Very recently, several novel conceptions of fractional derivatives have been proposed and employed to develop numerical simulations for a wide range of real-world configurations with memory, background, or non-local effects via an uncertainty parameter $ [0, 1] $ as a confidence degree of belief. Under the complexities of the uncertainty parameter, the major goal of this paper is to develop and examine the Atangana-Baleanu derivative in the Caputo sense for a convoluted glucose-insulin regulating mechanism that possesses a memory and enables one to recall all foreknowledge. However, as compared to other existing derivatives, this is a vitally important point, and the convenience of employing this derivative lessens the intricacy of numerical findings. The Atangana-Baleanu derivative in the Caputo sense of fuzzy valued functions (FVF) in parameterized interval representation is established initially in this study. Then, it is leveraged to demonstrate that the existence and uniqueness of solutions were verified using the theorem suggesting the Banach fixed point and Lipschitz conditions under generalized Hukuhara differentiability. In order to explore the regulation of plasma glucose in diabetic patients with impulsive insulin injections and by monitoring the glucose level that returns to normal in a finite amount of time, we propose an impulsive differential equation model. It is a deterministic mathematical framework that is connected to diabetes mellitus and fractional derivatives. The framework for this research and simulations was numerically solved using a numerical approach based on the Adams-Bashforth-Moulton technique. The findings of this case study indicate that the fractional-order model's plasma glucose management is a suitable choice.
Citation: Shao-Wen Yao, Saima Rashid, Mustafa Inc, Ehab E. Elattar. On fuzzy numerical model dealing with the control of glucose in insulin therapies for diabetes via nonsingular kernel in the fuzzy sense[J]. AIMS Mathematics, 2022, 7(10): 17913-17941. doi: 10.3934/math.2022987
Very recently, several novel conceptions of fractional derivatives have been proposed and employed to develop numerical simulations for a wide range of real-world configurations with memory, background, or non-local effects via an uncertainty parameter $ [0, 1] $ as a confidence degree of belief. Under the complexities of the uncertainty parameter, the major goal of this paper is to develop and examine the Atangana-Baleanu derivative in the Caputo sense for a convoluted glucose-insulin regulating mechanism that possesses a memory and enables one to recall all foreknowledge. However, as compared to other existing derivatives, this is a vitally important point, and the convenience of employing this derivative lessens the intricacy of numerical findings. The Atangana-Baleanu derivative in the Caputo sense of fuzzy valued functions (FVF) in parameterized interval representation is established initially in this study. Then, it is leveraged to demonstrate that the existence and uniqueness of solutions were verified using the theorem suggesting the Banach fixed point and Lipschitz conditions under generalized Hukuhara differentiability. In order to explore the regulation of plasma glucose in diabetic patients with impulsive insulin injections and by monitoring the glucose level that returns to normal in a finite amount of time, we propose an impulsive differential equation model. It is a deterministic mathematical framework that is connected to diabetes mellitus and fractional derivatives. The framework for this research and simulations was numerically solved using a numerical approach based on the Adams-Bashforth-Moulton technique. The findings of this case study indicate that the fractional-order model's plasma glucose management is a suitable choice.
