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Analysis of a Fractional-order Glucose-Insulin Biological System with Time Delay

Year 2022, Volume 4, Issue 1, 10 - 18, 30.03.2022
https://doi.org/10.51537/chaos.988758

Abstract

In the human glucose-insulin regulatory system, diverse metabolic issues can arise, including diabetes type I and type II, hyperinsulinemia, hypoglycemia, etc. Therefore, the analysis and characterization of such a biological system is a must. It is well known that mathematical models are an excellent option to study and predict natural phenomena to some extent. On the other hand, fractional-order calculus provides a generalization of derivatives and integrals to arbitrary orders giving us a framework to add memory properties and an extra degree of freedom to the mathematical models to approximate real-world phenomena with higher accuracy. In this work, we introduce a fractional-order version of a mathematical model of the glucose-insulin regulatory system. Using the fractional-order Caputo derivative, we can investigate different concentration rates among insulin, glucose, and healthy beta cells. Additionally, the model incorporates two time-lags to represent the elapsed time in insulin secretion in response to blood glucose level and the delay in glucose drop due to increased insulin concentration. Analytical results of the equilibrium points and their corresponding stability are given. Numerical results, including phase portraits and bifurcation diagrams, reveal that the fractional-order increases the chaotic regions, leading to potential metabolic problems. Vice versa, the system seems to work correctly when the behavior evolves to periodic windows.

