A Multi-Scale Model Coupling Population-Level Dynamics with Within-Host Physiology for Diabetes and Hypertension

The synergistic epidemic of type 2 diabetes mellitus (T2DM) and essential hyper-tension requires a departure from traditional, phenomenological compartmental modeling. This paper establishes a rigorous multi-scale mathematical framework that mechanistically couples the fast within-host physiological continuous dynamics (glucose-insulin homeostasis and vascular hemodynamics) with the slow macroscopic epidemiological transitions. Utilizing Geometric Singular Perturbation Theory (GSPT), we prove the existence of a normally hyperbolic invariant slow manifold, allowing for the rigorous separation of physiological and demographic time scales. We conduct a rigorous bifurcation analysis using center manifold reduction to demonstrate the existence of subcritical (backward) bifur-cations driven by micro-scale hysteresis in insulin sensitivity, proving that pushing the basic reproduction number below unity is insufficient for disease eradication in highly endemic regimes. Furthermore, we formulate an optimal control problem using Pontryagin’s Maximum Principle to identify optimal intervention trajec-tories. The theoretical results are corroborated by high-order implicit numerical schemes handling the system’s inherent stiffness. This work provides a deep, mathematically unified paradigm for understanding how localized physiological dysregulation cascades into population-level comorbidity regime shifts.