The Evolutionary Dynamics of Stochastic Epidemic Model with Nonlinear Incidence Rate

A stochastic SIRS epidemic model with nonlinear incidence rate and varying population size is formulated to investigate the effect of stochastic environmental variability on inter-pandemic transmission dynamics of influenza A. Sufficient conditions for extinction and persistence of the disease are established. In the case of persistence, the existence of endemic stationary distribution is proved and the distance between stochastic solutions and the endemic equilibrium of the corresponding deterministic system in the time mean sense is estimated. Based on realistic parameters of influenza A in humans, numerical simulations have been performed to verify/extend our analytical results. It is found that: (i) the deterministic threshold of the influenza A extinction \(R^{S}_{0}\) may exist and the threshold parameter will be overestimated in case of neglecting the impaction of environmental noises; (ii) the presence of environmental noises is capable of supporting the irregular recurrence of influenza epidemic, and the average level of the number of infected individuals I(t) always decreases with the increase in noise intensity; and (iii) if \(R^{S}_{0}>1\), the volatility of I(t) increases with the increase of noise intensity, while the volatility of I(t) decreases with the increase in noise intensity if \(R^{S}_{0}<1\).