A Two-Step Adams–bashforth Scheme for the Fractional-Order Glucose–insulin Model: Stability and Convergence

The regulation of blood glucose and insulin in the human body is a highly nonlinear physiological process influenced by hereditary memory, feedback mechanisms, and intrinsic delays. Classical integer-order models often fail to capture these memory-dependent and nonlocal effects. In this study, we introduce a novel fractional-order glucose–insulin regulatory model formulated with the Caputo–Fabrizio (CF) derivative, which employs a nonsingular exponential kernel to describe fading memory effects more realistically. The model accounts for essential physiological parameters and incorporates the interplay between blood glucose concentration, insulin secretion, and an auxiliary control variable representing insulin action or resistance. We establish the well-posedness of the proposed CF-based system by proving existence and uniqueness of solutions via fixed-point theory under Lipschitz continuity and boundedness conditions on the nonlinear operators. To approximate the solution, we design a two-step Adams–Bashforth scheme tailored to the CF operator and provide a rigorous analysis of its convergence and stability. Numerical simulations confirm the accuracy and robustness of the method, demonstrating its ability to reproduce physiologically consistent glucose–insulin trajectories that outperform classical integer-order models. The results highlight the effectiveness of Caputo–Fabrizio fractional calculus in capturing hereditary and fading-memory effects in glucose–insulin dynamics, offering a robust mathematical framework for parameter estimation, optimal control, and personalized strategies in diabetes management.