The threshold of a stochastic SIRS epidemic model in a population with varying size

In this paper, a stochastic susceptible-infected-removed-susceptible (SIRS) epidemic model in a population with varying size is discussed. A new threshold ~R0 is identified which determines the outcome of the disease. When the noise is small, if ~R0 < 1, the infected proportion of the population disappears, so the disease dies out, whereas if ~R0 > 1, the infected proportion persists in the mean and we derive that the disease is endemic. Furthermore, when R0 > 1 and subject to a condition on some of the model parameters, we show that the solution of the stochastic model oscillates around the endemic equilibrium of the corresponding deterministic system with threshold R0, and the intensity of fluctuation is proportional to that of the white noise. On the other hand, when the noise is large, we find that a large noise intensity has the effect of suppressing the epidemic, so that it dies out. These results are illustrated by computer simulations