| [1] | F. Chee, T. Fernando, Closed-loop control of blood glucose, Berlin, Heidelberg: Springer, 2007. https://doi.org/10.1007/978-3-540-74031-5 |
| [2] |
S. Sakulrang, E. J. Moore, S. Sungnul, A. de Gaetano, A fractional differential equation model for continuous glucose monitoring data, Adv. Differ. Equ., 2017 (2017), 1–11. https://doi.org/10.1186/s13662-017-1207-1 doi: 10.1186/s13662-017-1207-1
|
| [3] |
Y. Reznik, Continuous subcutaneous insulin infusion (CSII) using an external insulin pump for the treatment of type 2 diabetes, Diabetes Metab., 36 (2010), 415–421. https://doi.org/10.1016/j.diabet.2010.08.002 doi: 10.1016/j.diabet.2010.08.002
|
| [4] |
M. Z. Huang, J. X. Li, X. Y. Song, H. J. Guo, Modeling impulsive injections of insulin: towards artificial pancreas, SIAM J. Appl. Math., 72 (2012), 1524–1548. https://doi.org/10.1137/110860306 doi: 10.1137/110860306
|
| [5] |
X. Y. Song, M. Z. Huang, J. X. Li, Modeling impulsive insulin delivery in insulin pump with time delays, SIAM J. Appl. Math., 74 (2014), 1763–1785. https://doi.org/10.1137/130933137 doi: 10.1137/130933137
|
| [6] |
S. Z. Liu, M. Z. Huang, X. Y. Song, X. Y. Shi, Finite-time control of plasma glucose in insulin therapies for diabetes, Adv. Differ. Equ., 2018 (2018), 1–16. https://doi.org/10.1186/s13662-018-1532-z doi: 10.1186/s13662-018-1532-z
|
| [7] |
M. Farman, M. U. Saleem, M. O. Ahmad, A. Ahmad, Stability analysis and control of glucose insulin glucagon system in humans, Chinese. J. Phys., 56 (2018), 1362–1369. https://doi.org/10.1016/j.cjph.2018.03.037 doi: 10.1016/j.cjph.2018.03.037
|
| [8] |
M. U. Saleem, M. Farman, A. Ahmad, E. U. Haque, M. O. Ahmad, A Caputo Fabrizio fractional order model for control of glucose in insulin therapies for diabetes, Ain Shamas Eng. J., 11 (2020), 1309–1316. https://doi.org/10.1016/j.asej.2020.03.006 doi: 10.1016/j.asej.2020.03.006
|
| [9] |
M. U. Saleem, M. Farman, M. Rizwan, M. O. Ahmad, A. Ahmad, Controllability and observability of glucose insulin glucagon systems in human, Chin. J. Phys., 56 (2018), 1909–1916. https://doi.org/10.1016/j.cjph.2018.09.005 doi: 10.1016/j.cjph.2018.09.005
|
| [10] |
M. Farman, M. U. Saleem, A. Ahmad, S. Imtiaz, M. F. Tabassum, S. Akramd, et al., A control of glucose level in insulin therapies for the development of artificial pancreas by Atangana Baleanu derivative, Alex. Eng. J., 59 (2020), 2639–2648. https://doi.org/10.1016/j.aej.2020.04.027 doi: 10.1016/j.aej.2020.04.027
|
| [11] |
M. U. Saleem, M. Aslam, A. Akgül, M. Farman, R. Bibi, Controllability of PDEs model for type 1 diabetes, Math. Methods Appl. Sci., 2021. https://doi.org/10.1002/mma.7279 doi: 10.1002/mma.7279
|
| [12] |
N. Debbouchea, A. O. Almatroud, A. Ouannas, I. M. Batiha, Chaos and coexisting attractors in glucose-insulin regulatory system with incommensurate fractional-order derivatives, Chaos Solitons Fract., 143 (2021), 110575. https://doi.org/10.1016/j.chaos.2020.110575 doi: 10.1016/j.chaos.2020.110575
|
| [13] | I. M. Batiha1, J. Oudetallah, A. Ouannas, A. A. Al-Nana, I. H. Jebril, Tuning the fractional-order PID-controller for blood glucose level of diabetic patients, Int. J. Advance Soft Compu. Appl., 13 (2021), 1–10. |
| [14] | M. Caputo, Elasticita e dissipazione, Bologna: ZaniChelli, 1969. |
| [15] |
