  Consider an epidemic model with a constant flux of susceptibles, in a
situation where the corresponding deterministic epidemic model has a unique
stable endemic equilibrium. For 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 paper, we discuss how the Central Limit
Theorem, Moderate and Large Deviations allow us to give estimates of the
extinction time of the epidemic, depending upon the size of the population.
