Threshold analysis of a stochastic epidemic model with delay and temporary immunity

Abstract

A stochastic susceptible-infected-recovered model is formulated and investigated
when the temporary immunity is fixed for the population in this paper. The existence and
uniqueness of the global positive solution has been checked with probability one for any initial
value. And the sufficient conditions for the extinction and the persistence of the stochastic
epidemic model with temporary immunity are derived by constructing Lyapunov functions and
the generalized Ito’s formula, where the threshold of the persistence does not depend on the
temporary immunity, while the densities of the infected and recovered are obviously dependent on
the temporary immunity when given a perturbation. Illustrative examples and simulations show
that the perturbations make the properties of the stochastic epidemic model different from the
deterministic one.