Startseite Dynamical analysis of fractional-order of IVGTT glucose–insulin interaction
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

Dynamical analysis of fractional-order of IVGTT glucose–insulin interaction

  • Mansoor H. Alshehri , Sayed Saber EMAIL logo und Faisal Z. Duraihem
Veröffentlicht/Copyright: 26. April 2021
Veröffentlichen auch Sie bei De Gruyter Brill
Informationen für Autor*innen
International Journal of Nonlinear Sciences and Numerical Simulation
Aus der Zeitschrift International Journal of Nonlinear Sciences and Numerical Simulation Band 24 Heft 3

Abstract

This paper proposes a fractional-order model of glucose–insulin interaction. In Caputo’s meaning, the fractional derivative is defined. This model arises in Bergman’s minimal model, used to describe blood glucose and insulin metabolism, after intravenous tolerance testing. We showed that the established model has existence, uniqueness, non-negativity, and boundedness of fractional-order model solutions. The model’s local and global stability was investigated. The parametric conditions under which a Hopf bifurcation occurs in the positive steady state for a proposed model are studied. Moreover, we present a numerical treatment for solving the proposed fractional model using the generalized Euler method (GEM). The model’s local stability and Hopf bifurcation of the proposed model in sense of the GEM are presented. Finally, numerical simulations of the model using the Adam–Bashforth–Moulton predictor corrector scheme and the GEM have been presented to support our analytical results.

Keywords: fractional-order; glucose–insulin interaction; IVGTT; mathematical modeling
Corresponding author: Sayed Saber, Department of Mathematics and Computer Science, Faculty of Science, Beni-Suef University, Beni Suef, Egypt; and Department of Mathematics, Faculty of Sciences and Arts in Baljurashi, Albaha University, Al Bahah, Saudi Arabia, E-mail:

Funding source: King Saud University

Award Identifier / Grant number: RG-1441-439

Acknowledgment

The authors extend their appreciation to the Deanship of Scientific Research at King Saud University for funding this work through research group no. RG-1441-439.

  1. Author contribution: All authors read and approved the final manuscript.

  2. Research funding: The authors extend their appreciation to the Deanship of Scientific Research at King Saud University for funding this work through research group no. RG-1441-439.

  3. Conflict of interest statement: The authors declare that they have no competing interests.

References

[1] V. W. Bolie, “Coefficients of normal blood glucose regulation,” J. Appl. Physiol., vol. 16, pp. 783–788, 1961. https://doi.org/10.1152/jappl.1961.16.5.783.Suche in Google Scholar PubMed

[2] 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,” Comput. Biomed. Res., vol. 2, pp. 1–14, 1968. https://doi.org/10.1016/0010-4809(68)90003-7.Suche in Google Scholar PubMed

[3] R. N. Bergman, Y. Z. Ider, C. R. Bowden, and C. Cobelli, “Quantitative estimation of insulin sensitivity,” Am. J. Physiol., vol. 236, pp. E667–E677, 1979. https://doi.org/10.1152/ajpendo.1979.236.6.e667.Suche in Google Scholar

[4] A. De Gaetano and O. Arino, “Mathematical modelling of the intravenous glucose tolerance test,” J. Math. Biol., vol. 40, pp. 136–168, 2000. https://doi.org/10.1007/s002850050007.Suche in Google Scholar PubMed

[5] S. Saber, E. B. M. Bashier, S. M. Alzahrani, and I. A. Noaman, “A mathematical model of glucose-insulin interaction with time delay,” J. Comput. Appl. Math., vol. 7, no. 3, p. 416, 2018.Suche in Google Scholar

[6] A. Caumo, C. Cobelli, and M. Omenetto, “Over estimation of minimal model glucose effectiveness in presence of insulin response is due to under modeling,” Am. J. Physiol., vol. 278, pp. 481–488, 1999.Suche in Google Scholar

[7] G. Toffolo, R. N. Bergman, D. T. Finegood, C. R. Bowden, and C. Cobelli, “Quantitative estimation of beta cell sensitivity to glucose in the intact organism,” Diabetes, vol. 29, pp. 979–990, 1980. https://doi.org/10.2337/diab.29.12.979.Suche in Google Scholar PubMed

[8] G. Pacini and R. N. Bergman, “MINMOD: A computer program to calculate insulin sensitivity and pancreatic responsivity from the frequently sampled intravenous tolerance test,” Comput. Methods Progr. Biomed., vol. 23, pp. 113–122, 1986. https://doi.org/10.1016/0169-2607(86)90106-9.Suche in Google Scholar PubMed

