Infinite Horizon Optimal Control of a SIR Epidemic Under an ICU Constraint

The aim of this paper is to provide a rigorous mathematical analysis of an optimal control problem of a SIR epidemic on an infinite horizon. A state constraint related to intensive care units (ICU) capacity is imposed and the objective functional linearly depends on the state and the control. After preliminary asymptotic and viability analyses, a Γ-convergence argument is developed to reduce the problem to a finite horizon allowing to use a state constrained version of Pontryagin's theorem to characterize the structure of the optimal controls. Illustrating examples and numerical simulations are given according to the available data on Covid-19 epidemic in Italy.