An HIV stochastic model with cell-to-cell infection, B-cell immune response and distributed delay.

In this study, a delayed HIV stochastic model with virus-to-cell infection, cell-to-cell transmission and B-cell immune response is proposed. We first transform the stochastic differential equation with distributed delay into a high-dimensional degenerate stochastic differential equation, and then theoretically analyze the dynamic behaviour of the degenerate model. The unique global solution of the model is given by rigorous analysis. By formulating suitable Lyapunov functions, the existence of the stationary Markov process is obtained if the stochastic B-cell-activated reproduction number is greater than one. We also use the law of large numbers theorem and the spectral radius analysis method to deduce that the virus can be cleared if the stochastic B-cell-inactivated reproduction number is less than one. Through uncertainty and sensitivity analysis, we obtain key parameters that determine the value of the stochastic B-cell-activated reproduction number. Numerically, we examine that low level noise can maintain the number of the virus and B-cell populations at a certain range, while high level noise is helpful for the elimination of the virus. Furthermore, the effect of the cell-to-cell infection on model behaviour, and the influence of the key parameters on the size of the stochastic B-cell-activated reproduction number are also investigated.