# Dynamical Behaviors of the Tumor-immune System in a Stochastic Environment

## TL;DR

This paper analyzes the stochastic dynamics of the tumor-immune system, focusing on stability, boundary behaviors, and the effects of environmental noise through mathematical and numerical methods.

## Contribution

It introduces a stochastic model for tumor-immune interactions and provides rigorous analysis of its stability, boundary behavior, and stationary distributions, supported by simulations.

## Key findings

- Existence and uniqueness of global positive solutions
- Conditions for stochastic permanence of the system
- Numerical verification of theoretical results

## Abstract

This paper investigates dynamic behaviors of the tumor-immune system perturbed by environmental noise. The model describes the response of the cytotoxic T lymphocyte (CTL) to the growth of an immunogenic tumour. 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.