Averaging principle and optimal control for a two-time-scale stochastic reaction-diffusion HIV model

Considering the fact that free viruses replicate and spread much faster than healthy CD4+ T-cells, we propose a two-time-scale stochastic reaction-diffusion HIV model in this paper. Using the Ascoli-Arzelà theorem, we prove the tightness of the slow variable and the existence of an invariant measure that has ergodic property for the fast variable. Under certain conditions, the slow variable solution converges strongly to the solution of the averaged HIV system in the sense of $L^{p}$-norm ($p\geq 1$) as the time scale $\varepsilon \to 0$. For stochastic optimal controls of the fast-slow HIV system, the first order necessary optimality conditions are given by using the classical variational analysis approach. Simulation results show that the control strategy is effective, which can reduce the number of free viruses and increase the number of healthy CD4+ T-cells.