An Efficient Discretization Scheme for a Time‐Delay Variable‐Order Fractional Diabetes Model
Muner M. Abou Hasan
Source abstract
ABSTRACT In this work, we introduce a mathematical model for diabetes that uses time‐delay and variable‐order fractional derivatives, aiming to better reflect the complex and memory‐dependent behavior of glucose and insulin dynamics. The model is built using the Caputo definition of variable‐order derivatives. We explore the system's equilibrium points and examine their stability to understand how the system's behavior changes with different parameters. We also study the positivity and boundedness of the proposed system. To solve the model numerically, we design an effective method that combines a nonstandard finite difference scheme with the Grünwald–Letnikov operator. We analyze the proposed scheme and prove that the approximated solutions remain nonnegative and bounded. Through numerical simulations and comparisons, we demonstrate the reliability and practical advantages of our approach. The results highlight the crucial impact of time‐delay and variable‐order fractional dynamics on diabetes progression and treatment. Delayed insulin response and memory effects in glucose–insulin interaction are effectively modeled. This enhances the realism and personalization of blood sugar regulation analysis.
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