[9] A. Mahata, S. P. Mondal, S. Alam, and B. Roy, “Mathematical model of glucose–insulin regulatory system on diabetes mellitus in fuzzy and crisp environment,” Ecol. Genet. Genom., vol. 2, pp. 25–34, 2017. https://doi.org/10.1016/j.egg.2016.10.002.Suche in Google Scholar

[10] M. Farman, M. U. Saleem, M. Tabassum, A. Ahmad, and M. Ahmad, “A linear control of composite model for glucose insulin glucagon pump,” Ain Shams Eng. J., vol. 10, no. 4, pp. 867–872, 2019.10.1016/j.asej.2019.04.001Suche in Google Scholar

[11] P. S. Shabestari, S. Panahi, B. Hatef, S. Jafari, and J. C. Sprott, “A new chaotic model for glucose–insulin regulatory system,” Chaos, Solit. Fractals, vol. 112, pp. 44–51, 2018. https://doi.org/10.1016/j.chaos.2018.04.029.Suche in Google Scholar

[12] I. Podlubny, Fractional Differential Equations, New York, NY, USA, Academic Press, 1999.Suche in Google Scholar

[13] J. Singh, D. Kumar, and D. Baleanu, “On the analysis of fractional diabetes model with exponential law,” Adv. Differ. Equ., vol. 2018, no. 1, p. 231, 2018. https://doi.org/10.1186/s13662-018-1680-1.Suche in Google Scholar

[14] A. Rocco and B. J. West, “Fractional calculus and the evolution of fractal phenomena,” Physica A, vol. 265, p. 535, 1999. https://doi.org/10.1016/s0378-4371(98)00550-0.Suche in Google Scholar

[15] U. Khan, R. Ellahi, R. Ullah, et al.., “Correction to: extracting new solitary wave solutions of Benny-Luke equation and Phi-4 equation of fractional order by using (G’/G)-expansion method,” Opt. Quant. Electron., vol. 50, p. 146, 2018. https://doi.org/10.1007/s11082-018-1421-4.Suche in Google Scholar

[16] M. Ali Dokuyucu, “Caputo and Atangana-Baleanu-Caputo fractional derivative applied to garden equation,” Turk. J. Med. Sci., vol. 5, no. 1, pp. 1–7, 2020.Suche in Google Scholar

[17] M. Ali Dokuyucu, E. Celik, H. Bulut, and H. Mehmet Baskonus, “Cancer treatment model with the Caputo-Fabrizio fractional derivative,” Eur. Phys. J. Plus, vol. 133, no. 3, pp. 1–6, 2018. https://doi.org/10.1140/epjp/i2018-11950-y.Suche in Google Scholar

[18] M. Ali Dokuyucu and H. Dutta, “A fractional order model for Ebola Virus with the new Caputo fractional derivative without singular kernel,” Chaos, Solit. Fractals, vol. 134, no. 1, p. 109717, 2020. https://doi.org/10.1016/j.chaos.2020.109717.Suche in Google Scholar

[19] M. Ali Dokuyucu, “A fractional order alcoholism model via Caputo Fabrizio derivative,” AIMS Math., vol. 5, no. 2, pp. 781–797, 2020. https://doi.org/10.3934/math.2020053.Suche in Google Scholar

[20] R. Ullah, R. Ellahi, S. M. Sait, and S. T. Mohyud-Din, “On the fractional-order model of HIV-1 infection of CD4+ T-cells under the influence of antiviral drug treatment,” J. Taibah Univ. Sci., vol. 14, no. 1, pp. 50–59, 2020. https://doi.org/10.1080/16583655.2019.1700676.Suche in Google Scholar

[21] R. Ullah, R. Ellahi, and U. K. Syed Tauseef Mohyud-Din, “Exact traveling wave solutions of fractional order Boussinesq-like equations by applying Exp-function method,” Results Phys., vol. 8, pp. 114–120, 2018.10.1016/j.rinp.2017.11.023Suche in Google Scholar

[22] T. M. Sabri Thabet, S. Mohammed Abdo, K. Shah, and T. Abdeljawad, “Study of transmission dynamics of COVID-19 mathematical model under ABC fractional order derivative,” Results Phys., vol. 19, p. 103507, 2020. https://doi.org/10.1016/j.rinp.2020.103507.Suche in Google Scholar PubMed PubMed Central

