In this paper, we introduce the functional framework and the necessary conditions for the well-posedness of an inverse problem arising from 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 final time. The unknowns of the inverse problem are the 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 the 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.
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