Immune Modulation in the Tumor Microenvironment: Bifurcation Analysis of Cancer-CTL-Monocyte Dynamics

We present a mathematical model describing the interactions between cancer cells, cytotoxic T lymphocytes (CTLs), and monocytes within the tumor microenvironment. The model incorporates key immunological mechanisms, including tumor antigenicity, the Allee effect, and monocyte-mediated immune activation via MHCI cross-dressing. Using systems of nonlinear ordinary differential equations (ODEs), we derive analytical expressions for equilibrium points, evaluate their stability, and characterize bifurcations, such as saddle-node, Hopf, Bogdanov-Takens, and Bautin. A reduced model via quasi-steady-state approximation (QSSA) is also proposed, preserving the core dynamic structure to facilitate bifurcation analysis. A central finding of our study is the critical role of the monocyte-mediated T cell activation rate, denoted by the parameter $\beta$ , which encapsulates the immunostimulatory potential of inflammatory monocytes presenting tumor antigens via MHCI cross-dressing. Numerical continuation corroborates the existence of multiple codimension-two organizing centers, delineating parameter regimes of tumor clearance, immune-mediated control, bistability, sustained oscillations, and inevitable escape. Our results quantitatively characterize the critical role of the monocyte-T-cell activation rate ( $\beta$ ) and the Allee threshold ( $\gamma$ ) in tipping the balance between immune surveillance and tumor persistence. This framework provides actionable bifurcation-based criteria for designing combination immunotherapies that enhance antigen presentation or monocyte functionality to shift the system toward tumor-eliminating attractors.