Stochastic analysis of temporal tumor evolution with macrophage effects.

In this paper, we investigate the dynamic behavior of tumors interacting with macrophages under the influence of environmental noise. We conduct a stochastic analysis to examine how random fluctuations affect the model's dynamics. Using Lyapunov methods, we establish the existence of a unique positive global solution and derive sufficient conditions for the persistence and extinction of tumor cells. Numerical simulations are performed to validate these theoretical results and to illustrate the impact of noise compared to the deterministic system. This study provides some valuable insights into the complex interactions among random environmental factors and the tumor microenvironment with important implications for therapeutic strategies and cancer immunotherapy.