References

  • ADA, A. D. A., 2020 Classification and diagnosis of diabetes: Standards of medical care in diabetes. Diabetes Care 43: S14–S31.
  • Al-Hussein, A.-B. A., F. Rahma, L. Fortuna, M. Bucolo, M. Frasca, et al., 2020 A new time-delay model for chaotic glucose-insulin regulatory system. International Journal of Bifurcation and Chaos 30: 2050178.
  • Aram, Z., S. Jafari, J. Ma, J. C. Sprott, S. Zendehrouh, et al., 2017 Using chaotic artificial neural networks to model memory in the brain. Communications in Nonlinear Science and Numerical Simulation 44: 449–459.
  • Assadi, I., A. Charef, D. Copot, R. D. Keyser, T. Bensouici, et al., 2017 Evaluation of respiratory properties by means of fractional order models. Biomedical Signal Processing and Control 34: 206 – 213.
  • Baghdadi, G., S. Jafari, J. C. Sprott, F. Towhidkhah, and M. H. Golpayegani, 2015 A chaotic model of sustaining attention problem in attention deficit disorder. Communications in Nonlinear Science and Numerical Simulation 20: 174–185.
  • Bajaj, J., G. S. Rao, J. S. Rao, and R. Khardori, 1987 A mathematical model for insulin kinetics and its application to protein-deficient (malnutrition-related) diabetes mellitus (PDDM). Journal of Theoretical Biology 126: 491 – 503.
  • Bertram, R. and M. Pernarowski, 1998 Glucose diffusion in pancreatic islets of langerhans. Biophysical Journal 74: 1722 – 1731.
  • Chinnathambi, R., F. A. Rihan, and H. J. Alsakaji, 2021 A fractionalorder model with time delay for tuberculosis with endogenous reactivation and exogenous reinfections. Mathematical Methods in the Applied Sciences 44: 8011–8025.
  • Chuedoung, M.,W. Sarika, and Y. Lenbury, 2009 Dynamical analysis of a nonlinear model for glucose–insulin system incorporating delays and β -cells compartment. Nonlinear Analysis: Theory, Methods & Applications 71: e1048 – e1058.
  • Diethelm, K., 2010 The analysis of fractional differential equations: An application-oriented exposition using differential operators of Caputo type. Springer Science & Business Media.
  • Emerging Risk Factors Collaboration et al., 2010 Diabetes mellitus, fasting blood glucose concentration, and risk of vascular disease: a collaborative meta-analysis of 102 prospective studies. Lancet 375: 2215–22.
  • Forrest, E. and H. M. Payne Robinson, 1925 Insulin, growth hormone and carbohydrate tolerance in jamaican children rehabilitated from severe malnutrition.
  • Ionescu, C., A. Lopes, D. Copot, J. Machado, and J. Bates, 2017 The role of fractional calculus in modeling biological phenomena: A review. Communications in Nonlinear Science and Numerical Simulation 51: 141 – 159.
  • Jafari, A., I. Hussain, F. Nazarimehr, S. M. R. H. Golpayegani, and S. Jafari, 2021 A simple guide for plotting a proper bifurcation diagram. International Journal of Bifurcation and Chaos 31: 2150011.
  • Kroll, M. H., 1999 Biological variation of glucose and insulin includes a deterministic chaotic component. Biosystems 50: 189– 201.
  • Lakshmanan, M. and D. Senthilkumar, 2010 Dynamics of Nonlinear Time-Delay Systems. Springer-Verlag Berlin Heidelberg.
  • Lazarevi´c, M., 2011 Stability and stabilization of fractional order time delay systems. Scientific Technical Review 61: 31–45.
  • Lenbury, Y., S. Ruktamatakul, and S. Amornsamarnkul, 2001 Modeling insulin kinetics: responses to a single oral glucose administration or ambulatory-fed conditions. Biosystems 59: 15 – 25.
  • Li, J., Y. Kuang, and C. C. Mason, 2006 Modeling the glucose– insulin regulatory system and ultradian insulin secretory oscillations with two explicit time delays. Journal of theoretical biology 242: 722–735.
  • Lozano, J. A., 2006 Diabetes mellitus. Offarm 25 (10): 66–78. Muñoz-Pacheco, J. M., 2019 Infinitely many hidden attractors in a new fractional-order chaotic system based on a fracmemristor.
  • The European Physical Journal Special Topics 228: 2185–2196. Munoz-Pacheco, J. M., C. Posadas-Castillo, and E. Zambrano- Serrano, 2020 The effect of a non-local fractional operator in an asymmetrical glucose-insulin regulatory system: Analysis, synchronization and electronic implementation. Symmetry 12: 1395.
  • Naifar, O., A. M. Nagy, A. B. Makhlouf, M. Kharrat, and M. A. Hammami, 2019 Finite-time stability of linear fractional-order time-delay systems. International Journal of Robust and Nonlinear Control 29: 180–187.
  • Palumbo, P., S. Panunzi, and A. De Gaetano, 2007 Qualitative behavior of a family of delay-differential models of the glucoseinsulin system. Discrete and Continuous Dynamical Systems. Series B 2.
  • Petráˆs, I., 2011 Fractional-Order Nonlinear Systems. Springer. Prager, R., P. Wallace, and J. M. Olefsky, 1986 In vivo kinetics of insulin action on peripheral glucose disposal and hepatic glucose output in normal and obese subjects. The Journal of Clinical Investigation 78: 472–481.
  • Rajagopal, K., V.-T. Pham, F. R. Tahir, A. Akgul, H. R. Abdolmohammadi, et al., 2018 A chaotic jerk system with non-hyperbolic equilibrium: Dynamics, effect of time delay and circuit realisation. Pramana 90: 1–8.
  • Rihan, F., A. Arafa, R. Rakkiyappan, C. Rajivganthi, and Y. Xu, 2021 Fractional-order delay differential equations for the dynamics of hepatitis c virus infection with ifn-α treatment. Alexandria Engineering Journal 60: 4761–4774.
  • Rihan, F. A., 2013 Numerical modeling of fractional-order biological systems. Abstract and Applied Analysis 2013: 11.
  • R.Rosalba, B.Ana, A.Carlos, Z.Emiliano, V.Salvador, et al., 2018 Prevalencia de diabetes por diagnóstico médico previo en méxico. Salud Pública de México 60: 1–9.
  • Sano, M. and Y. Sawada, 1985 Measurement of the lyapunov spectrum from a chaotic time series. Physical review letters 55: 1082.
  • Sarika, W., Y. Lenburya, K. Kumnungkit, and W. Kunphasuruang, 2008 Modelling glucose-insulin feedback signal interchanges involving β-cells with delays. ScienceAsia 34: 77–86.
  • Shabestari, P. S., K. Rajagopal, B. Safarbali, S. Jafari, and P. Duraisamy, 2018 A novel approach to numerical modeling of metabolic system: Investigation of chaotic behavior in diabetes mellitus. Complexity 2018.
  • Shafiei, M., F. Parastesh, M. Jalili, S. Jafari, M. Perc, et al., 2019 Effects of partial time delays on synchronization patterns in izhikevich neuronal networks. The European Physical Journal B 92: 1–7.
  • Singh, H. K. and D. N. Pandey, 2021 Stability analysis of a fractional-order delay dynamical model on oncolytic virotherapy. Mathematical Methods in the Applied Sciences 44: 1377–1393.
  • Sprott, J. C., 2015 Strange attractors with various equilibrium types. The European Physical Journal Special Topics 224: 1409–1419.
  • Statista, 2019 Countries with the highest number of diabetics worldwide in 2019. https://www.statista.com/statistics/281082/ countries-with-highest-number-of-diabetics/, Accessed: 2021-07- 30.
  • Tasaka, Y., F. Nakaya, H. Matsumoto, and Y. Omori, 1994 Effects of aminoguanidine on insulin release from pancreatic islets. Endocrine journal 41: 309–313.
  • Teka, W. W., R. K. Upadhyay, and A. Mondal, 2018 Spiking and bursting patterns of fractional-order izhikevich model", journal = "communications in nonlinear science and numerical simulation 56: 161 – 176.
  • Yao, Z. and B. Tang, 2021 Further results on bifurcation for a fractional-order predator-prey system concerning mixed time delays. Discrete Dynamics in Nature and Society 2021.
  • Zambrano-Serrano, E., J. M. Munoz-Pacheco, L. C. Gómez- Pavón, A. Luis-Ramos, and G. Chen, 2018 Synchronization in a fractional-order model of pancreatic β -cells. The European Physical Journal Special Topics 227: 907–919.

