Infinite horizon optimal control of a SIR epidemic under an ICU constraint

TL;DR

This paper rigorously analyzes an infinite horizon optimal control problem for a SIR epidemic model with ICU capacity constraints, using advanced mathematical techniques and applying findings to Covid-19 data in Italy.

Contribution

It introduces a novel mathematical framework combining asymptotic, viability, and $\Gamma$-convergence analyses to characterize optimal controls under ICU constraints.

Findings

Optimal control strategies respecting ICU limits are characterized.

Numerical simulations demonstrate the model's applicability to Covid-19 data.

The approach provides insights into sustainable epidemic management.

Abstract

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 $\Gamma$-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 numeric simulations are given according to the available data on Covid-19 epidemic in Italy.