Stochastic analysis of an HIV model with various infection stages

In this study, we develop a stochastic model that captures the dynamics of HIV infection, encompassing susceptible individuals, asymptomatic HIV-positive individuals, and those exhibiting symptoms. Initially, we examine the existence and stability of both disease-free and endemic equilibria within the deterministic version of the model. Our analytical findings indicate that the basic reproduction number, $\mathcal{R}_{0}$ , is a pivotal factor in determining the uniqueness and global stability of these equilibria. Furthermore, we explore the impact of environmental noise on the HIV disease model, identifying two critical thresholds, $\mathcal{R}_{1}^{s}$ and $\mathcal{R}_{2}^{s}$ (with $\mathcal{R}_{2}^{s} < \mathcal{R}_{1}^{s}$ ). If $\mathcal{R}_{1}^{s}$ is less than unity, the disease is likely to be eradicated; conversely, if $\mathcal{R}_{2}^{s}$ exceeds unity, the disease will persist, and a unique stationary distribution will emerge. Additionally, our numerical simulations reveal that when $\mathcal{R}_{2}^{s} < 1 < \mathcal{R}_{1}^{s}$ , the disease may still face extinction. From an epidemiological viewpoint, our observations suggest that a decrease in environmental noise intensity results in a reduction of the oscillation amplitude in the disease dynamics. Conversely, an increase in noise intensity is associated with a lower mean of infectious individuals and a left-skewed distribution.