Dynamic behavior of a stochastic SAIR model with saturated treatment and logarithmic Ornstein–Uhlenbeck process

Based on the effectiveness of vaccine inoculation and hospital treatment in controlling infectious diseases, and accounting for the complexity of epidemic transmission mechanisms and environmental uncertainties in population dynamics, in this paper, we investigate the dynamics of a stochastic SAIR model with vaccination and saturated treatment, where the transmission rate is governed by the logarithmic Ornstein–Uhlenbeck (O–U) process. First, we demonstrate that for any given initial value, there exists a global positive solution. Next, the stationary distribution of the system is obtained through developing complex Lyapunov functions and applying the ergodicity theory of the O–U process. Then, to uncover the statistical properties, the precise expression of the probability density function close to the quasi-equilibrium is given by applying the Routh–Hurwitz criterion and solving the four-dimensional Fokker–Planck equation. We also define sufficient criteria for disease extinction at the local level. Lastly, computer simulations verify that the O–U process has a substantial effect on the dynamics of the model. Notably, our results demonstrate that higher mean-reversion rates coupled with lower volatility intensities substantially mitigate infection outbreaks, potentially informing future therapeutic strategies.