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Bifurcation analysis in a delay model of IVGTT glucose–insulin interaction

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Abstract

In this paper, a delayed differential model based on the intravenous glucose tolerance test is considered. The conditions to determine stability or instability of the model’s steady state are obtained. We obtain the necessary conditions for the appearance of a bifurcation, and we investigate the direction and stability of the local bifurcation. For this purpose, the normal form theory is used. In addition, the numerical diagrams in the direction of theoretical results are drawn.

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References

  • Ackerman E, Rosevear JW, McGuckin WF (1964) A mathematical model of the glucose tolerance test. Phys Med Biol 9(2):203

    Article  Google Scholar 

  • Ackerman E, Gatewood LC, Rosevear JW, Molnar GD (1965) Model studies of blood-glucose regulation. Bull Math Biophys 27:21–37

    Article  CAS  Google Scholar 

  • Balachandran B, Kalmar-Nagy T, Gilsinn DE (2009) Delay differential equations. Springer, Berlin

    Google Scholar 

  • Bennett DL, Gourley SA (2004) Asymptotic properties of a delay differential equation model for the interaction of glucose with plasma and interstitial insulin. Appl Math Comput 151:189–207

    Google Scholar 

  • Bergman RN, Cobelli C (1980) Minimal modelling, partition analysis and the estimation of insulin sensitivity. Fed Proc 39:110–115

    CAS  PubMed  Google Scholar 

  • Bergman RN, Ider YZ, Bowden CR, Cobelli C (1979) Quantitative estimation of insulin sensitivity. Am J Physiol 236:667–677

    Google Scholar 

  • Bianca C, Ferrara M, Guerrini L (2013) The Cai model with time delay: existence of periodic solutions and asymptotic analysis. Appl Math Inf Sci 7:21–27

    Article  Google Scholar 

  • Bolie VW (1961) Coefficients of normal blood glucose regulation. J Appl Physiol 16:783–788

    Article  CAS  Google Scholar 

  • Caumo A, Bergman RN, Cobelli C (2000) Insulin sensitivity from meal tolerance tests in normal subjects: a minimal model index. J Clin Endocrinol Metab 85(11):4396–402

    Article  CAS  Google Scholar 

  • Cooke KL, van den Driessche P (1986) On zeroes of some transcendental equations. Funkcial Ekvac 29:77–90

    Google Scholar 

  • De Gaetano A, Arino O (2000) Mathematical modeling of the intravenous glucose tolerance test. J Math Biol 40:136–168

    Article  Google Scholar 

  • Engelborghs K, Lemaire V, Bélair J, Roose D (2001) Numerical bifurcation analysis of delay differential equations arising from physiological modeling. J Math Biol 42:361–385

    Article  CAS  Google Scholar 

  • Gatewood LC, Ackerman E, Rosevear JW, Molnar GD (1968) Test of a mathematical model of the blood-glucose regulatory system. Comput Biomed Res 2:1–14

    Article  CAS  Google Scholar 

  • Gresl TA, Colman RJ, Havighurst TC, Byerley LO, Allison DB, Schoeller DA, Kemnitz JW (2003) Insulin sensitivity and glucose effectiveness from three minimal models: effects of energy restriction and body fat in adult male rhesus monkeys. Am J Physiol Regul Integr Comput Physiol 285:R1340–54

    Article  CAS  Google Scholar 

  • Hale JK, Verduyn Lunel SM (1993) Introduction to functional differential equations. Springer, New York

    Book  Google Scholar 

  • Hassard B, Wan YH (1978) Bifurcation formulae derived from center manifold theory. J Math Anal Appl 63:297–312

    Article  Google Scholar 

  • Hassard BD, Kazarinoff ND, Wan YH (1981) Theory and applications of Hopf bifurcations. Cambridge University Press, Cambridge

    Google Scholar 

  • Kuang Y (1993) Delay differential equations with applications in population dynamics, vol 191. Mathematics in science and engineering. Academic Press, Boston

    Google Scholar 

  • Kuznetsov YA (1995) Elements of applied bifurcation theory. Springer, New York

    Book  Google Scholar 

  • Li J, Kuang Y (2007) Analysis of a model of the glucose–insulin regulatory system with two delays. SIAM J Appl Math 67(3):757–776

    Article  CAS  Google Scholar 

  • Li J, Kuang Y, Li B (2001) Analysis of IVGTT glucose–insulin interaction models with time delay. Discrete Contin Dyn Syst Ser B 1:103–124

    Google Scholar 

  • Li J, Kuang Y, Mason CC (2006) Modeling the glucose–insulin regulatory system and ultradian insulin secretory oscillations with two explicit time delays. J Theor Biol 242:722–735

    Article  CAS  Google Scholar 

  • Li J, Wang M, De Gaetano A, Palumbo P, Panunzi S (2012) The range of time delay and the global stability of the equilibrium for an IVGTT model. Math Biosci 235:128–137

    Article  Google Scholar 

  • Mukhopadhyay A, De Gaetano A, Arino O (2004) Modelling the intra-venous glucose tolerance test: a global study for a single distributed delay model. Discrete Contin Dyn Syst Ser B 4(2):407–417

    Google Scholar 

  • Palumbo P, Panunzi S, De Gaetano A (2007) Qualitative behavior of a family of delay-differential models of the glucose–insulin system. Discrete Contin Dyn Syst Ser B 7(2):399–424

    Google Scholar 

  • Panunzi S, Palumbo P, De Gaetano A (2007) A discrete single delay model for the intra-venous glucose tolerance test. Theor Biol Med Model 4(35):1

    Google Scholar 

  • Panunzi S, De Gaetano A, Mingrone G (2010) Advantages of the single delay model for the assessment of insulin sensitivity from the intra-venous glucose tolerance test. Theor Biol Med Model 7:9

    Article  Google Scholar 

  • Shi X, Kuang Y, Makroglou A, Mokshagundam S, Li J (2017) Oscillatory dynamics of an intravenous glucose tolerance test model with delay interval. Chaos 27:114324

    Article  Google Scholar 

  • Steil GM, Volund A, Kahn SE, Bergman RN (1993) Reduced sample number for calculation of insulin sensitivity and glucose effectiveness from the minimal model. Diabetes 42:250–756

    Article  CAS  Google Scholar 

  • Sturis J, Polonsky KS, Mosekilde E, Van Cauter E (1991) Computer-model for mechanisms underlying ultradian oscillations of insulin and glucose. Am J Physiol 260:801–809

    Google Scholar 

  • Toffolo G, Bergman RN, Finegood DT, Bowden CR, Cobelli C (1980) Quantitative estimation of beta cells sensitivity to glucose in the intact organism: a minimal model of insulin kinetics in dog. Diabetes 29:979–990

    Article  CAS  Google Scholar 

  • Wang H, Li J, Kuang Y (2009) Enhanced modelling of the glucose–insulin system and its application in insulin therapies. J Biol Dyn 3(1):22–38

    Article  CAS  Google Scholar 

  • Wiggins S (1990) Introduction to applied nonlinear dynamical systems and chaos. Springer, New York

    Book  Google Scholar 

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Correspondence to MohammadReza Molaei.

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Mohabati, F., Molaei, M. Bifurcation analysis in a delay model of IVGTT glucose–insulin interaction. Theory Biosci. 139, 9–20 (2020). https://doi.org/10.1007/s12064-019-00298-y

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