Dynamical analysis of a tumor model with immunotherapy treatment

Among the recent advancements in cancer treatment, immunotherapy has emerged as a highly promising approach for managing and potentially curing malignant tumors by enhancing the body’s own immune response. Inspired by prior research work (Amin and Yu, 2025), the present work extends three-dimensional mathematical models to investigate the interactions among CD4+ T cells, cytokines, and tumor cells, with particular focus on their collective role in tumor regression. Within this framework, we analyze therapies involving CD4+ T cells, cytokine interventions, and polytherapy for comparative understanding of treatment outcomes. The models are examined to identify equilibrium points and characterize their stability, while bifurcation analysis highlights the critical thresholds at which qualitative changes in system dynamics arise. To capture nonlinear effects, normal form theory is applied, offering explicit insights into stability of oscillatory solutions generated through Hopf bifurcations. Remarkably, the results reveal the existence of multiple coexisting limit cycles, induced by generalized Hopf bifurcation, which give rise to rich oscillatory patterns and complex behaviors in tumor–immune interactions. These dynamics include bistability, where the system may settle into either tumor control (stable equilibrium) or sustained oscillatory immune responses (stable limit cycles), reflecting biologically relevant outcomes such as immune-induced tumor dormancy or relapse. Overall, this study demonstrates that Hopf bifurcation serves as a fundamental cause of oscillatory patterns and nonlinear transitions in tumor–immune dynamics, providing valuable theoretical insights into treatment design and long-term therapeutic effectiveness.