Year 2022, Volume 4, Issue 1, 10 - 18, 30.03.2022
https://doi.org/10.51537/chaos.988758

Abstract

References

  • ADA, A. D. A., 2020 Classification and diagnosis of diabetes: Standards of medical care in diabetes. Diabetes Care 43: S14–S31.
  • Al-Hussein, A.-B. A., F. Rahma, L. Fortuna, M. Bucolo, M. Frasca, et al., 2020 A new time-delay model for chaotic glucose-insulin regulatory system. International Journal of Bifurcation and Chaos 30: 2050178.
  • Aram, Z., S. Jafari, J. Ma, J. C. Sprott, S. Zendehrouh, et al., 2017 Using chaotic artificial neural networks to model memory in the brain. Communications in Nonlinear Science and Numerical Simulation 44: 449–459.
  • Assadi, I., A. Charef, D. Copot, R. D. Keyser, T. Bensouici, et al., 2017 Evaluation of respiratory properties by means of fractional order models. Biomedical Signal Processing and Control 34: 206 – 213.
  • Baghdadi, G., S. Jafari, J. C. Sprott, F. Towhidkhah, and M. H. Golpayegani, 2015 A chaotic model of sustaining attention problem in attention deficit disorder. Communications in Nonlinear Science and Numerical Simulation 20: 174–185.
  • Bajaj, J., G. S. Rao, J. S. Rao, and R. Khardori, 1987 A mathematical model for insulin kinetics and its application to protein-deficient (malnutrition-related) diabetes mellitus (PDDM). Journal of Theoretical Biology 126: 491 – 503.
  • Bertram, R. and M. Pernarowski, 1998 Glucose diffusion in pancreatic islets of langerhans. Biophysical Journal 74: 1722 – 1731.
  • Chinnathambi, R., F. A. Rihan, and H. J. Alsakaji, 2021 A fractionalorder model with time delay for tuberculosis with endogenous reactivation and exogenous reinfections. Mathematical Methods in the Applied Sciences 44: 8011–8025.
  • Chuedoung, M.,W. Sarika, and Y. Lenbury, 2009 Dynamical analysis of a nonlinear model for glucose–insulin system incorporating delays and β -cells compartment. Nonlinear Analysis: Theory, Methods & Applications 71: e1048 – e1058.
  • Diethelm, K., 2010 The analysis of fractional differential equations: An application-oriented exposition using differential operators of Caputo type. Springer Science & Business Media.
  • Emerging Risk Factors Collaboration et al., 2010 Diabetes mellitus, fasting blood glucose concentration, and risk of vascular disease: a collaborative meta-analysis of 102 prospective studies. Lancet 375: 2215–22.
  • Forrest, E. and H. M. Payne Robinson, 1925 Insulin, growth hormone and carbohydrate tolerance in jamaican children rehabilitated from severe malnutrition.
  • Ionescu, C., A. Lopes, D. Copot, J. Machado, and J. Bates, 2017 The role of fractional calculus in modeling biological phenomena: A review. Communications in Nonlinear Science and Numerical Simulation 51: 141 – 159.
  • Jafari, A., I. Hussain, F. Nazarimehr, S. M. R. H. Golpayegani, and S. Jafari, 2021 A simple guide for plotting a proper bifurcation diagram. International Journal of Bifurcation and Chaos 31: 2150011.
  • Kroll, M. H., 1999 Biological variation of glucose and insulin includes a deterministic chaotic component. Biosystems 50: 189– 201.
  • Lakshmanan, M. and D. Senthilkumar, 2010 Dynamics of Nonlinear Time-Delay Systems. Springer-Verlag Berlin Heidelberg.
  • Lazarevi´c, M., 2011 Stability and stabilization of fractional order time delay systems. Scientific Technical Review 61: 31–45.
  • Lenbury, Y., S. Ruktamatakul, and S. Amornsamarnkul, 2001 Modeling insulin kinetics: responses to a single oral glucose administration or ambulatory-fed conditions. Biosystems 59: 15 – 25.
  • Li, J., Y. Kuang, and C. C. Mason, 2006 Modeling the glucose– insulin regulatory system and ultradian insulin secretory oscillations with two explicit time delays. Journal of theoretical biology 242: 722–735.
  • Lozano, J. A., 2006 Diabetes mellitus. Offarm 25 (10): 66–78. Muñoz-Pacheco, J. M., 2019 Infinitely many hidden attractors in a new fractional-order chaotic system based on a fracmemristor.
