Modeling and analyzing glucose-insulin interactions during diabetes through fractional dynamics in presence of glucagon

Introduction and objective: Diabetes is a chronic metabolic disorder characterized by higher blood glucose levels due to insufficient insulin production, weakened insulin action, or both. This study aims to construct and analyze a fractional-order mathematical model using the Anatanga-Baleanu derivative to investigate how glucose levels in the human body are controlled through interactions with insulin and glucagon. The objective is to thoroughly understand these interactions, which could improve current treatments and medications. Methods: A fractional-order diabetes model has been developed. Fixed points and their Ulam-Hyers stability are analyzed to ensure the mathematical reliability of the model. The qualitative dynamics are examined to capture glucose-insulin-glucagon interactions in detail. The explicit fractional Euler method is utilized to demonstrate the efficiency of the model for numerical results. Also, the existence and uniqueness of the solutions are obtained. Finally, numerical simulations validate the significance of the results, underscoring their relevance to diabetes research and treatment strategies. Results: The findings reveal the dynamic interactions among glucose, insulin, and glucagon in diabetes, showing that higher parameter values increase peak amplitudes and prolong transients. The fractional-order glucose–insulin model, with greater flexibility than classical derivatives, captures memory effects that influence diabetes progression and highlights potential therapeutic targets for improved disease management. Conclusion: This study provides a deeper understanding of diabetes through a detailed analysis of the proposed fractional-order glucose-insulin interactive model by addressing its well-posedness, steady-state behavior, and Ulam-Hyers stability. It bridges theoretical models and experimental observations, clarifying insulin-glucagon interactions. Further, it provides a robust framework for exploring glucose regulation mechanisms and may guide researchers and clinicians in developing more effective therapeutic strategies for diabetes and related metabolic disorders.