Dynamic analysis of an HIV stochastic time-delay differential equations incorporating two infection pathways and CTL immune response.

We propose a five-dimensional stochastic time-delay differential equation model that includes virus-to-cell infection, silent and active cell-cell transmissions and CTL immune response. Using mathematical methods, the non-degenerate five-dimensional stochastic delay differential equation model is transformed into a degenerate eight-dimensional stochastic differential equation model. The existence of a unique global positive solution is demonstrated. Then, by establishing a suitable Lyapunov function, we obtain the existence of a stationary Markov process when the stochastic CTL-activated reproduction number is greater than one. Additionally, we derive a critical condition for virus extinction using spectral radius analysis method and the law of large numbers theorem. Finally, through numerical simulations, we explore the impacts of random perturbations and cell-cell transmission on the dynamic behaviour of the model, and investigate the effect of time delays on T-cell count and viral load. Furthermore, ensemble simulations quantify viral clearance probabilities relative to the clinical detection limit, revealing that CTL-mediated immunity utilizes environmental stochasticity more efficiently than B-cell-mediated mechanisms to accelerate viral clearance.