Analysis of an Impulsive Tumor-Chemotherapy Model With Michaelis-Menten Nonlinear Drug Degradation

While chemotherapeutic drugs can rapidly debulk tumor populations, they also induce irreversible harm to healthy tissues. Therefore, precise control of both dosage and timing of administration is critical. In this study, we establish a class of tumor–chemotherapy models incorporating a nonlinear drug elimination rate, characterized by a Michaelis–Menten function. For the tumor-extinction subsystem, we employ blue principal branch of the Lambert W function to derive an analytical solution for the chemotherapy dose by solving the Michaelis-Menten equation, and construct an equivalent difference equation to establish the global stability of the fixed point. By combining the differential equation comparison theorem with Floquet multiplier analysis, we obtain explicit conditions for the local and global asymptotic stability of the tumor-free periodic solution and the permanence of the system. Additionally, using the bifurcation theorem, we establish explicit conditions for the existence of a stable positive periodic solution under the mechanism of system permanence. Numerical simulations further show that changes in the impulsive chemotherapy dose can drive a transition in the system dynamics, causing the tumor cells to evolve from a persistent state to an extinction state, thereby highlighting the decisive role of dose intensity in repeated chemotherapy. Moreover, small perturbations in the initial state can switch the long-term dynamics between tumor extinction and permanence, revealing bistability induced by the interaction between fixed-dose pulsed administration and nonlinear pharmacokinetics. Overall, this work provides a rigorous mathematical framework for analyzing dose–time effects in chemotherapy and offers valuable guidance for the design of clinical treatment regimens.