[23] H. Alrabaiah, A. Zeb, E. Alzahrani, and K. Shah, “Dynamical analysis of fractional-order tobacco smoking model containing snuffing class,” Alexandria Eng. J., vol. 60, no. 4, pp. 3669–3678, 2021. https://doi.org/10.1016/j.aej.2021.02.005.Suche in Google Scholar

[24] M. Bahar Ali Khan, T. Abdeljawad, K. Shah, G. Ali, H. Khan, and A. Khan, “Study of a nonlinear multi-terms boundary value problem of fractional pantograph differential equations,” Adv. Differ. Equ., vol. 2021, p. 143, 2021. https://doi.org/10.1186/s13662-021-03313-z.Suche in Google Scholar

[25] A. Ali, K. Shah, H. Alrabaiah, Z. Shah, G. Ur Rahman, and S. Islam, “Computational modeling and theoretical analysis of nonlinear fractional order prey-predator system,” Fractals, vol. 29, no. 1, 2021, Art no. 2150001. https://doi.org/10.1142/S0218348X21500018.Suche in Google Scholar

[26] S. T. M. Thabet, M. S. Abdo, and K. Shah, “Theoretical and numerical analysis for transmission dynamics of COVID-19 mathematical model involving Caputo–Fabrizio derivative,” Adv. Differ. Equ., vol. 2021, no. 1, pp. 1–17, 2021. https://doi.org/10.1186/s13662-021-03316-w.Suche in Google Scholar PubMed PubMed Central

[27] Z. Ali, F. Rabiei, K. Shah, and Z. A. Majid, “Dynamics of SIR mathematical model for COVID-19 outbreak in Pakistan under Fractal-fractional derivative,” Accepted in Fractals, 2021. https://doi.org/10.1142/s0218348x21501206.Suche in Google Scholar

[28] M. Sinan, A Ali, K. Shah, T. A. Assiri, and T. A. Nofal, “Stability analysis and optimal control of Covid-19 pandemic SEIQR fractional mathematical model with harmonic mean type incidence rate and treatment,” Results Phys., vol. 22, p. 103873, 2021. https://doi.org/10.1016/j.rinp.2021.103873.Suche in Google Scholar PubMed PubMed Central

[29] M. Arfan, K. Shah, A. Ullah, M. Shutaywi, P. Kumam, and Z. Shah, “On fractional order model of tumor dynamics with drug interventions under nonlocal fractional derivative,” Results Phys., vol. 21, p. 103783. https://doi.org/10.1016/j.rinp.2020.103783.Suche in Google Scholar

[30] Y. Yu, M. Shi, H. Kang, et al.., “Hidden dynamics in a fractional-order memristive Hindmarsh–Rose model,” Nonlinear Dyn., vol. 100, pp. 891–906, 2020. https://doi.org/10.1007/s11071-020-05495-9.Suche in Google Scholar

[31] Y. Yu, B. Han, M. Shi, B. Bao, Y. Chen, and M. Chen, “Complex dynamical behaviors of a fractional-order system based on a locally active memristor,” Complexity, 2019, Art no. 2051053, 13 pages.10.1155/2019/2051053Suche in Google Scholar

[32] M. W. Khan, M. Abid, and A. Qayyum Khan, “Fractional order Bergman’s minimal model-a better representation of blood glucose-insulin system” in 2019 International Conference on Applied and Engineering Mathematics (ICAEM), 27–29 Aug. 2019, https://doi.org/10.1109/icaem.2019.8853741.Suche in Google Scholar

[33] B. S. Alkahtani, O. J. Algahtani, R. S. Dubey, and P. Goswami, “The solution of modified fractional Bergman’s minimal blood glucose–insulin model,” Entropy, vol. 19, no. 114, pp. 1–11, 2017. https://doi.org/10.3390/e19050114.Suche in Google Scholar

[34] S. Sakulrang, E. J. Moore, S. Sungnul, and A. de Gaetano, “A fractional differential equation model for continuous glucose monitoring data,” Adv. Differ. Equ., vol. 2017, no. 1, p. 150, 2017. https://doi.org/10.1186/s13662-017-1207-1.Suche in Google Scholar

[35] M. H. Alshehri, F. Z. Duraihem, A. Ahmad, and S. Saber, “A Caputo (discretization) fractional-order model of glucose-insulin interaction: numerical solution and comparisons with experimental data,” J. Taibah Univ. Sci., vol. 15, no. 1, pp. 26–36, 2021. https://doi.org/10.1080/16583655.2021.1872197.Suche in Google Scholar

