Stability analysis and numerical simulations of IVGTT glucose-insulin interaction models with two time delays
Abstract
This paper presents a systematic study of a mathematical model of glucose and insulin interaction with two time delays, with a focus on analytical studies, bifurcation analysis, and very well numerical simulations. This model based on the Intra-Venous Glucose Tolerance Test (IVGTT) and is presented with two time delays. One delay is the insulin response time to an increase in glucose concentration, and the hepatic glucose production time delay is the other. Then, we establish results on positivity, boundedness, and persistence. We also provide sufficient stability analysis conditions for both local and global asymptotic stability of the proposed models. For the latter, two different strategies are used: stability bifurcation analysis and Lyapunov-Krasovskii functionals. We investigate different regions of parameter space using two approaches, that yield different sets of sufficient conditions for global stability. The bifurcation graphs generated from our extensive and carefully designed simulations complement and confirm these analytical results. The insulin concentration level peaks after the glucose concentration level, according to the numerical simulations.
Keyword : glucose-insulin regulatory system, insulin secretion, ultradian oscillation, delay differential equation model
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
E. Ackerman, J.W. Rosevear and W.F. McGuckin. A mathematical model of the glucose-tolerance test. Physics in Medicine and Biology, 9(2):203–213, 1964. https://doi.org/10.1088/0031-9155/9/2/307
R.N. Bergman and C. Cobelli. Minimal modeling, partition analysis, and the estimation of insulin sensitivity. Fed Proc., 39(1):110–115, 1980.
R.N. Bergman, Y.Z. Ider, C.R. Bowden and C. Cobelli. Quantitative estimation of insulin sensitivity. American Journal of Physiology-Endocrinology and Metabolism, 236(6):E667, 1979. https://doi.org/10.1152/ajpendo.1979.236.6.E667
V.W. Bolie. Coefficients of normal blood glucose regulation. Journal of Applied Physiology, 16(5):783–788, 1961. https://doi.org/10.1152/jappl.1961.16.5.783
A. Caumo, R.N. Bergman and C. Cobelli. Insulin Sensitivity from Meal Tolerance Tests in Normal Subjects: A Minimal Model Index. The Journal of Clinical Endocrinology & Metabolism, 85(11):4396–4402, 2000. https://doi.org/10.1210/jcem.85.11.6982
A. De Gaetano and O. Arino. Mathematical modelling of the intravenous glucose tolerance test. Journal of Mathematical Biology, 40:136–168, 2000. https://doi.org/10.1007/s002850050007
L.C. Gatewood, E. Ackerman, J.W. Rosevear, G.D. Molnar and T.W. Burns. Tests of a mathematical model of the blood-glucose regulatory system. Computers and Biomedical Research, 2(1):1–14, 1968. https://doi.org/10.1016/0010-4809(68)90003-7
T.A. Gresl, R.J. Colman, T.C. Havighurst, L.O. Byerley, D. B. Allison, D.A. Schoeller and J. W. Kemnitz. Insulin sensitivity and glucose effectiveness from three minimal models: effects of energy restriction and body fat in adult male rhesus monkeys. American Journal of Physiology-Regulatory, Integrative and Comparative Physiology, 285(6):R1340–R1354, 2003. https://doi.org/10.1152/ajpregu.00651.2002
Y. He, M. Wu and J.-H. She. Delay-dependent stability criteria for linear systems with multiple time delays. IEE Proceedings - Control Theory and Applications, 153(4):447–452, 2006. https://doi.org/10.1049/ip-cta:20045279
W.M. Hirsch, H. Hanisch and J.-P. Gabriel. Differential equation models of some parasitic infections: Methods for the study of asymptotic behavior. Communications on Pure and Applied Mathematics, 38(6):733–753, 1985. https://doi.org/10.1002/cpa.3160380607
Y. Kuang. Delay differential equations: With applications in population dynamics. Boston, Academic Press, 1993.
C.P. Li and F.R. Zhang. A survey on the stability of fractional differential equations. The European Physical Journal Special Topics, 193:27–47, 2011. https://doi.org/10.1140/epjst/e2011-01379-1
J. Li and Y. Kuang. Analysis of a model of the glucoseinsulin regulatory system with two delays. SIAM Journal on Applied Mathematics, 67(3):757–776, 2007. https://doi.org/10.1137/050634001
J. Li, Y. Kuang and B. Li. Analysis of IVGTT glucose-insulin interaction models with time delay. Discrete and Continuous Dynamical Systems - B, 1(1):103–124, 2001. https://doi.org/10.3934/dcdsb.2001.1.103
J. Li, Y. Kuang and C.C. Mason. Modeling the glucoseinsulin regulatory system and ultradian insulin secretory oscillations with two explicit time delays. Journal of Theoretical Biology, 242(3):722–735, 2006. https://doi.org/10.1016/j.jtbi.2006.04.002
J. Li, M. Wang, A. De Gaetano, P. Palumbo and S. Panunzi. The range of time delay and the global stability of the equilibrium for an IVGTT model. Mathematical Biosciences, 235(2):128–137, 2012. https://doi.org/10.1016/j.mbs.2011.11.005
G. Pacini and R.N. Bergman. MINMOD: a computer program to calculate insulin sensitivity and pancreatic responsivity from the frequently sampled intravenous glucose tolerance test. Computer Methods and Programs in Biomedicine, 23(2):113–122, 1986. https://doi.org/10.1016/0169-2607(86)90106-9
S. Saber, E.B.M. Bashier, S.M. Alzahrani and I.A. Noaman. A mathematical model of glucose-insulin interaction with time delay. Journal of Applied & Computational Mathematics, 3(7):416–421, 2018. https://doi.org/10.4172/2168-9679.1000416
G.M. Steil, A. Volund, S.E. Kahn and R.N. Bergman. Reduced Sample Number for Calculation of Insulin Sensitivity and Glucose Effectiveness From the Minimal Model: Suitability for Use in Population Studies. Diabetes, 42(2):250–256, 1993. https://doi.org/10.2337/diab.42.2.250