Modelling a stochastic HIV model with logistic target cell growth and nonlinear immune response function

A stochastic HIV viral model with both logistic target cell growth and nonlinear immune response function is formulated to investigate the effect of white noise on each population. The existence of the global solution is verified. By employing a novel combination of Lyapunov functions, we obtain the existence of the unique stationary distribution for small white noises. We also derive the extinction of the virus for large white noises. Numerical simulations are performed to highlight the effect of white noises on model dynamic behaviour under the realistic parameters. It is found that the small intensities of white noises can keep the irregular blips of HIV virus and CTL immune response, while the larger ones force the virus infection and immune response to lose efficacy.