Dynamical Behaviors of the Tumor-Immune System in a Stochastic Environment

This paper investigates dynamical behaviors of the tumor-immune system perturbed by environmental noise. The model describes the response of the cytotoxic T lymphocyte to the growth of an immunogenic tumor. The main methods are stochastic Lyapunov analysis, comparison theorem for stochastic differential equations (SDEs), and strong ergodicity theorem. Firstly, we prove the existence and uniqueness of the global positive solution for the tumor-immune system. Then we go a further step to study the boundaries of moments for tumor cells and effector cells and the asymptotic behavior in the boundary equilibrium points. Furthermore, we discuss the existence and uniqueness of stationary distribution and stochastic permanence of the tumor-immune system. Finally, we give several examples and numerical simulations to verify our results.