  In this paper we introduce the functional framework and the necessary
conditions for the well-posedness of an inverse problem arising in the
mathematical modeling of disease transmission. The direct problem is given by
an initial boundary value problem for a reaction diffusion system. The inverse
problem consists in the determination of the disease and recovery transmission
rates from observed measurement of the direct problem solution at the end 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.
Moreover, we establish the uniqueness up an additive constant 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.
