Dynamics and density function of a stochastic epidemic model with Ornstein–Uhlenbeck process and treatment

In this paper, we assume that two key parameters follow Ornstein–Uhlenbeck processes and incorporate treatment interventions into the model to gain further insight into the transmission dynamics. We first prove the existence and uniqueness of a positive solution. By constructing a suitable Lyapunov function and exploiting the ergodicity of the Ornstein–Uhlenbeck process, we derive sufficient conditions for the existence of a stationary distribution. Under the same conditions that guarantee local asymptotic stability of the endemic equilibrium, we then obtain an explicit expression for the covariance matrix of the probability density around the quasi-endemic equilibrium of the stochastic system. We also establish sufficient conditions for extinction. Numerical simulations are performed to validate the theoretical results.