = [S_{H}, E^{s}_{H}, E^{l}_{H}, I_{H}, S_{M}, I_{M}]^{T},
U = [u_{1}, u_{2}]^{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 (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):
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 = {(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 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
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 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 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 also
x(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 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 λ_{1}(t), λ_{2}(t), λ_{3}(t), λ_{4}(t), λ_{5}(t) and λ_{6}(t). and λ_{6}(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).