A Nonlinear Quantum-Deformation-Fractional Model for Memory-Dependent Glucose–Insulin–Glucagon Dynamics With Real-Data Validation

This study develops a nonlinear (q,τ)-fractional biophysical model for memory-dependent glucose–insulin–glucagon regulation. The model is motivated by the fact that metabolic regulation is governed by delayed insulin action, counter-regulatory glucagon response, nonlinear feedback, and scale-dependent physiological adaptation. A (q,τ)-Caputo fractional operator is used to incorporate nonlocal memory and deformation effects, where the fractional orders describe the intensity of metabolic memory, whereas the parameters q and τ control kernel deformation and scale sensitivity. The proposed system couples glucose, insulin, and glucagon through nonlinear insulin-mediated glucose uptake, glucagon-driven counter-regulation, external glucose input, and threshold-dependent activation during low-glucose states. Analytical properties of the model are investigated through a Volterra-type integral representation, well-posedness analysis, parameter optimization, and local stability of the optimized equilibrium. A memory-weighted numerical scheme is then constructed for simulating the nonlinear fractional system. Real-data validation is performed using metabolic and brain-metabolic datasets, including OhioT1DM, ShanghaiT2DM, and ADNI-derived variables. The results indicate that the proposed (q,τ)-fractional structure improves the representation of delayed glucose decay, insulin-response dynamics, counter-regulatory effects, and Alzheimer-related brain-metabolic impairment compared with classical and ordinary fractional formulations. The framework therefore provides a flexible computational biophysical approach for modeling nonlinear memory-dependent metabolic regulation and for analyzing real physiological signals across systemic and brain-related glucose dynamics.