Threshold dynamics of deterministic and stochastic avian influenza epidemic models with weak Allee effects and general incidence rates

Based on WHO’s legitimated fear that there will be an avian influenza virus strain capable of mutating and mathematical modeling methods used to describe the transmission dynamics of influenza have been widely used, but the Allee effect of population due to limited resources has not been profoundly studied. In this study, we first develop a novel deterministic SI-SIR type model with weak Allee effects to depict the spread of influenza. We prove that there is a unique global positive solution to the model with any positive initial value, then we analyze the local asymptotical stability of the possible equilibria. After that, we develop a stochastic counterpart on the basis of the deterministic model where the disease transmission rates are assumed to be affected by the Ornstein-Uhlenbeck process to depict the effects of environmental noise. By developing a stochastic Lyapunov function, we establish sufficient criteria for the existence of a stationary distribution of the stochastic model which implies that all the populations in a community will coexist for a long time. Further, we also give some conditions for the exponential extinction of the infective avian population. Meanwhile, the threshold for extinction and persistence of the infective avian population is also derived. Finally, numerical simulations are presented to illustrate the theoretical results.