Deviations From the Law of Large Numbers and Extinction of an Endemic Disease

Consider an endemic disease, which corresponds to an epidemic model with a constant flux of susceptibles, in a situation where the corresponding deterministic epidemic model has a unique stable endemic equilibrium. If we consider the associated stochastic model, whose law of large numbers limit is the deterministic model, the disease free equilibrium is an absorbing state, which is reached soon or later by the process. However, for a large population size, i.e. when the stochastic model is close to its deterministic limit, the time needed for the stochastic perturbations to stop the epidemic may be enormous. In this presentation, we discuss how the Central Limit Theorem, Moderate and Large Deviations allow us to try to estimate the extinction time of the epidemic.