Malwina Luczak: Extinction Times in the Stochastic Logistic Epidemic

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.