Extinction time for the weaker of two competing SIS epidemics

TL;DR

This paper analyzes the extinction time of the weaker virus strain in a stochastic SIS epidemic model with two competing strains, providing precise asymptotic results and a novel fluid limit approximation approach.

Contribution

It introduces a new method for fluid limit approximation near stable fixed points in Markov chains and derives detailed asymptotic results for strain extinction times.

Findings

Extinction time of weaker strain is asymptotically characterized.

New fluid limit approximation method near stable fixed points.

Results extend to cases where reproductive ratios converge as N increases.

Abstract

We consider a simple stochastic model for the spread of a disease caused by two virus strains in a closed homogeneously mixing population of size N. The spread of each strain in the absence of the other one is described by the stochastic logistic SIS epidemic process, and we assume that there is perfect cross-immunity between the two strains, that is, individuals infected by one are temporarily immune to re-infections and infections by the other. For the case where one strain has a strictly larger basic reproductive ratio than the other, and the stronger strain on its own is supercritical (that is, its basic reproductive ratio is larger than 1), we derive precise asymptotic results for the distribution of the time when the weaker strain disappears from the population, that is, its extinction time. We further extend our results to certain parameter values where the difference between the two reproductive ratios may tend to 0 as $N \to \infty$. In proving our results, we illustrate a new approach to a fluid limit approximation for a sequence of Markov chains in the vicinity of a stable fixed point of the limit.