  • The European Physical Journal Special Topics 228: 2185–2196. Munoz-Pacheco, J. M., C. Posadas-Castillo, and E. Zambrano- Serrano, 2020 The effect of a non-local fractional operator in an asymmetrical glucose-insulin regulatory system: Analysis, synchronization and electronic implementation. Symmetry 12: 1395.
  • Naifar, O., A. M. Nagy, A. B. Makhlouf, M. Kharrat, and M. A. Hammami, 2019 Finite-time stability of linear fractional-order time-delay systems. International Journal of Robust and Nonlinear Control 29: 180–187.
  • Palumbo, P., S. Panunzi, and A. De Gaetano, 2007 Qualitative behavior of a family of delay-differential models of the glucoseinsulin system. Discrete and Continuous Dynamical Systems. Series B 2.
  • Petráˆs, I., 2011 Fractional-Order Nonlinear Systems. Springer. Prager, R., P. Wallace, and J. M. Olefsky, 1986 In vivo kinetics of insulin action on peripheral glucose disposal and hepatic glucose output in normal and obese subjects. The Journal of Clinical Investigation 78: 472–481.
  • Rajagopal, K., V.-T. Pham, F. R. Tahir, A. Akgul, H. R. Abdolmohammadi, et al., 2018 A chaotic jerk system with non-hyperbolic equilibrium: Dynamics, effect of time delay and circuit realisation. Pramana 90: 1–8.
  • Rihan, F., A. Arafa, R. Rakkiyappan, C. Rajivganthi, and Y. Xu, 2021 Fractional-order delay differential equations for the dynamics of hepatitis c virus infection with ifn-α treatment. Alexandria Engineering Journal 60: 4761–4774.
  • Rihan, F. A., 2013 Numerical modeling of fractional-order biological systems. Abstract and Applied Analysis 2013: 11.
  • R.Rosalba, B.Ana, A.Carlos, Z.Emiliano, V.Salvador, et al., 2018 Prevalencia de diabetes por diagnóstico médico previo en méxico. Salud Pública de México 60: 1–9.
  • Sano, M. and Y. Sawada, 1985 Measurement of the lyapunov spectrum from a chaotic time series. Physical review letters 55: 1082.
  • Sarika, W., Y. Lenburya, K. Kumnungkit, and W. Kunphasuruang, 2008 Modelling glucose-insulin feedback signal interchanges involving β-cells with delays. ScienceAsia 34: 77–86.
  • Shabestari, P. S., K. Rajagopal, B. Safarbali, S. Jafari, and P. Duraisamy, 2018 A novel approach to numerical modeling of metabolic system: Investigation of chaotic behavior in diabetes mellitus. Complexity 2018.
  • Shafiei, M., F. Parastesh, M. Jalili, S. Jafari, M. Perc, et al., 2019 Effects of partial time delays on synchronization patterns in izhikevich neuronal networks. The European Physical Journal B 92: 1–7.
  • Singh, H. K. and D. N. Pandey, 2021 Stability analysis of a fractional-order delay dynamical model on oncolytic virotherapy. Mathematical Methods in the Applied Sciences 44: 1377–1393.
  • Sprott, J. C., 2015 Strange attractors with various equilibrium types. The European Physical Journal Special Topics 224: 1409–1419.
  • Statista, 2019 Countries with the highest number of diabetics worldwide in 2019. https://www.statista.com/statistics/281082/ countries-with-highest-number-of-diabetics/, Accessed: 2021-07- 30.
  • Tasaka, Y., F. Nakaya, H. Matsumoto, and Y. Omori, 1994 Effects of aminoguanidine on insulin release from pancreatic islets. Endocrine journal 41: 309–313.
  • Teka, W. W., R. K. Upadhyay, and A. Mondal, 2018 Spiking and bursting patterns of fractional-order izhikevich model", journal = "communications in nonlinear science and numerical simulation 56: 161 – 176.
  • Yao, Z. and B. Tang, 2021 Further results on bifurcation for a fractional-order predator-prey system concerning mixed time delays. Discrete Dynamics in Nature and Society 2021.
  • Zambrano-Serrano, E., J. M. Munoz-Pacheco, L. C. Gómez- Pavón, A. Luis-Ramos, and G. Chen, 2018 Synchronization in a fractional-order model of pancreatic β -cells. The European Physical Journal Special Topics 227: 907–919.

