Analysis of a Fractional-order Glucose-Insulin Biological System with Time Delay

Abstract

In the human glucose-insulin regulatory system, diverse metabolic issues can arise, including diabetes type I and type II, hyperinsulinemia, hypoglycemia, etc. Therefore, the analysis and characterization of such a biological system is a must. It is well known that mathematical models are an excellent option to study and predict natural phenomena to some extent. On the other hand, fractional-order calculus provides a generalization of derivatives and integrals to arbitrary orders giving us a framework to add memory properties and an extra degree of freedom to the mathematical models to approximate real-world phenomena with higher accuracy. In this work, we introduce a fractional-order version of a mathematical model of the glucose-insulin regulatory system. Using the fractional-order Caputo derivative, we can investigate different concentration rates among insulin, glucose, and healthy beta cells. Additionally, the model incorporates two time-lags to represent the elapsed time in insulin secretion in response to blood glucose level and the delay in glucose drop due to increased insulin concentration. Analytical results of the equilibrium points and their corresponding stability are given. Numerical results, including phase portraits and bifurcation diagrams, reveal that the fractional-order increases the chaotic regions, leading to potential metabolic problems. Vice versa, the system seems to work correctly when the behavior evolves to periodic windows.