Bifurcations in a delayed fractional model of glucose\u2013insulin interaction with incommensurate orders

Natchapon Lekdee,Sekson Sirisubtawee,Sanoe Koonprasert

doi:10.1186/s13662-019-2262-6

Abstract

This paper proposes a delayed fractional-order model of glucose–insulin interaction in the sense of the Caputo fractional derivative with incommensurate orders. This fractional-order model is developed from the first-order model of glucose–insulin interaction. Firstly, we investigate the non-negativity and the boundedness of solutions of the fractional-order model. Secondly, the stability and the bifurcation of the model are studied by separating the associated characteristic equation of the model into its real and imaginary parts and taking a time delay as the bifurcation parameter. The asymptotic stability and the Hopf bifurcation are discussed via the condition of creation of the bifurcation. Furthermore, it is shown that the onset of the bifurcation is related to the fractional orders of the model. Finally, some numerical simulations of the model using the Adam–Bashforth–Moulton predictor corrector scheme are demonstrated to support our obtained theoretical results.

Highlights

Several mathematicians have developed mathematical models describing the relationship between insulin and glucose in human beings
3 Fractional-order model description we develop a delayed fractional-order glucose–insulin interaction model, which is based on the generalized dynamical model (3), using the Caputo fractional derivative
7 Conclusion In this article, we have proposed the delayed fractional differential equation model for glucose–insulin interaction using the Caputo fractional derivative with incommensurate orders

Summary

Introduction

Several mathematicians have developed mathematical models describing the relationship between insulin and glucose in human beings. Fractionalorder differential equations currently play a significant role in generalizing integer-order mathematical models for glucose–insulin dynamics so that some mentioned advantages of fractional-order derivatives are carried on with the classical models. After taking care of the units for both sides of the equations, we obtain the delayed fractionalorder model of glucose–insulin interaction with incommensurate orders as follows: CDqa G(t). As mentioned in the generalized MINMOD Millennium model [21], the terms β1–q1 and β1–q2 can be physiologically considered as the parameters describing the rheological behavior in enhancing the muscular and liver sensibility to the action of insulin. The issue of a Hopf bifurcation for the proposed fractional-order system will be carried out when the time delay τ is used as a bifurcation parameter

Non-negative and bounded solutions

Conclusion