Asymptotic behavior of a stochastic delayed avian influenza model with saturated incidence rate

In this paper, we establish a stochastic delayed avian influenza model with saturated incidence rate. Firstly, we prove the existence and uniqueness of the global positive solution with any positive initial value. Then, we study the asymptotic behaviors of the disease-free equilibrium and the endemic equilibrium by constructing some suitable Lyapunov functions and applying the Young's inequality and Hölder's inequality. If $\mathscr{R}_0 < 1$, then the solution of stochastic system is going around disease-free equilibrium while the solution of stochastic system is going around endemic equilibrium as $\mathscr{R}_0 >1$. Finally, some numerical examples are carried out to illustrate the accuracy of the theoretical results.