A Mathematical Analysis of the Diabetes Model Using Fractal-Fractional Derivatives

Abstract

This work presents a mathematical model for diabetes utilizing the Caputo fractional derivative to depict the complex dynamics of glucose and insulin interactions within the human body. Conventional integer-order differential equations often inadequately represent the memory and hereditary components fundamental to physiological processes. The Caputo fractional-fractional derivative is an innovative non-integer order derivative characterized by a power-law kernel and various practical uses. This unique derivative illustrates the dynamics of diabetes mellitus, since the operator may be employed to develop models that encapsulate the dynamics with memory effects. Diabetes mellitus, a prevalent condition globally, is a significant contributor to the progression of numerous life-threatening disorders. Over time, diabetes, a chronic metabolic disease marked by elevated blood glucose, seriously harms the heart, blood vessels, eyes, kidneys, and nerves. The current study uses fractional-fractal derivatives to model and analyze the diabetes mellitus model without genetic components. After examining the diabetes mellitus model's critical points, the existence and uniqueness of the model's solutions under the fractional-fractal operator are examined using Picard's theorem. The dynamic behavior of the model was validated by simulation studies over a range of fractal-fractional parameters determined by the Caputo operator.