A Mathematical Framework for Glucose-Insulin Dynamics in Type 1 Diabetes under BGM/MDI Therapy

We present a rigorous mathematical analysis of glucose-insulin regulation inType 1 Diabetes under the clinically prevalent but mathematically under-studiedcombination of Multiple Daily Injections (MDI) and sparse Blood Glucose Moni-toring (BGM). The physiological dynamics are captured by an 11-dimensional hy-brid impulsive ODE system that explicitly includes a discontinuous renal-clearancethreshold at 180 mg/dL. We prove well-posedness via a Filippov existence argu-ment for the associated differential inclusion, together with a one-sided Lipschitzuniqueness argument that exploits monotonicity of the switch; global asymptoticstability of the continuous-infusion equilibrium follows from a Lyapunov-ISS anal-ysis. Structural identifiability is examined in the basal-only, no-meal regime usingtransfer-function techniques, revealing seven globally identifiable parameter combi-nations — including a lumped DC-gain term — while confirming that the insulindistribution volume is not separately identifiable. Under the sparse BGM schedule(6 finger-stick readings per day) we prove a practical identifiability rank bound ofat most 6 estimable directions per day, and conjecture that this bound saturatesunder periodic dosing. The full computational and empirical validation program— parameter recovery on virtual patients, numerical testing of the rank-saturationconjecture, and a large-scale study of 10,000 virtual patients compared against pub-lished clinical ranges — is planned future work and has not yet been carried out;this paper’s contribution is the analytical framework and its proofs.