Bifurcation Analysis of Cancer-Immunity Cycle
Despite major advances in cancer immunotherapy, many patients still experience immune escape and disease progression. Understanding the dynamical mechanisms governing the transition between tumor control and immune escape is therefore essential for improving therapeutic strategies. Bifurcation theory and stability analysis provide a mathematical framework for explaining how gradual parameter changes can produce sudden qualitative transitions in biological systems. In the context of tumor-immune interactions, such transitions may correspond to critical thresholds separating immune-limited tumor control from immune-escape behavior. In this study, we investigate a discrete-time model of the cancer-immunity cycle applying the local stability analysis to the system equilibria based on the trace and determinant of the Jacobian matrices. The model incorporates immune suppression through a parameter representing tumor-mediated immune evasion, including mechanisms related to immune checkpoint pathways such as PD-1/PD-L1 signaling. The system exhibits a saddle-node bifurcation associated with the appearance and disappearance of nontrivial equilibria, under weak immune conditions. In addition, under strong immune conditions, the model demonstrates a distinct stability transition in which a stable spiral equilibrium (spiral sink) loses stability and becomes an unstable spiral (spiral source), resulting in the loss of stable tumor-control dynamics before equilibrium disappearance occurs. Additional mathematical analysis and numerical investigations indicate that the system does not generate stable non-equilibrium attractors such as limit cycles over the parameter ranges considered. Consequently, the stable equilibrium remains the only stable attractor in the model, emphasizing the importance of maintaining equilibrium stability for effective immune-mediated tumor suppression. Also, further parameter analyses reveal that both equilibrium existence and stability are highly sensitive near bifurcation boundaries, reflecting the delicate balance between tumor proliferation and immune activation. Overall, this work provides a mathematical framework for distinguishing equilibrium existence from effective tumor control in discrete tumor-immune systems. Beyond the specific model considered, the results highlight the importance of stability analysis in understanding immune escape and may contribute to future approaches in adaptive and personalized immunotherapy modeling.