  This paper is concerned with the well-posedness and optimal control problem
of a reaction-diffusion system for an epidemic
Susceptible-Infected-Recovered-Susceptible (SIRS) mathematical model in which
the dynamics develops in a spatially heterogeneous environment. Using as
control variables the transmission rates $u_{i}$ and $u_{e}$ of contagion
resulting from the contact with both asymptomatic and symptomatic persons,
respectively, we optimize the number of exposed and infected individuals at a
final time $T$ of the controlled evolution of the system. More precisely, we
search for the optimal $u_{i}$ and $u_{e}$ such that the number of infected
plus exposed does not exceed at the final time a threshold value $\Lambda$,
fixed a priori. We prove here the existence of optimal controls in a proper
functional framework and we derive the first-order necessary optimality
conditions in terms of the adjoint variables.
