Optimality of vaccination for an SIR epidemic with an ICU constraint

TL;DR

This paper analyzes optimal vaccination strategies in SIR epidemic models with ICU capacity constraints, showing that optimal controls are bang-bang with at most one switch, and providing conditions for vaccination implementation or cessation.

Contribution

It characterizes the structure of optimal vaccination controls under ICU constraints using Pontryagin's theorem and extends the analysis to infinite horizon scenarios.

Findings

Optimal controls are bang-bang with at most one switch.

Maximum vaccination effort is applied initially until ICU constraints are met.

Conditions for implementing or stopping vaccination are explicitly characterized.

Abstract

This paper studies an optimal control problem for a class of SIR epidemic models, in scenarios in which the infected population is constrained to be lower than a critical threshold imposed by the ICU (intensive care unit) capacity. The vaccination effort possibly imposed by the health-care deciders is classically modeled by a control input affecting the epidemic dynamic. After a preliminary viability analysis the existence of optimal controls is established, and their structure is characterized by using a state-constrained version of Pontryagin's theorem. The resulting optimal controls necessarily have a bang-bang regime with at most one switch. More precisely, the optimal strategies impose the maximum-allowed vaccination effort in an initial period of time, which can cease only once the ICU constraint can be satisfied without further vaccination. The switching times are characterized in order to identify conditions under which vaccination should be implemented or halted. The uniqueness of the optimal control is also discussed. Numerical examples illustrate our theoretical results and the corresponding optimal strategies. The analysis is eventually extended to the infinite horizon by $\Gamma$-convergence arguments.