Mathematical analysis of an inverse problem for a reaction-diffusion system originated in epidemiology

In this article we focus our interest on the study of an inverse problem arising in the mathematical modeling of disease transmission. The mathematical model is given by an initial boundary value problem for a reaction diffusion system. Meanwhile, the inverse problem consists in the determination of the disease and recovery transmission rates from observed measurement of the direct problem solution at some fixed time. The unknowns of the inverse problem are coefficients of the reaction term. We formulate the inverse problem as an optimization problem for an appropriate cost functional. Then, the existence of solutions of the inverse problem is deduced by proving the existence of a minimizer for the cost functional. We establish the uniqueness of identification problem. The uniqueness is a consequence of the first order necessary optimality condition and a stability of the inverse problem unknowns with respect to the observations. Moreover, we develop a numerical approximation and simulations of the inverse problem.