Analysis of the deterministic and stochastic SIRS epidemic models with nonlinear incidence

• The deterministic and stochastic SIRS epidemic models are studied.
• If R0≤1R0≤1, then the disease-free equilibrium is globally asymptotically stable.
• If R0>1R0>1, then stochastic perturbations may lead the disease to extinction.
• We show that there is a unique stationary distribution to stochastic system.

In this paper, the deterministic and stochastic SIRS epidemic models with nonlinear incidence are introduced and investigated. For deterministic system, the basic reproductive number R0R0 is obtained. Furthermore, if R0≤1R0≤1, then the disease-free equilibrium is globally asymptotically stable and if R0>1R0>1, then there is a unique endemic equilibrium which is globally asymptotically stable. For stochastic system, to begin with, we verify that there is a unique global positive solution starting from the positive initial value. Then when R0>1R0>1, we prove that stochastic perturbations may lead the disease to extinction in scenarios where the deterministic system is persistent. When R0≤1R0≤1, a result on fluctuation of the solution around the disease-free equilibrium of deterministic model is obtained under appropriate conditions. At last, if the intensity of the white noise is sufficiently small and R0>1R0>1, then there is a unique stationary distribution to stochastic system.