The diffusion identification in a SIS reaction-diffusion system

This article is concerned with the determination of the diffusion matrix in the reaction-diffusion mathematical model arising from the spread of an epidemic. The mathematical model that we consider is a susceptible-infected-susceptible model with diffusion, which was deduced by assuming the following hypotheses: The total population can be partitioned into susceptible and infected individuals; a healthy susceptible individual becomes infected through contact with an infected individual; there is no immunity, and infected individuals can become susceptible again; the spread of epidemics arises in a spatially heterogeneous environment; the susceptible and infected individuals implement strategies to avoid each other by staying away. The spread of the dynamics is governed by an initial boundary value problem for a reaction-diffusion system, where the model unknowns are the densities of susceptible and infected individuals and the boundary condition models the fact that there is neither emigration nor immigration through their boundary. The reaction consists of two terms modeling disease transmission and infection recovery, and the diffusion is a space-dependent full diffusion matrix. The determination of the diffusion matrix was conducted by considering that we have experimental data on the infective and susceptible densities at some fixed time and in the overall domain where the population lives. We reformulated the identification problem as an optimal control problem where the cost function is a regularized least squares function. The fundamental contributions of this article are the following: The existence of at least one solution to the optimization problem or, equivalently, the diffusion identification problem; the introduction of first-order necessary optimality conditions; and the necessary conditions that imply a local uniqueness result of the inverse problem. In addition, we considered two numerical examples for the case of parameter identification.