An eradication time problem for the SIR model

TL;DR

This paper formulates a time-optimal control problem for the SIR epidemic model with vaccination, deriving conditions for optimal vaccination strategies to minimize infection duration.

Contribution

It introduces a novel control framework for the SIR model, providing necessary and sufficient conditions for optimal vaccination timing.

Findings

Derived dynamic programming equations for the control problem

Established conditions for optimal vaccination strategies

Provided a mathematical framework for epidemic control timing

Abstract

We consider a susceptible, infected, and recovered infectious disease model which incorporates a vaccination rate. In particular, we study the problem of choosing the vaccination rate in order to reduce the number of infected individuals to a given threshold as quickly as possible. This is naturally a problem of time-optimal control. We interpret the optimal time as a solution of two dynamic programming equations and give necessary and sufficient conditions for a vaccination rate to be optimal.