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 \(\beta \) $\beta$ 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 \(R_0^s > 1\) $R_0^s>1$ , 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 \(R_0^e\) $R_0^e$ for disease extinction, and when \(R_0^e < 1\) $R_0^e<1$ , the disease extinguishes at an exponential rate. Finally, our conclusions are verified through numerical simulations.