Complete classification of the long-term behavior of a tumor–immune model with impulsive therapeutic interventions

We propose and analyze a stochastic tumor–immune interaction model subject to periodic impulsive treatment, capturing the combined effects of chemotherapy and immunotherapy. The model extends previous deterministic frameworks by incorporating environmental noise and treatment-induced state-dependent impulses, leading to a hybrid stochastic differential system. We rigorously classify the long-term behavior of the system, providing sharp conditions for the extinction and persistence of tumor cells. Unlike earlier studies that treat impulses deterministically and yield limited insight, our framework accounts for random fluctuations in treatment efficacy and immune response. We derive explicit threshold conditions that determine the effectiveness of the combined therapies, offering practical guidelines for improving treatment outcomes. Our analysis framework can be extended to study a broad class of stochastic hybrid models recently introduced in tumor–immune dynamics.