Global stability and periodicity in a glucose-insulin regulation model with a single delay

Maia Angelova; Gleb Beliakov; Anatoli Ivanov; Sergiy Shelyag

doi:10.1016/j.cnsns.2020.105659

Global stability and periodicity in a glucose-insulin regulation model with a single delay

Maia Angelova^*, Gleb Beliakov, Anatoli Ivanov, Sergiy Shelyag

^*Corresponding author for this work

Aston Digital Futures Institute

Research output: Contribution to journal › Article › peer-review

3 Citations (Scopus)

Abstract

A two-dimensional system of differential equations with delay modelling the glucose-insulin interaction processes in the human body is considered. Sufficient conditions are derived for the unique positive equilibrium in the system to be globally asymptotically stable. They are given in terms of the global attractivity of the fixed point in a limiting interval map. The existence of slowly oscillating periodic solutions is shown in the case when the equilibrium is unstable. The mathematical results are supported by extensive numerical simulations. It is deduced that typical behaviour in the system is the convergence to either a stable periodic solution or to the unique stable equilibrium. The coexistence of several periodic solutions together with the stable equilibrium is demonstrated as a possibility.

Original language	English
Article number	105659
Number of pages	18
Journal	Communications in Nonlinear Science and Numerical Simulation
Volume	95
Early online date	7 Dec 2020
DOIs	https://doi.org/10.1016/j.cnsns.2020.105659
Publication status	Published - Apr 2021

Funding

This work was initiated during AI’s visit to Deakin University, Burwood Campus, in December 2017. He would like to express his appreciation of the accommodation and support from the School of Information Technology, Faculty of Science, and of the hospitality and collegiality from staff and his coauthors. This research was undertaken with the assistance of resources and services from the National Computational Infrastructure (NCI), which is supported by the Australian Government. MA thanks Newton Advanced Fellowship (UK Royal Society) / Academy of Medical Sciences UK for partial funding to develop this research.

Funders	Funder number
Academy of Medical Sciences
School of Information Technology, Faculty of Science
National Computational Infrastructure
Australian Government
Royal Society

Keywords

Delay differential equations
Diabetes
Existence of periodic solutions
Global asymptotic stability
Limiting interval maps
Linearization
Stability analysis

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10.1016/j.cnsns.2020.105659

https://arxiv.org/abs/2008.11019Licence: CC BY 4.0

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title = "Global stability and periodicity in a glucose-insulin regulation model with a single delay",

abstract = "A two-dimensional system of differential equations with delay modelling the glucose-insulin interaction processes in the human body is considered. Sufficient conditions are derived for the unique positive equilibrium in the system to be globally asymptotically stable. They are given in terms of the global attractivity of the fixed point in a limiting interval map. The existence of slowly oscillating periodic solutions is shown in the case when the equilibrium is unstable. The mathematical results are supported by extensive numerical simulations. It is deduced that typical behaviour in the system is the convergence to either a stable periodic solution or to the unique stable equilibrium. The coexistence of several periodic solutions together with the stable equilibrium is demonstrated as a possibility.",

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author = "Maia Angelova and Gleb Beliakov and Anatoli Ivanov and Sergiy Shelyag",

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AU - Angelova, Maia

AU - Beliakov, Gleb

AU - Ivanov, Anatoli

AU - Shelyag, Sergiy

PY - 2021/4

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N2 - A two-dimensional system of differential equations with delay modelling the glucose-insulin interaction processes in the human body is considered. Sufficient conditions are derived for the unique positive equilibrium in the system to be globally asymptotically stable. They are given in terms of the global attractivity of the fixed point in a limiting interval map. The existence of slowly oscillating periodic solutions is shown in the case when the equilibrium is unstable. The mathematical results are supported by extensive numerical simulations. It is deduced that typical behaviour in the system is the convergence to either a stable periodic solution or to the unique stable equilibrium. The coexistence of several periodic solutions together with the stable equilibrium is demonstrated as a possibility.

AB - A two-dimensional system of differential equations with delay modelling the glucose-insulin interaction processes in the human body is considered. Sufficient conditions are derived for the unique positive equilibrium in the system to be globally asymptotically stable. They are given in terms of the global attractivity of the fixed point in a limiting interval map. The existence of slowly oscillating periodic solutions is shown in the case when the equilibrium is unstable. The mathematical results are supported by extensive numerical simulations. It is deduced that typical behaviour in the system is the convergence to either a stable periodic solution or to the unique stable equilibrium. The coexistence of several periodic solutions together with the stable equilibrium is demonstrated as a possibility.

KW - Delay differential equations

KW - Diabetes

KW - Existence of periodic solutions

KW - Global asymptotic stability

KW - Limiting interval maps

KW - Linearization

KW - Stability analysis

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Global stability and periodicity in a glucose-insulin regulation model with a single delay

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