Optimal control for a SIR epidemic model with limited quarantine

TL;DR

This paper investigates optimal control strategies for a SIR epidemic model with constraints on quarantine duration and intensity, deriving conditions for optimal interventions and analyzing their structure through both theoretical and numerical methods.

Contribution

It introduces a framework for optimal quarantine control in SIR models with bounded duration and cost considerations, including bang-bang solutions and numerical analysis.

Findings

Optimal control is bang-bang when intervention cost is linear.

Characterization of quarantine start and end times.

Numerical results support theoretical analysis.

Abstract

We study first order necessary conditions for an optimal control problem of a Susceptible-Infected-Recovered (SIR) model with limitations on the duration of the quarantine. The control is done by means of the reproduction number, i.e., the number of secondary infections produced by a primary infection, which represents an external intervention that we assume time-dependent. Moreover, the control function can only be applied over a finite time interval, and the duration of the most strict quarantine (smallest possible reproduction number) is also bounded. We consider a maximization problem where the cost functional has two terms: one is the number of susceptible individuals in the long-term and the other depends on the cost of interventions. When the intervention term is linear with respect to the control, we obtain that the optimal solution is bang-bang, and we characterize the times to begin and end the strict quarantine. In the general case, when the cost functional includes the term that measures the intervention cost, we analyze the optimality of controls through numerical computations.