Stability analysis of a nonlinear tumor–healthy–immune cell model under radiotherapy and chemotherapy

Abstract

This paper develops an analytical framework for a nonlinear dynamical model describing interactions among healthy cells, tumor cells, quiescent tumor cells, and immune cells under radiotherapy and chemotherapy. The system is formulated as a set of nonlinear ordinary differential equations with therapeutic inputs represented as time-dependent functions. The analysis begins by establishing positivity, boundedness, and an invariant region that confines all solutions to biologically meaningful states. Two equilibrium points are identified: the tumor-free equilibrium and the endemic equilibrium. The basic reproduction number is derived using the Next Generation Matrix approach. The local stability of the equilibrium points is examined using the Jacobian matrix and the Routh–Hurwitz criteria. Global stability is proved with Lyapunov’s direct method. Sensitivity analysis is performed using the normalized forward sensitivity index to determine the parameters that most influence. The results show that the tumor growth rate and the transformation rate promote tumor persistence. Radiotherapy efficacy and the immune killing rate suppress tumor growth. When the system converges to the tumor-free equilibrium, it represents effective disease control. The findings demonstrate how mathematical stability and sensitivity analysis support the design of treatment protocols. They also provide a basis for evaluating combined radiotherapy–chemotherapy strategies and how these can shift the tumor–immune balance toward recovery.