B. Shiri, D. Baleanu, A general fractional pollution model for lakes, Commun. Appl. Math. Comput., 4 (2022), 1105–1130. https://doi.org/10.1007/s42967-021-00135-4 doi: 10.1007/s42967-021-00135-4
|
| [16] |
B. Shiri, I. Perfilieva, Z. Alijani, Classical approximation for fuzzy Fredholm integral equation, Fuzzy Sets. Syst., 404 (2021), 159–177. https://doi.org/10.1016/j.fss.2020.03.023 doi: 10.1016/j.fss.2020.03.023
|
| [17] |
Z. Alijani, D. Baleanu, B. Shirid, G. C. Wu, Spline collocation methods for systems of fuzzy fractional differential equations, Chaos Solitons Fract., 131 (2020), 109510. https://doi.org/10.1016/j.chaos.2019.109510 doi: 10.1016/j.chaos.2019.109510
|
| [18] |
K. Karthikeyan, P. Karthikeyan, H. M. Baskonus, K. Venkatachalam, Y. M. Chu, Almost sectorial operators on $\Psi$-Hilfer derivative fractional impulsive integro-differential equations, Math. Methods Appl. Sci., 2021. https://doi.org/10.1002/mma.7954 doi: 10.1002/mma.7954
|
| [19] |
S. Rashid, S. Sultana, Y. Karaca, A. Khalid, Y. M. Chu, Some further extensions considering discrete proportional fractional operators, Fractals, 30 (2022), 1–12. https://doi.org/10.1142/S0218348X22400266 doi: 10.1142/S0218348X22400266
|
| [20] |
S. N. Hajiseyedazizi, M. E. Samei, J. Alzabut, Y. M. Chu, On multi-step methods for singular fractional $q$-integro-differential equations, Open Math., 19 (2021), 1378–1405. https://doi.org/10.1515/math-2021-0093 doi: 10.1515/math-2021-0093
|
| [21] |
F. Jin, Z. S. Qian, Y. M. Chu, M. ur Rahman, On nonlinear evolution model for drinking behavior under Caputo-Fabrizio derivative, J. Appl. Anal. Comput., 12 (2022), 790–806. https://doi.org/10.11948/20210357 doi: 10.11948/20210357
|
| [22] |
S. Rashid, E. I. Abouelmagd, A. Khalid, F. B. Farooq, Y. M. Chu, Some recent developments on dynamical $\hbar$-discrete fractional type inequalities in the frame of nonsingular and nonlocal kernels, Fractals, 30 (2022), 1–15. https://doi.org/10.1142/S0218348X22401107 doi: 10.1142/S0218348X22401107
|
| [23] | I. Podlubny, Fractional differential equations, New York: Academic Press, 1999. |
| [24] |
M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Prog. Fract. Differ. Appl., 1 (2015), 73–85. https://doi.org/10.12785/pfda/010201 doi: 10.12785/pfda/010201
|
| [25] |
M. Bohner, O. Tunç, C. Tunç, Qualitative analysis of Caputo fractional integro-differential equations with constant delays, Comput. Appl. Math., 40 (2021), 1–17. https://doi.org/10.1007/s40314-021-01595-3 doi: 10.1007/s40314-021-01595-3
|
| [26] |
J. R. Graef, C. Tunç, H. Şevli, Razumikhin qualitative analyses of Volterra integro-fractional delay differential equation with Caputo derivatives, Commun. Nonlinear Sci. Numer. Simul., 103 (2021), 106037. https://doi.org/10.1016/j.cnsns.2021.106037 doi: 10.1016/j.cnsns.2021.106037
|
| [27] |
Z. U. A. Zafar, S. Zaib, M. T. Hussainc, C. Tunç, S. Javeed, Analysis and numerical simulation of tuberculosis model using different fractional derivatives, Chaos Solitons Fract., 160 (2022), 112202. https://doi.org/10.1016/j.chaos.2022.112202 doi: 10.1016/j.chaos.2022.112202
|
| [28] |
F. Z. Wang, M. N. Khan, I. Ahmad, H. Ahmad, H. Abu-Zinadah, Y. M. Chu, Numerical solution of traveling waves in chemical kinetics: Time-fractional fishers equations, Fractals, 30 (2022), 1–11. https://doi.org/10.1142/S0218348X22400515 doi: 10.1142/S0218348X22400515
|
| [29] |
