Stationary distribution and extinction of a stochastic S-DI-A epidemic model with log-normal standard Ornstein–Uhlenbeck process

In the current work, a stochastic epidemic model for the dynamics of transmission of Human Immunodeficiency Virus (HIV) infection is developed and analysed. The construction considers nonlinear modes of disease transmission and incorporation of environmental variability in order to improve the reality of the modelling approach beyond deterministic hypotheses. The parameter R0 is obtained and applied to establish sufficient criteria for the existence and uniqueness of a stationary distribution of the system through the Markov semigroup theory and Lyapunov function methods. In addition, sufficient criteria for stochastic elimination of the disease are given under some mild conditions. Finally, numerical simulations are conducted using stochastic schemes to analyse system dynamics under different noise intensities and average transmission rates. The results imply that stochastic disturbances play a fundamental role in describing the evolution of HIV infection. Sensitivity analysis shows that transmission parameters are the key driving factors of disease spread, whereas stochastic impacts significantly affect variability without changing the stability structure. This work advances a better quantitative understanding of disease transmission mechanisms and offers predictive insights into epidemic dynamics and supports the design of intervention strategies.