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Bifurcation dynamics of the modified physiological model of artificial pancreas with insulin secretion delay

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Abstract

In this paper, we modify the original physiological model of artificial pancreas by introducing the insulin secretion time delay. The non-resonant double Hopf bifurcation is analyzed by the Center Manifold Theorem and Normal Form Method. Numerical results supporting the theoretical analysis are presented in some typical parameter regions. It is shown that the critical value of technological delay and the area of death island of the non-resonant double Hopf bifurcation in the modified model are far less than those in the original model. This implies that when the secretion delay appears, the smaller technological delay can induce the double Hopf bifurcation. In addition, the region IV with complex coexisting bi-stability also decreases sharply. Furthermore, the rich dynamics such as various period, quasi-period and chaotic behaviors are found when some key parameters are changed. The obtained results can have important theoretical guidance for the diagnosis and treatment of diabetes patients.

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References

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

    Article  MATH  MathSciNet  Google Scholar 

  2. The Diabetes Control and Complications Trial Research Group: The effect of intensive treatment of diabetes on the development and progression of long-term complications in insulin dependent diabetes mellitus. New England J. Med. 329(14), 977–986 (1993)

    Article  Google Scholar 

  3. Pfeiffer, E.F., Thum, C., Clemens, A.H.: The artificial beta cella continuous control of blood sugar by external regulation of insulin. Horm. Metab. Res. 6(7), 339–342 (1974)

    Article  Google Scholar 

  4. Mirouze, J., Selam, J.L., Pham, T.C., Cavadore, D.: Evaluation of exogenous insulin homoeostasis by the artificial pancreas in insulin-dependent diabetes. Diabetologia 13(3), 273–278 (1977)

    Article  Google Scholar 

  5. Heinemann, L., Ampudia-Blasco, F.J.: Glucose clamps with the biostator: a critical reappraisal. Horm. Metab. Res. 26(12), 579–583 (1994)

    Article  Google Scholar 

  6. Sternberg, F., et al.: Subcutaneous glucose concentration in humans: real estimation and continuous monitoring. Diabetes Care 18(9), 1266–1269 (1995)

    Article  Google Scholar 

  7. Tamada, J.A., Bohannon, N.J., Potts, R.O.: Measurement of glucose in diabetic subjects using noninvasive transdermal extraction. Nat. Med. 1(11), 1198–1201 (1995)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. Makroglou, A., Li, J., Kuang, Y.: Mathematical models and software tools for the glucose-insulin regulatory system and diabetes: an overview. Appl. Numer. Math. 56, 559–573 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  11. Xu, J., Pei, L.: Effects of technological delay on insulin and blood glucose in a physiological model. Int. J. Non-Linear Mech. 45(6), 628–633 (2010)

    Article  Google Scholar 

  12. Takens, F.: Singularities of vector fields. Publ. Math. deLIHE S 43(1), 47–100 (1974)

    MathSciNet  Google Scholar 

  13. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York (1983)

    MATH  Google Scholar 

  14. Xu, J., Chung, K.W.: Effects of time delayed position feedback on a van der Pol–Duffing oscillator. Physica D 180(1–2), 17–39 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  15. Pei, L.J., Xu, J.: Nonresonant double Hopf bifurcation in delayed Stuart–Landau system. J. Vib. Eng. 18(1), 1–6 (2005)

    Google Scholar 

  16. Xu, J., Chung, K.W., Chan, C.L.: An efficient method for studying resonant double Hopf bifurcation in nonlinear systems with delayed feedbacks. SIAM J. Appl. Dyn. Syst. 6(1), 29–60 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  17. Xu, J., Pei, L.J.: The nonresonant double Hopf bifurcation in delayed neural network. Int. J. Comput. Math. 85(1) (2008)

  18. Hale, J.K.: Theory of Functional Differential Equations. Springer, New York (1977)

    MATH  Google Scholar 

  19. Nayfeh, A.H.: Method of Normal Forms. Wiley, New York (1993)

    Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Mathematics, Zhengzhou University, Zhengzhou, Henan, 450001, P.R. China

    Lijun Pei & Hongtao Shi

  2. Department of Dynamics and Control, Beihang University, Beijing, 100191, P.R. China

    Qingyun Wang

Authors
  1. Lijun Pei
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  2. Qingyun Wang
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  3. Hongtao Shi
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Correspondence to Qingyun Wang.

Additional information

This work is supported by National Natural Science Foundation of China (Nos. 10702065, 10972001 and 10832006).

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Pei, L., Wang, Q. & Shi, H. Bifurcation dynamics of the modified physiological model of artificial pancreas with insulin secretion delay. Nonlinear Dyn 63, 417–427 (2011). https://doi.org/10.1007/s11071-010-9812-5

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  • DOI: https://doi.org/10.1007/s11071-010-9812-5

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