[36] N. Lekdee, S. Sirisubtawee, and S. Koonprasert, “Bifurcations in a delayed fractional model of glucose–insulin interaction with incommensurate orders,” Adv. Differ. Equ., vol. 2019, p. 318, 2019. https://doi.org/10.1186/s13662-019-2262-6.Suche in Google Scholar

[37] W. Lin, “Global existence theory and chaos control of fractional differential equations,” J. Math. Anal. Appl., vol. 332, pp. 709–726, 2007. https://doi.org/10.1016/j.jmaa.2006.10.040.Suche in Google Scholar

[38] H.-L. Li, L. Zhang, C. Hu, Y.-L. Jiang, and Z. Teng, “Dynamical analysis of a fractional-order predator-prey model incorporating a prey refuge,” J. Appl. Math. Comput., vol. 54, no. 9, pp. 435–449, 2017. https://doi.org/10.1007/s12190-016-1017-8.Suche in Google Scholar

[39] A. E. Matouk, “Chaos, feedback control and synchronization of a fractional-order modified autonomous Van der Pol-Duffing circuit,” Commun. Nonlinear Sci. Numer. Simul., vol. 16, pp. 975–986, 2016.10.1016/j.cnsns.2010.04.027Suche in Google Scholar

[40] E. Ahmeda and A. S. Elgazzar, “On fractional order differential equations model for nonlocal epidemics,” Physica A, vol. 379, pp. 607–614, 2007. https://doi.org/10.1016/j.physa.2007.01.010.Suche in Google Scholar PubMed PubMed Central

[41] C. Vargas-De-León, “Volterra-type Lyapunov functions for fractional-order epidemic systems,” Commun. Nonlinear Sci. Numer. Simul., vol. 24, pp. 75–85, 2015. https://doi.org/10.1016/j.cnsns.2014.12.013.Suche in Google Scholar

[42] J. Huo, H. Zhao, and L. Zhu, “The effect of vaccines on backward bifurcation in a fractional order HIV model,” Nonlinear Anal. Real World Appl., vol. 26, pp. 289–305, 2015. https://doi.org/10.1016/j.nonrwa.2015.05.014.Suche in Google Scholar

[43] J. P. C dos Santos, E. Monteiro, and J. C. Valverde, “Global stability of fractional SIR epidemic model,” Proc. Ser. Braz. Soc. Comput. Appl. Math., vol. 5, no. 1, pp. 1–7, 2017.10.5540/03.2017.005.01.0019Suche in Google Scholar

[44] M. S. Tavazoei, M. Haeri, M. Attari, S. Bolouki, and M. Siami, “More details on analysis of fractional-order Van der Pol oscillator,” J. Vib. Control, vol. 15, no. 6, pp. 803–819, 2009. https://doi.org/10.1177/1077546308096101.Suche in Google Scholar

[45] Z. M. Odibat and S. Momani, “An algorithm for the numerical solution of differential equations of fractional order,” J. Appl. Math. Inform., vol. 26, nos. 1–2, pp. 15–27, 2008.Suche in Google Scholar

[46] Z. Odibat and N. Shawagfeh, “Generalized Taylor’s formula,” Appl. Math. Comput., vol. 186, pp. 286–293, 2007. https://doi.org/10.1016/j.amc.2006.07.102.Suche in Google Scholar

[47] J. Li, Y. Kuang, and B. Li, “Analysis of IVGTT glucose-insulin interaction models with time delay,” Discrete Contin. Dyn. Syst. Ser. B., vol. 1, no. 1, 2001.10.3934/dcdsb.2001.1.103Suche in Google Scholar

[48] A. Mukhopadhyay, A. De Gaetano, and O. Arino, “Modelling the intra-venous glucose tolerance test: A global study for a single distributed delay model,” Discrete Contin. Dyn. Syst. Ser. B., vol. 4, no. 2, 2004.10.3934/dcdsb.2004.4.407Suche in Google Scholar

[49] S. Panunzi, P. Palumbo, and A. De Gaetano, “Modeling IVGTT data with delay differential equations,” IASI-CNR Research Report, vol. 625, 2004.Suche in Google Scholar
Received: 2020-08-28
Revised: 2021-04-01
Accepted: 2021-04-12
Published Online: 2021-04-26

