A Novel Crossover Dynamics of Variable-Order Fractal-Fractional Stochastic Diabetes Model: Numerical Simulations

Abstract

Recent research has introduced piecewise fractional differential equations—particularly within deterministic-stochastic frameworks—to better model complex and real-world phenomena. However, their application to modeling the progression and treatment of diabetes remains limited. This paper proposes a novel mathematical framework for diabetes by incorporating piecewise stochastic-deterministic differential equations that employ variable-order fractional derivatives and fractal-fractional operators. This integrated modeling approach allows for the simultaneous representation of memory effects, heterogeneous dynamics, and random fluctuations inherent in biological systems such as glucose–insulin interactions. To solve the proposed models, we develop two specialized numerical schemes: the nonstandard Grünwald–Letnikov finite difference method for the deterministic fractional component and a nonstandard Milstein method for handling the stochastic part. These techniques are specifically designed to preserve the qualitative behavior of the system, ensuring both stability and accuracy. Theoretical analysis establishes the stability properties of the proposed methods, while extensive numerical simulations—performed on a piecewise-structured diabetes model—demonstrate their effectiveness. The results reveal significant insights into disease progression under varying physiological conditions and random perturbations. This comprehensive modeling framework thus provides a more realistic and personalized representation of diabetes dynamics, with potential applications in predicting treatment responses and optimizing individualized therapeutic strategies.