Modeling tumor-immune interactions with Beddington-Deangelis response: equilibrium analysis and state-dependent impulsive control

Dynamical tumor-immune interaction models support precise adaptive anti-tumor therapies. This paper investigates a Beddington-DeAngelis type tumor-immune model incorporating immune interference and tumor refuge via two stages. First, we analyze the untreated system, derive existence and stability criteria for multiple equilibria alongside forward/backward bifurcations, and characterize parameter spaces leading to stable hightumor equilibria (progressive cancer). Second, to reverse tumor progression, we design a tumor-load-triggered state-dependent impulsive control: effector immune supplementation strength is set to a 1 for severe tumors and a 2 for mild ones, modulated by real-time immune status. Based on semi-continuous dynamical system theory, we prove the existence of order-1 periodic solutions and homoclinic orbits, and obtain sufficient conditions for their orbital asymptotic stability. Numerical simulations reveal that immunotherapy dosages a 1 and a 2 dominate therapeutic efficacy for mild and severe cases; moderate impulsive combined interventions effectively suppress tumor growth.