A fractional-order tumor–immune system model with combined vitamin and chemotherapy: analysis, stability, and control insights
摘要
In this study, we develop and analyze a fractional-order tumor–immune interaction model incorporating the Caputo fractional derivative to investigate the impact of vitamin supplementation on immune enhancement and tumor suppression. The model captures the dynamics between tumor cells, immune effector cells, and the effects of chemotherapeutic drugs and vitamins, aiming to understand their combined influence on tumor progression. To ensure biological realism, the model parameters were calibrated against experimental tumor growth data, establishing consistency. A central objective is to determine optimal therapeutic strategies that minimize drug dosage while maximizing tumor eradication and preserving patient health. The optimal control problem is formulated and analyzed using Pontryagin’s Maximum Principle to characterize the necessary conditions for optimal treatment. We establish the well-posedness of the model by proving the existence, uniqueness, positivity, and boundedness of solutions. The local asymptotic stability of equilibrium points is examined through linearization techniques, while global stability and persistence are rigorously analyzed using Lyapunov function methods. Sensitivity analysis is conducted to evaluate the influence of key parameters on system dynamics, and the potential for Hopf bifurcation is investigated to explore oscillatory behavior under specific conditions. Numerical simulations, implemented via an efficient Adams-type predictor–corrector scheme, are provided to support the analytical results and illustrate the complex interplay between treatment parameters and tumor–immune dynamics. The findings highlight the critical role of immune support and dosage optimization in improving treatment efficacy and patient outcomes.