Optimal Control of a SIR Epidemic With ICU Constraints and Target Objectives

TL;DR

This paper rigorously analyzes an optimal control problem for SIR epidemic models, incorporating ICU capacity constraints and immunity targets, deriving explicit feedback controls and illustrating their behavior through numerical simulations.

Contribution

It provides a comprehensive mathematical analysis of Pontryagin extremals and explicit feedback controls for SIR models with ICU and immunity constraints, using viability tools.

Findings

Explicit closed-loop feedback controls derived

Characterization of different zones of interest in the control problem

Numerical illustrations demonstrating the control strategies

Abstract

The aim of this paper is to provide a rigorous mathematical analysis of an optimal control problem with SIR dynamics. The main feature of our study is the presence of state constraints (related to intensive care units ICU capacity) and strict target objectives (related to the immunity threshold). The first class of results provides a comprehensive description of different zones of interest using viability tools. The second achievement is a thorough mathematical analysis of Pontryagin extremals for the aforementioned problem allowing to obtain an explicit closed-loop feedback optimal control. All our theoretical results are numerically illustrated for a further understanding of the geometrical features and scenarios.