Pre-Publicación 2026-10
Aníbal Coronel, Fernando Huancas, Mauricio Sepúlveda:
Identification of a power-like reaction term in a reaction-diffusion SIS model
Abstract:
In this paper, we introduce the necessary conditions for the reaction term determination in a class of reaction-diffusion systems, based on knowledge of an approximation of the state variables at the end of the processes. The system considered is a generalization of the susceptible-infected-susceptible (SIS) model of disease transmission, assuming spatial displacement of individuals. We consider that the reaction term is defined by a power incidence function, and the coefficients are space-time-dependent functions modeling the disease and recovery rates. We introduce a formulation of the inverse problem as a constrained optimization problem for an appropriate cost functional. The main results of the paper are the existence of a minimizer for the cost functional; the definition of a first-order necessary optimality condition; the proofs of stability of the state and adjoint equations with respect to the coefficients, the power of the incidence, and the observations; and the introduction of a biological consistent numerical approximation of the optimal control problem. Moreover, we present some numerical examples of the parameter identification problem.
