Dynamical behaviors of stochastic virus dynamic models with saturation responses

•The stochastic virus model with saturation functional responses is assumed to be influenced by white noises.•White noises influence the HIV virus system in two different ways. Therefore, two different stochastic virus models with saturation functional responses are considered.•If the influence intensity of white noises is weaker, similar to the deterministic HIV model, it is viruses with larger infection rates that are always dominant in the stochastic HIV model. We consider a stochastic virus model with saturation functional responses, in which the death rate of the uninfected CD4+T, the infected CD4+T, and the HIV virus particles are influenced by white noises. We prove the global existence of a positive solution of the system. Although this system does not have any equilibrium points compared with the corresponding deterministic model, by Itô’s formula and by constructing a proper Lyapunov function, we prove that the solutions of the stochastic virus model fluctuate randomly around the uninfected equilibrium of the deterministic virus model if the crucial value R0 1 and that the stochastic perturbations (of standard white noise type) influence the rate of change of the uninfected CD4+T, the infected CD4+T, and the HIV virus particles directly by strength proportional to the distances between T¯ and T, T¯* and T*(t), and V¯ and V(t), respectively. Here (T¯,T¯*,V¯) is the infected equilibrium of not only the stochastic system but also the corresponding deterministic system. We obtain the sufficient conditions which guarantee the stochastic asymptotical stability of the infected equilibrium. Finally, we present some numerical simulations to verify our results, and discuss our results.