We consider an influenza A epidemic model that incorporates vaccination and asymptomatic patients and investigate its dynamics. The stability analysis of the equilibria in the deterministic model is conducted using the Routh-Hurwitz stability criterion. To account for the significant impact of environmental disturbance on disease transmission, the transmission rate is modeled as a logarithm Ornstein-Uhlenbeck process, resulting in a stochastic model. The existence of the unique global positive solution in the model is verified, and Lyapunov functions are employed to establish sufficient conditions for the presence of a stationary distribution. The paper also examines the conditions for the extinction or persistence of asymptomatic and infected individuals. Lastly, an exact expression for the probability density function of the stochastic model near the quasi-equilibrium is derived, and numerical simulations are performed to support the proposed theoretical findings.