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Original Research Articles
  3. One-dimensional optimal system and similarity transformations for the 3 + 1 Kudryashov–Sinelshchikov equation
  4. Hopf bifurcations in a network of FitzHugh–Nagumo biological neurons
  5. Gravity modulation effect on weakly nonlinear thermal convection in a fluid layer bounded by rigid boundaries
  6. A new numerical formulation for the generalized time-fractional Benjamin Bona Mohany Burgers’ equation
  7. Chebyshev spectral method for solving a class of local and nonlocal elliptic boundary value problems
  8. Adaptive extragradient methods for solving variational inequalities in real Hilbert spaces
  9. Elastic trend filtering
  10. A new approach to representations of homothetic motions in Lorentz space
  11. Numerical simulation of variable-order fractional differential equation of nonlinear Lane–Emden type appearing in astrophysics
  12. Stability analysis and numerical simulations of the fractional COVID-19 pandemic model
  13. Numerical solutions of the Bagley–Torvik equation by using generalized functions with fractional powers of Laguerre polynomials
  14. N-fold Darboux transformation and exact solutions for the nonlocal Fokas–Lenells equation on the vanishing and plane wave backgrounds
  15. Closed form soliton solutions to the space-time fractional foam drainage equation and coupled mKdV evolution equations
  16. Master-slave synchronization in the Duffing-van der Pol and Φ6 Duffing oscillators
  17. Singularity analysis and analytic solutions for the Benney–Gjevik equations
  18. Trajectory controllability of nonlinear fractional Langevin systems
  19. Similarity transformations for modified shallow water equations with density dependence on the average temperature
  20. Non-equidistant partition predictor–corrector method for fractional differential equations with exponential memory
  21. Dynamical analysis of fractional-order of IVGTT glucose–insulin interaction
  22. Dynamic response of Mathieu–Duffing oscillator with Caputo derivative
  23. Non-Newtonian fluid flow having fluid–particle interaction through a porous zone in a channel with permeable walls
  24. A class of computationally efficient Newton-like methods with frozen inverse operator for nonlinear systems
https://doi.org/10.1515/ijnsns-2020-0201
Schlagwörter für diesen Artikel
fractional-order; glucose–insulin interaction; IVGTT; mathematical modeling

Artikel in diesem Heft

  1. Frontmatter
  2. Original Research Articles
  3. One-dimensional optimal system and similarity transformations for the 3 + 1 Kudryashov–Sinelshchikov equation
  4. Hopf bifurcations in a network of FitzHugh–Nagumo biological neurons
  5. Gravity modulation effect on weakly nonlinear thermal convection in a fluid layer bounded by rigid boundaries
  6. A new numerical formulation for the generalized time-fractional Benjamin Bona Mohany Burgers’ equation
  7. Chebyshev spectral method for solving a class of local and nonlocal elliptic boundary value problems
  8. Adaptive extragradient methods for solving variational inequalities in real Hilbert spaces
  9. Elastic trend filtering
  10. A new approach to representations of homothetic motions in Lorentz space
  11. Numerical simulation of variable-order fractional differential equation of nonlinear Lane–Emden type appearing in astrophysics
  12. Stability analysis and numerical simulations of the fractional COVID-19 pandemic model
  13. Numerical solutions of the Bagley–Torvik equation by using generalized functions with fractional powers of Laguerre polynomials
  14. N-fold Darboux transformation and exact solutions for the nonlocal Fokas–Lenells equation on the vanishing and plane wave backgrounds
  15. Closed form soliton solutions to the space-time fractional foam drainage equation and coupled mKdV evolution equations
  16. Master-slave synchronization in the Duffing-van der Pol and Φ6 Duffing oscillators
  17. Singularity analysis and analytic solutions for the Benney–Gjevik equations
  18. Trajectory controllability of nonlinear fractional Langevin systems
  19. Similarity transformations for modified shallow water equations with density dependence on the average temperature
  20. Non-equidistant partition predictor–corrector method for fractional differential equations with exponential memory
  21. Dynamical analysis of fractional-order of IVGTT glucose–insulin interaction
  22. Dynamic response of Mathieu–Duffing oscillator with Caputo derivative
  23. Non-Newtonian fluid flow having fluid–particle interaction through a porous zone in a channel with permeable walls
  24. A class of computationally efficient Newton-like methods with frozen inverse operator for nonlinear systems
Jetzt anmelden und 20% sparen
Institutioneller Zugang
Wie funktioniert Authentifizierung?
Haben Sie eine Idee, wie wir unsere Website verbessern können?
Bitte schreiben Sie uns.
© 2025 De Gruyter Brill
Heruntergeladen am 18.9.2025 von https://www.degruyterbrill.com/document/doi/10.1515/ijnsns-2020-0201/html
Button zum nach oben scrollen