On a minimum eradication time for the SIR model with time-dependent coefficients

TL;DR

This paper investigates the minimum eradication time in a time-varying SIR epidemic model with vaccination, addressing challenges due to oscillations in infection levels and establishing mathematical properties of the associated control problem.

Contribution

It introduces two definitions of eradication time for the time-dependent SIR model and proves their equivalence under certain thresholds, along with analyzing the Hamilton-Jacobi equation governing the problem.

Findings

Two definitions of eradication time coincide under suitable thresholds.

The minimum eradication time satisfies a well-posed Hamilton-Jacobi equation.

The value function is locally semiconcave under specific conditions.

Abstract

We study the minimum eradication time problem for controlled Susceptible-Infected-Recovered (SIR) epidemic models that incorporate vaccination control and time-varying infected and recovery rates. Unlike the SIR model with constant rates, the time-varying model is more delicate as the number of infectious individuals can oscillate, which causes ambiguity for the definition of the eradication time. We accordingly introduce two definitions that describe the minimum eradication time, and we prove that for a suitable choice of the threshold, the two definitions coincide. We also study the well-posedness of time-dependent Hamilton-Jacobi equation that the minimum eradication time satisfies in the viscosity sense and verify that the value function is locally semiconcave under certain conditions.