Speaker
Description
Type-2 Diabetes Mellitus (T2DM) is characterised by dysregulated glucose levels involving complex interactions between insulin and glucagon. While delay differential equation (DDE) models of glucose-insulin dynamics reproduce ultradian oscillations \cite{bridgewater2020}, glucagon is often omitted despite its physiological role, and its dynamics are not well understood. In T2DM, $\alpha$-cell dysfunction leads to impaired glucagon suppression and hyperglycemia \cite{unger2012}.
To address this gap, we extend the Cohen--Li glucose--insulin--glucagon model \cite{cohenli2021} by introducing a delay $\tau_3$ representing insulin-mediated glucagon suppression, alongside delays for insulin secretion ($\tau_1$) and hepatic glucose production ($\tau_2$). The resulting three-delay DDE system is analysed by deriving the equilibrium and the quasi-polynomial characteristic equation governing linear stability. We use Hopf bifurcation analysis with respect to $\tau_3$ to assess the role of glucagon feedback in oscillatory behaviour.
Our numerical simulations reproduce ultradian oscillations and support the analytical results. The analysis indicates that glucagon feedback modulates oscillatory stability while having a limited role in generating oscillations independently. This model provides a framework for studying delayed hormonal regulation and contributes to the development of digital-twin models for investigating diabetes
progression.
Bibliography
@article{bridgewater2020,
title = {Amplitude and {Frequency} {Variation} in {Nonlinear} {Glucose} {Dynamics} with {Multiple} {Delays} via {Periodic} {Perturbation}},
volume = {30},
issn = {0938-8974, 1432-1467},
url = {http://link.springer.com/10.1007/s00332-020-09612-1},
doi = {10.1007/s00332-020-09612-1},
abstract = {Abstract
Characterising the glycemic response to a glucose stimulus is an essential tool for detecting deficiencies in humans such as diabetes. In the presence of a constant glucose infusion in healthy individuals, it is known that this control leads to slow oscillations as a result of feedback mechanisms at the organ and tissue level. In this paper, we provide a novel quantitative description of the dependence of this oscillatory response on the physiological functions. This is achieved through the study of a model of the ultradian oscillations in glucose-insulin regulation which takes the form of a nonlinear system of equations with two discrete delays. While studying the behaviour of solutions in such systems can be mathematically challenging due to their nonlinear structure and non-local nature, a particular attention is given to the periodic solutions of the model. These arise from a Hopf bifurcation which is induced by an external glucose stimulus and the joint contributions of delays in pancreatic insulin release and hepatic glycogenesis. The effect of each physiological subsystem on the amplitude and period of the oscillations is exhibited by performing a perturbative analysis of its periodic solutions. It is shown that assuming the commensurateness of delays enables the Hopf bifurcation curve to be characterised by studying roots of linear combinations of Chebyshev polynomials. The resulting expressions provide an invaluable tool for studying the interplay between physiological functions and delays in producing an oscillatory regime, as well as relevant information for glycemic control strategies.},
language = {en},
number = {3},
urldate = {2026-03-10},
journal = {Journal of Nonlinear Science},
author = {Bridgewater, Adam and Huard, Benoit and Angelova, Maia},
month = jun,
year = {2020},
pages = {737--766},
}
@article{unger2012,
title = {Glucagonocentric restructuring of diabetes: a pathophysiologic and therapeutic makeover},
volume = {122},
issn = {0021-9738},
shorttitle = {Glucagonocentric restructuring of diabetes},
url = {http://www.jci.org/articles/view/60016},
doi = {10.1172/JCI60016},
language = {en},
number = {1},
urldate = {2026-03-10},
journal = {Journal of Clinical Investigation},
author = {Unger, Roger H. and Cherrington, Alan D.},
month = jan,
year = {2012},
pages = {4--12},
}
@article{cohenli2021,
title = {A {Novel} {Model} and its {Analysis} on the {Metabolic} {Regulations} of {Glucose}, {Insulin}, and {Glucagon}},
volume = {81},
issn = {0036-1399, 1095-712X},
url = {https://epubs.siam.org/doi/10.1137/21M1390876},
doi = {10.1137/21M1390876},
language = {en},
number = {6},
urldate = {2026-03-10},
journal = {SIAM Journal on Applied Mathematics},
author = {Cohen, Reed B. and Li, Jiaxu},
month = jan,
year = {2021},
pages = {2684--2703},
}
