Bifurcation Analysis of Tumor-Immune System Models under Varying Treatment Intensities

Abstract: This study presents a comprehensive bifurcation analysis of a nonlinear tumor-immune interaction model under varying treatment intensities. The model, governed by two coupled differential equations, captures the complex dynamics between tumor growth, immune response, and treatment-induced cytotoxic effects. Stability and equilibrium analyses reveal critical treatment thresholds that separate tumor persistence, oscillatory remission, and eradication regimes. Numerical simulations, including nullcline plots, bifurcation diagrams, and sensitivity analyses, demonstrate how treatment intensity and immune system parameters influence system behavior. Results indicate that low treatment intensities lead to uncontrolled tumor growth, moderate levels induce oscillatory coexistence through Hopf bifurcation, and higher intensities stabilize the tumor-free equilibrium, signifying successful therapy. The model emphasizes the delicate balance between therapeutic efficacy and immune preservation, offering insights into optimizing treatment strategies for tumor suppression.

Keywords: Tumor-immune interaction, Bifurcation analysis, Nonlinear dynamics, Stability analysis, Hopf bifurcation, Treatment intensity, Mathematical oncology.