Extinction time for the weaker of two competing SIS epidemics

Abstract

We consider a simple Markov 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 SIS logistic 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 is strictly stronger than the other, and the stronger strain on its own is supercritical, we derive precise asymptotic results for the distribution of the time when the weaker strain disappears from the population. We further extend our results to certain parameter values where the difference between the basic reproductive ratios of the two strains may tend to 0 as $N\to\infty$. In our proofs, 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 differential equation, valid over long time intervals.