Dynamical Properties of a Stochastic Tumor–Immune System with Impulsive Perturbations and Regime Switching

Despite numerous clinical attempts to treat tumors, malignant tumors remain a significant threat to human health due to associated side effects. Consequently, researchers are dedicated to studying the dynamical evolution of tumors in order to provide guidance for therapeutic treatment. This paper presents a stochastic tumor–immune model to discover the role of the regime switching in microenvironments and analyze tumor evolution under comprehensive pulse effects. By selecting an appropriate Lyapunov function and applying Itô’s formula, the ergodicity theory of Markov chains, and inequality analysis methods, we undertake a systematic investigation of a tumor’s behavior, focusing on its extinction, its persistence, and the existence of a stationary distribution. Our detailed analysis uncovers a profound impact of environmental regime switching on the dynamics of tumor cells. Specifically, we find that when the system is subjected to a high-intensity white noise environment over an extended duration, the growth of tumor cells is markedly suppressed. This critical finding reveals the indispensable role of white noise intensity and exposure duration in the long-term evolution of tumors. The tumor cells exhibit a transition from persistence to extinction when the environmental regime switches between two states. Furthermore, the growth factor of the tumor has an essential influence on the steady-state distribution of the tumor evolution. The theoretical foundations in this paper can provide some practical insights to develop more effective tumor treatment strategies, ultimately contributing to advancements in cancer research and care.