Optimal control of a reaction–diffusion model related to the spread of COVID-19

Abstract

This paper is concerned with the well-posedness and optimal control problem of a reaction–diffusion system for an epidemic susceptible–exposed–infected–recovered–susceptible mathematical model in which the dynamics develops in a spatially heterogeneous environment. Using as control variables the transmission rates [Formula: see text] and [Formula: see text] 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 [Formula: see text] of the controlled evolution of the system. More precisely, we search for the optimal [Formula: see text] and [Formula: see text] such that the number of infected plus exposed does not exceed at the final time a threshold value [Formula: see text], 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.