FRACTIONAL DIFFERENTIAL EQUATIONS AND NUMERICAL SOLUTIONS FOR INSULIN-GLUCOSE REGULATORY DYNAMICS: A CAPUTO FRACTIONAL BERGMAN MINIMAL MODEL ANALYSIS

Abstract

This study develops and analyzes a fractional-order generalization of the Bergman minimal model for insulin-glucose regulatory dynamics. Classical integer-order differential equations inadequately capture the hereditary and memory-dependent characteristics inherent in endocrine metabolic processes, motivating the adoption of Caputo fractional derivatives of order α ∈ (0, 1]. This nonlinear fractional differential equation system that would model plasma glucose G(t), remote (interstitial) insulin action X(t), and plasma insulin concentration I(t) is numerically solved using the Adams-Bashforth-Moulton predictor-corrector algorithm. There are five different orders of simulations (0.75, 0.85, 0.90, 0.95, 1.00), where physiologically valid parameters are used based on oral glucose tolerance test (OGTT) protocols. Findings indicate that reduced fractional order ( 0.90 ) leads to significantly slower glucose clearance and extended insulin response, which is in line with the sub-diffusive dynamics anomaly in diabetic and pre-diabetic populations. Using the classical model (alpha = 1.00) the restoration of basal glucose occurs in 240 minutes, and in the 0.75 case at the same time the suprabasal glucose concentration is 135.68 mg/dL. Convergence analysis proves that the numerical scheme has O(h(2 -alpha)) accuracy. The fractional model provides a more biophysically realistic account of glucose homeostasis and has great potential in personalized glycemic predictions, optimization of insulin therapy, and classification of diabetes mellitus.