Global Solution and Optimal Control of an Epidemic Propagation with a Heterogeneous Diffusion

In this paper, we explore the solvability and the optimal control problem for a compartmental model based on reaction–diffusion partial differential equations describing a transmissible disease. The nonlinear model takes into account the disease spreading due to the human social diffusion, under a dynamic heterogeneity in infection risk. The analysis of the resulting system provides the existence proof for a global solution. Moreover, we discuss the conditions of optimality, which are of some help to reduce the concentration of the infected population in certain spatial areas.