Threshold dynamics and density function of a stochastic epidemic model with media coverage and mean-reverting Ornstein–Uhlenbeck process

Infectious disease transmission, mainly affected by media coverage and stochastic perturbations, has imposed great social financial burden on the community in the past few decades and even threatened public health. However, there are few studies devoted to the theoretical dynamics of epidemic models with media coverage and biologically reasonable stochastic effect yet. In this sense, this paper mainly formulates and studies a stochastic epidemic model with media coverage and two mean-reverting Ornstein–Uhlenbeck processes. We first illustrate the biological implication and mathematically reasonable assumption of introducing the Ornstein–Uhlenbeck process as stochastic effect. It is theoretically proved that the solution to the stochastic model is unique and global, as well as the existence of an ergodic stationary distribution. After that, by solving the corresponding Fokker–Planck equation and using our developed algebraic equation theory, it is derived that the above global solution around the endemic equilibrium follows a unique probability density function. For completeness, the sufficient criteria for extinction exponentially of the model are established. Finally, several numerical simulations are provided to verify our theoretical results. Besides, the impact of stochastic noises and media coverage on epidemic transmission is studied by comparison with the previous results of a deterministic model.