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 R̃₀ is identified which determines the outcome of the disease. When the noise is small, ifR̃₀1, 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.