S. Rashid, E. I. Abouelmagd, S. Sultana, Y. M. Chu, New developments in weighted $n$-fold type inequalities via discrete generalized ĥ-proportional fractional operators, Fractals, 30 (2022), 1–16. https://doi.org/10.1142/S0218348X22400564 doi: 10.1142/S0218348X22400564
|
| [30] |
S. A. Iqbal, M. G. Hafez, Y. M. Chu, C. Park, Dynamical analysis of nonautonomous RLC circuit with the absence and presence of Atangana-Baleanu fractional derivative, J. Appl. Anal. Comput., 12 (2022), 770–789. https://doi.org/10.11948/20210324 doi: 10.11948/20210324
|
| [31] |
A. Atangana, D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Therm. Sci., 20 (2016), 763–769. https://doi.org/10.2298/TSCI160111018A doi: 10.2298/TSCI160111018A
|
| [32] |
J. F. Goméz-Aguilar, A. Atangana, Retracted: Power and exponentials laws: theory and application, J. Comput. Appl. Math., 354 (2019), 52–65. https://doi.org/10.1016/j.cam.2019.01.003 doi: 10.1016/j.cam.2019.01.003
|
| [33] |
J. Singh, B. Ganbari, D. Kumar, D. Baleanu, Analysis of fractional model of guava for biological pest control with memory effect, J. Adv. Res., 32 (2021), 99–108. https://doi.org/10.1016/j.jare.2020.12.004 doi: 10.1016/j.jare.2020.12.004
|
| [34] |
V. P. Dubey, S. Dubey, D. Kumar, J. Singh, A computational study of fractional model of atmospheric dynamics of carbon dioxide gas, Chaos Solitons Fract., 142 (2021), 110375. https://doi.org/10.1016/j.chaos.2020.110375 doi: 10.1016/j.chaos.2020.110375
|
| [35] |
V. P. Dubey, J. Singh, A. M. Alshehri, S. Dubey, D. Kumar, A comparative analysis of two computational schemes for solving local fractional Laplace equations, Math. Methods Appl. Sci., 44 (2021), 13540–13559. https://doi.org/10.1002/mma.7642 doi: 10.1002/mma.7642
|
| [36] |
D. Baleanu, B. Shiri, Nonlinear higher order fractional terminal value problems, AIMS Math., 7 (2022), 7489–7506. https://doi.org/10.3934/math.2022420 doi: 10.3934/math.2022420
|
| [37] |
B. Shiri, G. C. Wu, D. Baleanu, Terminal value problems for the nonlinear systems of fractional differential equations, Appl. Numer. Math., 170 (2021), 162–178. https://doi.org/10.1016/j.apnum.2021.06.015 doi: 10.1016/j.apnum.2021.06.015
|
| [38] |
G. Yang, B. Shiri, H. Kong, G. C. Wu, Intermediate value problems for fractional differential equations, Comput. Appl. Math., 40 (2021), 1–20. https://doi.org/10.1007/s40314-021-01590-8 doi: 10.1007/s40314-021-01590-8
|
| [39] |
J. Singh, D. Kumar, Z. Hammouch, A. Atangana, A fractional epidemiological model for computer viruses pertaining to a new fractional derivative, Appl. Math. Comput., 316 (2018), 504–515. https://doi.org/10.1016/j.amc.2017.08.048 doi: 10.1016/j.amc.2017.08.048
|
| [40] |
J. Singh, D. Kumar, D. Baleanu, On the analysis of chemical kinetics system pertaining to a fractional derivative with Mittag-Leffler type kernel, Chaos, 27 (2017), 103113. https://doi.org/10.1063/1.4995032 doi: 10.1063/1.4995032
|
| [41] |
J. Singh, D. Kumar, D. Baleanu, On the analysis of fractional diabetes model with exponential law, Adv. Differ. Equ., 2018 (2018), 1–15. https://doi.org/10.1186/s13662-018-1680-1 doi: 10.1186/s13662-018-1680-1
|
| [42] |
S. Salahshour, T. Allahviranloo, S. Abbasbandy, Solving fuzzy fractional differential equations by fuzzy Laplace transforms, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 1372–1381. https://doi.org/10.1016/j.cnsns.2011.07.005 doi: 10.1016/j.cnsns.2011.07.005
