Numerical simulation of a fractional glucose-insulin model via successive approximation and ABM schemes

We developed a fractional-order glucose–insulin regulatory model in the Caputo sense to encode memory effects in metabolic dynamics. The three-equation nonlinear system employed component-wise fractional orders to represent heterogeneous memory depths across plasma glucose, insulin action, and secretion. We established well-posedness (existence, uniqueness), positivity, and boundedness, and assess local stability; oscillatory regimes were further examined via discrete-time Hopf conditions for the discretized dynamics. For computation, we implement the successive approximation method (SAM) and a fractional Adams–Bashforth–Moulton (ABM) predictor–corrector scheme. In head-to-head tests, ABM achieved lower residuals, better stability, and higher efficiency than SAM, with validation against frequently sampled intravenous glucose tolerance test (FSIGT) data and a global sensitivity analysis highlighting insulin responsiveness and glucose-threshold parameters as most influential. Residual analysis indicated that increasing the fractional order(s) toward the integer case reduced numerical error—for example, the representative state error $ \lvert\Delta u\rvert $ decreased from $ 129.6 $ at $ \nu = 0.5 $ to $ 34.1 $ at $ \nu = 0.9 $. These results supported the clinical relevance of fractional-order modeling for improved diabetes management, parameter tuning, and control strategy design.