Dynamics and Density Function of a Stochastic SICA Model of a Standard Incidence Rate with Ornstein–Uhlenbeck Process

In this paper, we study an SICA model with a standard incidence rate, where the contact rate \(\beta \) is controlled by the Ornstein–Uhlenbeck process. We first prove the existence and uniqueness of the global positive solution, and by constructing an appropriate Lyapunov function, we demonstrate that when \(R_0^s > 1\), the system has a stationary distribution. Furthermore, we obtain a concrete expression of the probability density function near the quasi-positive equilibrium point. By constructing another suitable Lyapunov function, we also derive a threshold value \(R_0^e\) for disease extinction, and when \(R_0^e < 1\), the disease extinguishes at an exponential rate. Finally, our conclusions are verified through numerical simulations.