Stability Analysis of Discretized Model of Glucose–Insulin Homeostasis

Year 2021, Volume: 17 Issue: 4, 369 - 377, 29.12.2021

İlkem Turhan Çetinkaya , Mehmet Kocabıyık , Mevlüde Yakıt Ongun

Abstract

In this paper, the mathematical model which describes the glucose-insulin homeostasis in healthy rats is considered. The model is discretized by constructing a nonstandard finite difference (NSFD) scheme to obtain the numerical solutions. The equilibrium point of the discretized model is determined and stability analysis of the discretized model is discussed. The effect of time step sizes on 4th order Runge-Kutta method and NSFD method is presented. Also, comparison of NSFD scheme solution, Runge-Kutta-Fehlberg Method (RKF45) solution and analytical solution are presented in graphical form. The effectiveness of the proposed method in the solution and interpretation of the model is observed.
Keywords: Glucose-Insulin homeostasis, nonstandard finite difference scheme, stability analysis.

Keywords

Glucose-Insulin homeostasis, nonstandard finite difference scheme, stability analysis

References

• 1. Chen, CL, Tsai, HW, Wong, SS. 2010. Modeling the physiological glucose–insulin dynamic system on diabetics. Journal of Theoretical Biology; 265: 314–322.
• 2. Dereouich, M, Boutayeb, A. 2002. The effect of physical exercise on the dynamics of glucose and insulin. Journal of Biomechanics; 35: 911–917.
• 3. Neatpisarnvanit, C, Boston, JR. 2002. Estimation of Plasma Insulin From Plasma Glucose. IEEE Transactions on biomedical engineering, 49(11): 1253-1259.
• 4. Li, J, Kuang,Y. Analysis of IVGTT glucose-insulin interaction models with time delay. 2001. Discrete and Continuous Dynamical Systems- Series B; 1(1): 103–124.
• 5. Hussain, J, Zadeng, D. 2014. A mathematical model of glucose-insulin interaction. Science Vision; 14(2): 84-88.
• 6. Wang, H, Li, J, Kuang, Y. 2007. Mathematical modeling and qualitative analysis of insulin therapies. Mathematical Biosciences; 210: 17–33.
• 7. Al-Kahby, H, Dannan, F, Elaydi, S. Non-standard Discretization Methods for Some Biological Models. 2000. Applications of Nonstandard Finite Difference Schemes; 155-180.
• 8. Mickens, RE. Difference Equations Theory and Applications, Atlanta, Ga, USA: Chapman & Hall, 1990.
• 9. Mickens, RE. Advances in the applications of Nonstandard Finite Difference Schemes, Singapore: Wiley-Interscience, 2005.
• 10. Mickens, RE. Nonstandard finite difference models of differential equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1994.
• 11. Mickens, RE. 2002. Nonstandard Finite Difference Schemes for Differential Equations. Journal of Difference Equations and Applications; 8(9): 823-847, DOI:10.1080/1023619021000000807.
• 12. Mickens, RE. Exact solutions to a finite-difference model of a nonlinear reaction-advection equation: implications for numerical analysis. 1989. Numerical Methods for Partial Differential Equations; 5(4): 313-325.
• 13. Arenas, AJ, Morano, JA, Cortes, JC. Non-standard numerical method for a mathematical model of RSV epidemiological transmission. 2008. Computers and Mathematics with Applications; 56: 670-678.
• 14. Khalsaraei MM, Khodadosti, F. 2014. Nonstandard Finite Difference Schemes for Differential Equations, Sahand Communications in Mathematical Analysis; 1(2): 47-54.
• 15. Yakıt Ongun, M, Turhan, İ. 2013. A Numerical Comparison for a Discrete HIV Infection of CD4+T-Cell Model Derived from Nontandard Numerical Scheme. Journal of Applied Mathematics; 2013.
• 16. Yakıt Ongun, M, Arslan, D, Garrappa, R. 2013. Nonstandard finite difference schemes for a fractional-order Brusselator system. 2013. Advances in Difference Equations; 2013(102).
• 17. Lombarte, M, Lupo, M, Campetelli, G, Basualdo, M, Rigalli, A. Mathematical model of glucose–insulin homeostasis in healthy rats. 2013. Mathematical Biosciences; 245: 269–277.
• 18. Gopal, M. Digital Control Engineering, New Delhi: New Age International (P) Limited, Publishers, 2003.
• 19. Gopal, M. Digital Control and State Variable Methods, Conventional and Intelligent Control Systems, New Delhi: Tata McGraw-Hill Publishing Company Limited, 2009.