Threshold Dynamics and Probability Density Function of a Stochastic Avian Influenza Epidemic Model with Nonlinear Incidence Rate and Psychological Effect

In this paper, we examine a stochastic avian influenza model with a nonlinear incidence rate within avian populations and the psychological effect within the human population, where susceptible humans reduce their contact with infected avians as the number of infected humans increases. For the deterministic model, the basic reproduction number R, possible equilibria, and related asymptotic stability are first studied. Then, for the stochastic model, we obtain a critical value R0S, which can determine the persistence and extinction of avian influenza. It is theoretically proved that the stochastic model has a unique stationary distribution ϖ(·) if R0S>1, but the disease will go to extinction when R0S<1. Taking stochasticity into account, a quasi-endemic equilibrium T¯ ^∗ related to the endemic equilibrium of the deterministic model is defined. We develop an important lemma for solving the special Fokker–Planck equation and derive the explicit expression of the density function of the distribution ϖ(·) around the equilibrium T¯ ^∗. Numerical simulations verify our theoretical results, and we study the impact of noise and the psychological effect on the transmission dynamics of avian influenza.