Stationary distribution and extinction of a multi-stage HIV model with nonlinear stochastic perturbation

In this paper, we investigate a stochastic multi-stage model to evaluate the influence of treatment intensification with the integrase inhibitor raltegravir on viral load and 2-LTR dynamics in HIV patients under suppressive therapy. Firstly, it is proven that the model has a unique global positive solution. Secondly, by constructing a Lyapunov function, we establish sufficient conditions for the existence of a unique ergodic stationary distribution if $$R_{0}^{S}>1$$ . Thirdly, we obtain sufficient criterions $$R_{0}^{s}<1$$ for disease extinction. Finally, the analytical results are demonstrated via two simulation examples. Our contribution also concentrates on proposing a method constructing Lyapunov function, which can be successfully used for the research about stationary distribution of epidemic model with nonlinear stochastic perturbation.