Fractional Differential Equations of Glucose and Insulin: Exposing Dynamic System Behavior Through Memory-Dependent Modeling

Abstract

The classical Bergman minimal model is a glucose regulation model that uses integer-order ordinary differential equations to model the dynamics of glucose regulation under the influence of insulin. Biological systems however are those that are hereditary and have memory effects that cannot be well represented by integer-order calculus. This paper constructs and studies a fractional-order extension of the Bergman minimal model using Caputo fractional derivatives with fractional orders of 0 10 and 0 10 respectively on the dynamics of glucose and insulin respectively. The Grunwald Letnikow numerical method is used to model the intravenous glucose tolerance test (IVGTT) protocol in a 240-minute time frame. Artificial data created at 13 time points of classical and fractional framework display different dynamic signatures. The fractional model (0.85, 0.90) has a slower initial glucose clearance and increased long-tail memory behavior, with the highest glucose concentration of 151.90 mg/dL at t 10 min compared to the classical model at 248.84mg/dL. A sensitivity analysis of 0.75, 0.85, 0.95 and 1.0 on 0.75 confirms that there is a monotonic relationship between the fractional order and the glucose clearance rate. In both models, the presence of global asymptotic stability of the basal state is confirmed by analysis of the stability at equilibrium. It yields RMSE = 34.27 mg/dL compared to the ODE baseline which is due to truly different physiological interpretations rather than fitting errors. These findings indicate that the memory-dependent regulatory processes that are not apparent in classical models are revealed using fractional-order models, which provide a more biologically realistic account of glucoregulatory processes applicable in modeling type 2 diabetes.