A new application of fractional glucose-insulin model and numerical solutions

Yıl 2024, Cilt: 42 Sayı: 2, 407 - 413, 30.04.2024

Zafer Öztürk , Halis Bilgil , Sezer Sorgun

Öz

Along with the developing technology, obesity and diabetes are increasing rapidly among people. The identification of diabetes diseases, modeling, predicting their behavior, conducting simulations, studying control and treatment methods using mathematical methods has be-come of great importance. In this paper, we have obtained numerical solutions by considering the glucose-insulin fractional model. This model consists of three compartments: the blood glucose concentration (G), the blood insulin concentration (I) and the ready-to-absorb glucose concentration (D) in the small intestine. The fractional derivative is used in the sense of Caputo. By performing mathematical analyzes for the Glucose-Insulin fractional mathematical model, numerical results were obtained with the help of the Euler method and graphs were drawn.

Anahtar Kelimeler

Fractional Glucose-Insulin Model, Mathematical Modeling, Euler Method, Caputo Derivative

Kaynakça

• [1] Makroglou, A., Li, J., Kuang, Y. Mathematical models and software tools for the glucose-insulin regulatory system and diabetes: an overview. Applied numerical mathematics, 2006;56(3-4):559-573.
• [2] Huard, B., Easton, J. F., Angelova, M. Investigation of stability in a two-delay model of the ultradian oscillations in glucose-insulin regulation. Communications in Nonlinear Science and Numerical Simulation, 2015;26(1-3):211-222.
• [3] Cobelli, C., Nucci, G., Del Prato, S. A physiological simulation model of the glucose-insulin system in type I diabetes. Diabetes nutrition and Metabolism, 1998; 11: 78-78.
• [4] I. Podlubny, Fractional Differential Equations, Academy Press, San Diego CA, 1999.
• [5] Bergman, R. N. The minimal model of glucose regulation: a biography. In Mathematical Modeling in Nutrition and the Health Sciences Springer, Boston, MA., 2003; (pp. 1-19).
• [6] Linda, J.S.A. An Introduction to Mathematical Biology. Pearson Education Ltd., USA, 2007; pp. 123-127.
• [7] Fabietti, P. G., Canonico, V., Federici, M. O., Benedetti, M. M., Sarti, E. Control oriented model of insulin and glucose dynamics in type 1 diabetics. Medical and Biological Engineering and Computing, 2006; 44(1):69-78.
• [8] H. Hethcote, M. Zhien, L. Shengbing. Effects of quarantine in six endemic models for infectious diseases, Mathematical Biosciences. 2002;180:141160.
• [9] Allen, L. J., Brauer, F., Van den Driessche, P., & Wu, J. Mathematical epidemiology (Vol. 1945). Berlin: Springer, 2008.
• [10] Lombarte, M., Lupo, M., Campetelli, G., Basualdo, M., Rigalli, A. Mathematical model of glucose-insulin homeostasis in healthy rats. Mathematical biosciences, 2013;245(2):269-277.
• [11] Allen L. J. S. An Introduction to Mathematical Biology, Department of Mathematics and Statistics, Texas Tech University, Pearson Education., 2007; 348.
• [12] Hovorka, R., Canonico, V., Chassin, L. J., Haueter, U., Massi-Benedetti, M., Federici, M. O., Wilinska, M. E. Nonlinear model predictive control of glucose concentration in subjects with type 1 diabetes. Physiological measurement, 2004;25(4): 905.
• [13] Öztürk, Z., Sorgun, S., Bilgil H. SIQRV Modeli ve Nümerik Uygulaması, Avrupa Bilim ve Teknoloji Dergisi, 2021;28:573-578.
• [14] W.O. Kermack and A.G. McKendrick. A contribution to the mathematical theory of epidemics, in: Proceedings of the Royal Society of London, Series A, Containing Papers of a Mathematical and Physical Character, 1927;115(772):700-721.
• [15] Yaro, D., Omari-Sasu, S. K., Harvim, P., Saviour, A. W., Obeng, B. A. Generalized Euler method for modeling measles with fractional differ ential equations. Int. J. Innovative Research and Development, 2015; 4.
• [16] Çetinkaya, İ. T., Kocabıyık, M., Ongun, M. Y. Stability Analysis of Discretized Model of Glucose-Insulin Homeostasis. Celal Bayar University Journal of Science, 2021;17(4):369-377.
• [17] Braun, M. and Golubitsky, M. Differential equations and their applications, 1983; (Vol. 1). New York: Springer-Verlag.
• [18] Öztürk, Z., Sorgun, S., Bilgil, H., Erdinç, Ü. New Exact Solutions of Conformable Time-Fractional Bad and Good Modified Boussinesq Equations. Journal of New Theory, 2021;37:8-25.
• [19] Öztürk, Z., Bilgil H., Sorgun, S., Kesir Mertebeden Glikoz-İnsülin Modeli. IV. International Turkic World Congress on Science and Engineering, 23-24 June 2022, Nigde Ömer Halis Demir University, Nigde, Turkey.
• [20] Sevindir, H. K., Çetinkaya, S., Demir, A. On Effects of a New Method for Fractional Initial Value Problems. Advances in Mathematical Physics 2021, 2021.
• [21] Cetinkaya, S., Demir, A. On the Solution of Bratu's Initial Value Problem in the Liouville-Caputo Sense by Ara Transform and Decomposition Method. Comptes rendus de l'Academie bulgare des Sciences, 2021; 74(12):1729-1738.
• [22] Cetinkaya, S., Demir, A., Sevindir, H. K. Solution of space-time-fractional problem by Shehu variational iteration method. Advances in Mathematical Physics 2021, 2021.
• [23] Abukhaled, M., Khuri, S., Rabah, F. Solution of a nonlinear fractional COVID-19 model. International Journal of Numerical Methods for Heat & Fluid Flow, 2022.
• [24] Yousef, A., Bozkurt, F., Abdeljawad, T., Emreizeeq, E. A mathematical model of COVID-19 and the multi fears of the community during the epidemiological stage. Journal of Computational and Applied Mathematics, 2023; 419:114624.
• [25] Öztürk, Z., Bilgil, H., Erdinç, Ü. An optimized continuous fractional grey model for forecasting of the time dependent real world cases. Hacettepe Journal of Mathematics and Statistics, 2022;1-19.
• [26] Erdinc, U., Bilgil, H., Ozturk, Z. A Novel Fractional Forecasting Model for Time Dependent Real World Cases: Accepted: May 2022. REVSTAT-Statistical Journal, 2022.
• [27] Sinan, M., Leng, J., Anjum, M., Fiaz, M. Asymptotic behavior and semi-analytic solution of a novel compartmental biological model. Mathematical Modelling and Numerical Simulation with Applications, 2022;2(2):88-107.