This study explores the combined dynamic impact of stochastic disturbances and public health alerts on the transmission mechanisms of avian-human influenza. A stochastic epidemic model with distributed delay and media coverage is developed, in which the contact rate of avian influenza is modeled using the logarithmic Ornstein-Uhlenbeck process. The existence and uniqueness of a global positive solution for the system are established. Using Hasminskii’s ergodic theory, the exponential ergodic stationary distribution of the system is also established. By constructing suitable stochastic Lyapunov functions, we derive sufficient criteria for both the persistence of infected individuals and the exponential extinction of the disease. Finally, numerical simulations are provided to validate the theoretical findings.