Analysis and Simulation of Fractional-Order Diabetes Model

In this article, we research the diabetes model and its consequences using the Caputo and Atangana Baleanu fractional derivatives. The presence and uniqueness are strongly mentored by the fixed point theorem and the approach to Picard - Lindelof. A deterministic mathematical model corresponding to the fractional derivative of diabetes mellitus. The Laplace transformation is used for the diagnostic structure of the diabetes model. Finally, numerical calculations are made to illustrate the effect of changing the fractional-order to obtain the theoretical results, and comparisons are made for the Caputo and Atangana Baleanu derivative. The results of the following work by controlling plasma glucose with the fractional-order model make it a suitable candidate for controlling human type 1 diabetes.

Keywords:

Fractional order glucose insulin system, stability Picard–Lindelof approach, Fixed point theorem, ABC derivative,

PDF

___

[1] L. Guariguata, D.R. Whiting, I. Hambleton, J. Beagley, U. Linnenkamp, and J. E. Shaw, Global estimates of diabetes prevalence for 2013 and projections for 2035, Diabetes Research and Clinical Practice, vol. 103, no. 2, pp. 137-149, 2014.
[2] C. Florkowski, HbA1c as a diagnostic test for diabetes mellitus-reviewing the evidence, Clinical Biochemist Reviews, vol. 34, no. 2, pp. 75-83, 2013.
[3] J. Nicholas, J. Charlton, A. Dregan, and M. C. Gulliford, Recent hba1c values and mortality risk in type 2 diabetes, population-based case-control study, PLoS One, vol. 8, no. 7, Article ID e68008, 2013.
[4] American Diabetes Association, Standards of medical care in diabetes-2015 abridged for primary care providers, Clinical Diabetes, vol. 33, no. 2, pp. 97-111, 2015.
[5] E.J. Lyons, Z.H. Lewis, B.G. Mayrsohn, and J.L. Rowland, Behavior change techniques implemented in electronic lifestyle activity monitors: a systematic content analysis, Journal of Medical Internet Research, vol. 16, no. 8, p. e192, 2014.
[6] W. Hamer, Epidemiology Old and New. London: Kegan Paul, 1928.
[7] R. Ross, The Prevention of Malaria, 1910.
[8] G.A. Leonov, N.V. Kuznetsov, T.N. Mokaev, Homoclinic orbits, and self-excited and hidden attractors in a Lo- renz-like system describing convective ?uid motion. Eur Phys J Spec Top, 224(8):1421-1458.
[9] A. Boutayeb, E.H. Twizell, K. Achouayb, A. Chetouani: A mathematical model for the burden of diabetes and its compli- cations. Biomed. Eng. Online 3(1), 20 (2004).
[10] C. Zecchin, A. Facchinetti, G. Sparacino, G. De Nicolao, C. Cobelli, A new neural network approach for short-term glucose prediction using continuous glucose monitoring time-series and meal information. Conf. Proc. IEEE Eng. Med. Biol. Soc. 2011, 5653-5656 (2011).
[11] A.A. Sharief, A. Sheta, Developing a mathematical model to detect diabetes using multigene genetic programming. Int. J. Adv. Res. in Artif. Intell. (IJARAI) 3(10), 54 (2014).
[12] Y.C. Rosado, Mathematical model for detecting diabetes. In: Proceedings of the National Conference on Undergraduate Research (NCUR), University of Wisconsin La-Crosse, La-Crosse (2009).
[13] E. Ackerman, I. Gatewood, J. Rosevear, G. Molnar, Blood glucose regulation and diabetes. In: Heinmets, F. (ed.) Concepts and Models of Biomathematics, pp. 131-156. Decker, New York (1969).
[14] M. Asif, et al. Numerical modeling of NPZ and SIR models with and without diffusion. Results in Physics 19 (2020): 103512.
[15] M. Asif, et al. Numerical simulation for solution of SEIR models by meshless and finite difference methods. Chaos, Solitons and Fractals 141 (2020): 110340.
[16] Ahmad, Shabir, et al. Fractional order mathematical modeling of COVID-19 transmission. Chaos, Solitons and Fractals 139 (2020): 110256.
[17] F.A. Rihan, et al. A fractional-order epidemic model with time-delay and nonlinear incidence rate. Chaos, Solitons and Fractals 126 (2019): 97-105.
[18] F. Haq, K. Shah, G. Rahman, M. Shahzad, Numerical solution of fractional order smoking model via laplace Adomian decomposition method, Alexandria Engineering Journal (2018) 57, 1061-1069.
[19] S. Kumar, A. Kumar, I K. Argyros, 2017, A new analysis for the Keller-Segel model of fractional order, Numer. Algorithms, 75 213-228.
[20] S. Kumar, M.M. Rashidi, 2014, New analytical method for gas dynamic equation arising in shock fronts, Comput. Phys. Commun., 185 1947-1954.
[21] Z. Odibat, A.S. Bataineh, 2015, An adaptation of homotopy analysis method for reliable treatment of strongly nonlinear problems: construction of homotopy polynomials, Math. Meth. Appl. Sci., 38 991-1000.
[22] A. Boutayeb, E.H. Twzell, K. Achouayti, A. Chetouan, A mathematical model for the burden of diabetes and its compli- cations Biomed Eng Online 3(1), 20(2004).
[23] C.P. Li, C.X. Tao, 2009, On the fractional Adams method, Comput. Math. Appl., 58 1573-1588.
[24] B.S.T. Alkahtani, 2016, Chua's circuit model with Atangana-Baleanu derivative with fractional order, Chaos Solitons Fractals, 89 547-551.