Extinction Times in the Stochastic Logistic Epidemic

Abstract:

The stochastic logistic process is a well-known birth-and-death process, often used to model the spread of an epidemic within a population of size N.  We survey some of the known results about the time to extinction for this model.  Our focus is on new results for the "subcritical" regime, where the recovery rate exceeds the infection rate by more than N^{-1/2}, and the epidemic dies out rapidly with high probability.  Previously, only a first-order estimate for the mean extinction time of the epidemic was known, even in the case where the recovery rate and infection rate are fixed constants: we derive precise asymptotics for the distribution of the extinction time throughout the subcritical regime.  In proving our results, we illustrate a technique for approximating certain types of Markov chain by differential equations over long time periods.

This is joint work with Graham Brightwell.