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The interaction between the immune system and tumors is complex, and macrophages play a dual role in promoting and suppressing tumor growth. This research proposes a mathematical model that incorporates a single feedback control strategy in a tumor-macrophage interaction system. The model examines how feedback regulation affects tumor dynamics using Caputo fractional-order derivatives. Existence and uniqueness of solutions are demonstrated using Banach’s fixed-point theorem. Stability analysis of the model identified four equilibrium points. According to stability analysis, the tumor-free equilibrium is always unstable. The tumor dominant equilibrium in the presence of a feedback control function is conditionally asymptotically stable. The analysis identified two interior equilibrium points that are stable under the particular parameter conditions. Numerical simulation of the proposed model is performed using a predictor-corrector scheme. Parametric investigation is conducted to examine how different feedback strengths affect system dynamics. Through numerical simulation, several steady-state behaviors are identified, including uncontrolled growth, stable oscillations, and tumor elimination. These outcomes vary based on the fractional order parameter and the intensity of feedback control. The results reveal essential threshold values for tumor stability, highlighting conditions that may lead to either tumor elimination or persistence.
Padder et al. (Thu,) studied this question.