Optimal Conditions for the Control Problem Associated to a Biomedical Process

This paper considers a mathematical model of infectious disease of SIS type. We will analyze the problem of minimizing the cost of diseases trough medical treatment. Mathematical modeling of this process leads to an optimal control problem with a finite horizon. The necessary conditions for optimality are given. Using the optimality conditions we prove the existence, uniqueness and stability of the steady state for a differential equations system.


THE MODEL
The mathematical model proposed in this paper is the simplest model of an epidemic diseases. It is based on the a simplified version of what is called an SIS model as in [3]. We denote by N(t) the total size of the population in period t. We suppose that the population will be divided up into two groups of people, those that have been infected by the disease and are infective, and those that are susceptible to being infected by the disease. We will label the number of those infected in moment t as I(t) and the number susceptible in moment t as S(t). Therefore we have The name SIS comes from two population groups Susceptible and Infected. Individuals go from being susceptible to a disease to being infected. And then they recover and again become susceptible. The number of individuals treated is denoted by M(t).
The SIS model with stationary populations, normalized to one consists of the following equations From the first equation of the system (2) we obtain S(t) = 1 − I(t), and we have In medical terms β is the transmission coefficient of the disease, λ is the spontaneous rate of recovery for an untreated infected person and δ is the rate of recovery with treatment. Treatment is assumed effective, hence δ > λ . Infection imposes a cost of treatment. The cost of treatment is C(M(t)), where this is interpreted as total social cost of treatment and is assumed to exhibit increasing marginal cost. These costs include materials, facilities and labor used in administering treatment.
Our economic problem is to choose M(t), such that to minimize the cost of treatment, on the finite time, given by

DETERMINATION OF OPTIMALITY CONDITIONS
The economic problem from the previous section leads us to the following mathematical optimization problem P * . The problem P * . To determine (I * , M * ) which minimize the following functional which verifies: where AC([0, T ], [0, 1]) is the class of absolutely continuous function. In our problem P * , I is the state variable and M is a control variable.
In order to solve the minimum problem P * we will transform it into the maximum problem P. The problem P. To determine (I * , M * ) which minimize the following functional where AC([0, T ], [0, 1] is the class of absolutely continuous function. Remark 1. One can prove that the above problems are equivalent.
In what following we will determine the necessary conditions for optimal problem P as in [1], [2], [4]. To do this we will apply the principle of Pontryagin. For this we define the function of Hamilton-Pontryagin given by Proof. Let (I * (t) , M * (t)) an optimal solution for P. The Hamilton function associated to the problem P is From the Pontryagin's principle there exists the adjoint absolutely continuous function µ (t) such thaṫ and Using the transformation µ(t) (t) = e −ρt q (t), the Hamilton function becomes The first and second derivatives of function H with respect to M are Using the property of the cost function we obtain that H is a concave function of M.
Because M * (t) maximizes (17) and H is a concave function of M we have thus Using again the transformation µ (t) = e −ρt q (t) , (15) becomeṡ for every admissible trajectory (I(t), M(t)) of the problem P.
In next theorem, we give the sufficient conditions for the solution of our optimal control problem P. Theorem 5. Let (I * (t), M * (t)) be an admissible trajectory in problem P. If there exists an absolutely continuous function q(t) such that for all t, the following conditions are satisfied and the transversality condition q(T ) = 0 holds, then (I * (t), M * (t)) is an optimal trajectory for the problem P. Proof. Let (I * (t), M * (t)) be an arbitrary admissible trajectory for P. We denote In the following, we simplify our notations and put I * Using the transformation q(t) = e ρt µ(t) in (13)