|
| [43] |
R. P. Agrawal, V. Lakshmikantham, J. J. Nieto, On the concept of solution for fractional differential equations with uncertainty, Nonlinear Anal., 72 (2010), 2859–2862. https://doi.org/10.1016/j.na.2009.11.029 doi: 10.1016/j.na.2009.11.029
|
| [44] |
B. Bede, S. G. Gal, Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations, Fuzzy Sets Syst., 151 (2005), 581–599. https://doi.org/10.1016/j.fss.2004.08.001 doi: 10.1016/j.fss.2004.08.001
|
| [45] |
B. Bede, I. J. Rudas, A. L. Bencsik, First order linear fuzzy differential equations under generalized differentiability, Inform. Sci., 177 (2007), 1648–1662. https://doi.org/10.1016/j.ins.2006.08.021 doi: 10.1016/j.ins.2006.08.021
|
| [46] | T. H. Zhao, O. Castillo, H. Jahanshahi, A. Yusuf, M. O. Alassafi, F. E. Alsaadi, et al., A fuzzy-based strategy to suppress the novel coronavirus (2019-NCOV) massive outbreak, Appl. Comput. Math., 20 (2021), 160–176. |
| [47] |
S. Ahmad, A. Ullah, K. Shah, S. Salahshour, A. Ahmadian, T. Ciano, Fuzzy fractional-order model of the novel coronavirus, Adv. Differ. Equ., 2020 (2020), 1–17. https://doi.org/10.1186/s13662-020-02934-0 doi: 10.1186/s13662-020-02934-0
|
| [48] |
T. Allahviranloo, M. B. Ahmadi, Fuzzy Laplace transforms, Soft Comput., 14 (2010), 235–243. https://doi.org/10.1007/s00500-008-0397-6 doi: 10.1007/s00500-008-0397-6
|
| [49] | H. J. Zimmermann, Fuzzy set theory–and its applications, Dordrecht: Springer, 2001. https://doi.org/10.1007/978-94-010-0646-0 |
| [50] |
L. A. Zadeh, Fuzzy sets, Inform. Control, 8 (1965), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X doi: 10.1016/S0019-9958(65)90241-X
|
| [51] | T. Allahviranloo, Fuzzy fractional differential operators and equations, Cham: Springer, 2021. https://doi.org/10.1007/978-3-030-51272-9 |
| [52] |
S. Salahshour, T. Allahviranloo, S. Abbasbandy, Solving fuzzy fractional differential equations by fuzzy Laplace transforms, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 1372–1381. https://doi.org/10.1016/j.cnsns.2011.07.005 doi: 10.1016/j.cnsns.2011.07.005
|
| [53] |
B. Bede, L. Stefanini, Generalized differentiability of fuzzy-valued functions, Fuzzy Sets Syst., 230 (2013), 119–141. https://doi.org/10.1016/j.fss.2012.10.003 doi: 10.1016/j.fss.2012.10.003
|
| [54] |
S. Rashid, F. Jarad, T. M. Jawa, A study of behaviour for fractional order diabetes model via the nonsingular kernel, AIMS Math., 7 (2022), 5072–5092. https://doi.org/10.3934/math.2022282 doi: 10.3934/math.2022282
|
| [55] |
K. Diethelm, N. J. Ford, A. D. Freed, Detailed error analysis for a fractional Adams method, Numer. Algorithms, 36 (2004), 31–52. https://doi.org/10.1023/B:NUMA.0000027736.85078.be doi: 10.1023/B:NUMA.0000027736.85078.be
|
| [56] |
K. Diethelm, N. J. Ford, Multi-order fractional differential equations and their numerical solution, Appl. Math. Comput., 154 (2004), 621–640. https://doi.org/10.1016/S0096-3003(03)00739-2 doi: 10.1016/S0096-3003(03)00739-2
|
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Shao-Wen Yao, Saima Rashid, Mustafa Inc, Ehab E. Elattar. On fuzzy numerical model dealing with the control of glucose in insulin therapies for diabetes via nonsingular kernel in the fuzzy sense[J]. AIMS Mathematics, 2022, 7(10): 17913-17941. doi: 10.3934/math.2022987
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