Complex bifurcation dynamics of tumor-immune interactions incorporating combination therapies

The complex nonlinear dynamics of tumor-immune interactions drive tumor heterogeneity and complicate treatment strategies. While bifurcation and multistability analyses of mathematical models can uncover critical system dynamics, few studies have explored bifurcation-particularly cusp bifurcation-in high-dimensional tumor-immune systems, as most existing analyses are limited to simplified two-dimensional models. In this study, we extend the tumor-immune model proposed by Anderson et al. (2024a) to incorporate combination therapy with immune checkpoint inhibitors (ICIs) and C-C chemokine receptor type 2 (CCR2) antagonists. By combining linear algebra methods, Sotomayor's theorem, projection singularity analysis, numerical simulation, and continuation techniques, we explicitly identify transcritical and saddle-node bifurcations, conduct a systematic cusp bifurcation analysis as a classical two-parameter problem, and derive explicit quantitative conditions. Calibration against four independent murine datasets demonstrates the model's ability to reproduce tumor growth dynamics across multiple tumor datasets. Bifurcation analysis characterizes the multiplicity of tumorous equilibria, and reveals the emergence of monostable and bistable regimes. In particular, the saddle-node bifurcation curve partitions the two-parameter space under combination therapy into monostable and bistable regions. Notably, a dose-dependent inverse correlation between ICIs and CCR2 antagonists emerges along this bifurcation boundary. Our results demonstrate that effective tumor control depends not only on treatment intensity but also on the patient's initial tumor burden. The corresponding critical thresholds are determined by bifurcation structure and the stable manifold (or characteristic space) of the saddle point, respectively. These findings provide theoretical guidance for treatment design under different tumor states. Importantly, the identification of a stable low-tumor state-being clinically acceptable and controllable-highlights the potential for sustained tumor control as an alternative therapeutic objective to complete tumor eradication.