Dynamical behaviors of stochastic virus dynamic models with saturation responses

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. Further, we assume that 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.