Identification of space distributed coefficients in an indirectly transmitted diseases model

Abstract

We consider the coefficients identification problem in a mathematical model for indirect transmission of a disease between two independent host populations living in two non-coincident spatial domains. The direct problem is given by an initial boundary value problem for a set of seven differential equations: a single equation for the dynamics of propagation of the contaminant and six equations governing the dynamics of disease in each host population under the susceptible-infected-removed approach. The different rates of disease transmission are space dependent functions and are the coefficients in the reaction terms. The identification problem consists of the determination of the coefficients in the reaction terms from an observation of the state variables at the final time of the process. We apply a methodology based on optimization with partial differential equations as constraints. We reformulate the inverse problem as an optimization problem for an appropriate cost function. Our main results are: the proof of existence of solutions for the optimization problem, the introduction of a necessary optimality conditions, the stability of direct problem solution with respect to the unknown coefficients, the stability of the adjoint system solution with respect to the unknown coefficients and the observations, and the uniqueness up to an additive constant of the identification problem.