Details

Primary Language English
Subjects Engineering, Multidisciplinary
Journal Section Research Articles
Authors

B. FERNÁNDEZ-CARREÓN
Benemérita Universidad Autónoma de Puebla
0000-0001-7272-4110
Mexico


J. M. MUÑOZ-PACHECO (Primary Author)
Benemérita Universidad Autónoma de Puebla
0000-0002-9106-6982
Mexico


E. ZAMBRANO-SERRANO This is me
Universidad Autónoma de Nuevo León
0000-0002-2115-0097
Mexico


O. G. FÉLİX-BELTRÁN This is me
Benemérita Universidad Autónoma de Puebla
0000-0003-4626-1954
Mexico

Supporting Institution VIEP-BUAP
Project Number 2021: BUAP-CA-276
Thanks This work was supported by 2021 VIEP-BUAP project.
Publication Date March 30, 2022
Published in Issue Year 2022, Volume 4, Issue 1

Cite

APA Fernández-carreón, B. , Muñoz-pacheco, J. M. , Zambrano-serrano, E. & Félix-beltrán, O. G. (2022). Analysis of a Fractional-order Glucose-Insulin Biological System with Time Delay . Chaos Theory and Applications , 4 (1) , 10-18 . DOI: 10.51537/chaos.988758

Chaos Theory and Applications in Applied Sciences and Engineering: An interdisciplinary journal of nonlinear science 23830