  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 and determines the
conditions of optimality to reduce the concentration of the infected population
in certain spatial areas.
