A hybrid approach to diabetes modeling: Fractional derivatives and stochastic analysis

Sayed Saber; Emad Solouma; A. F. Aljohani; Faisal Muteb K. Almalki; Sayed Saber; Emad Solouma; A. F. Aljohani; Faisal Muteb K. Almalki

doi:10.3934/math.2026206

AIMS Mathematics

2026, Volume 11, Issue 2: 5029-5061. doi: 10.3934/math.2026206

Previous Article Next Article

Research article Special Issues

A hybrid approach to diabetes modeling: Fractional derivatives and stochastic analysis

1.
Department of Mathematics, Faculty of Science, Al-Baha University, Alaqiq, 65779–7738, Saudi Arabia
2.
Department of Mathematics and Computer Science, Faculty of Science, Beni-Suef University, Egypt
3.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Saudi Arabia
4.
Department of Mathematics, Faculty of Science, University of Tabuk, Tabuk, Saudi Arabia

Received: 01 October 2025 Revised: 23 January 2026 Accepted: 26 January 2026 Published: 27 February 2026
MSC : 26A33, 34A08, 34F05, 65L05, 92C45, 93C95

In this study, the Michaelis-Menten function was incorporated into a fractional stochastic model of glucose-insulin degradation to describe insulin degradation. The existence, uniqueness, boundedness, and nonnegativity properties of the model were derived through rigorous mathematical analysis. This paper presented an analysis of stability, including asymptotic and Hyers-Ulam stability. By integrating stochastic perturbations into fractional differential equations with Caputo derivatives, we enhanced the realism of diabetes modeling. A stochastic Runge-Kutta method of type 4 was employed to solve the system, ensuring improved accuracy and stability in simulating glucose-insulin dynamics under random fluctuations. Through sensitivity analysis, four optimized control strategies, including insulin injection regimens and pharmaceutical interventions, were identified. An analysis of the dynamic behavior of the system under various physiological conditions was carried out using fractional Predictor-Evaluator-Corrector-Evaluator (PECE) methods of Adams-Bashford-Moulton type and Runge-Kutta type 4. Based on the results, it appeared that fractional stochastic modeling could capture long-term memory effects and inherent randomness in glucose metabolism, thereby providing a more comprehensive framework for diabetes management and prediction.
- fractional calculus,
- glucose-insulin model,
- stochastic differential equations,
- Michaelis-Menten kinetics,
- Hyers-Ulam stability,
- Predictor-corrector method,
- Runge-Kutta method,
- sensitivity analysis,
- diabetes management
Citation: Sayed Saber, Emad Solouma, A. F. Aljohani, Faisal Muteb K. Almalki. A hybrid approach to diabetes modeling: Fractional derivatives and stochastic analysis[J]. AIMS Mathematics, 2026, 11(2): 5029-5061. doi: 10.3934/math.2026206

Related Papers:

Abstract

In this study, the Michaelis-Menten function was incorporated into a fractional stochastic model of glucose-insulin degradation to describe insulin degradation. The existence, uniqueness, boundedness, and nonnegativity properties of the model were derived through rigorous mathematical analysis. This paper presented an analysis of stability, including asymptotic and Hyers-Ulam stability. By integrating stochastic perturbations into fractional differential equations with Caputo derivatives, we enhanced the realism of diabetes modeling. A stochastic Runge-Kutta method of type 4 was employed to solve the system, ensuring improved accuracy and stability in simulating glucose-insulin dynamics under random fluctuations. Through sensitivity analysis, four optimized control strategies, including insulin injection regimens and pharmaceutical interventions, were identified. An analysis of the dynamic behavior of the system under various physiological conditions was carried out using fractional Predictor-Evaluator-Corrector-Evaluator (PECE) methods of Adams-Bashford-Moulton type and Runge-Kutta type 4. Based on the results, it appeared that fractional stochastic modeling could capture long-term memory effects and inherent randomness in glucose metabolism, thereby providing a more comprehensive framework for diabetes management and prediction.

References

[1]	J. Li, Y. Kuang, C. C. Mason, Modeling the glucose-insulin regulatory system and ultradian insulin secretory oscillations with two time delays, J. Theor. Biol., 242 (2006), 722–735. https://doi.org/10.1016/j.jtbi.2006.04.002 doi: 10.1016/j.jtbi.2006.04.002
[2]	B. Topp, K. Promislow, G. Devries, R. M. Miura, D. T. Finegood, A model of $\beta$-cell mass, insulin, and glucose, J. Theor. Biol., 206 (2000), 605–619. https://doi.org/10.1006/jtbi.2000.2150 doi: 10.1006/jtbi.2000.2150
[3]	H. Wang, J. Li, Y. Kuang, Mathematical modeling and qualitative analysis of insulin therapies, Math. Biosci., 210 (2007), 17–33. https://doi.org/10.1016/j.mbs.2007.05.008 doi: 10.1016/j.mbs.2007.05.008
[4]	H. Wang, J. Li, Y. Kuang, Enhanced modelling of the glucose-insulin system and its applications in insulin therapies, J. Biol. Dyn., 3 (2009), 22–38. https://doi.org/10.1080/17513750802101927 doi: 10.1080/17513750802101927
[5]	M. Ma, J. Li, Dynamics of a glucose-insulin model, J. Biol. Dyn., 16 (2022), 733–745. https://doi.org/10.1080/17513758.2022.2146769 doi: 10.1080/17513758.2022.2146769
[6]	S. Saber, E. B. M. Bashier, S. M. Alzahrani, I. A. Noaman, A mathematical model of glucose-insulin interaction with time delay, J. Appl. Comput. Math., 7 (2018), 416–421. https://doi.org/10.4172/2168-9679.1000416 doi: 10.4172/2168-9679.1000416
[7]	M. H. Alshehri, F. Z. Duraihem, A. Alalyani, S. Saber, A Caputo (discretization) fractional-order model of glucose-insulin interaction: Numerical solution and comparisons with experimental data, J. Taibah Univ. Sci., 15 (2021), 26–36. https://doi.org/10.1080/16583655.2021.1872197 doi: 10.1080/16583655.2021.1872197
[8]	K. I. A. Ahmed, S. M. Mirgani, A. Seadawy, S. Saber, A comprehensive investigation of fractional glucose-insulin dynamics: Existence, stability, and numerical comparisons using residual power series and generalized Runge-Kutta methods, J. Taibah Univ. Sci., 19 (2025), 2460280. https://doi.org/10.1080/16583655.2025.2460280 doi: 10.1080/16583655.2025.2460280
[9]	M. H. Alshehri, S. Saber, F. Z. Duraihem, Dynamical analysis of fractional-order of IVGTT glucose-insulin interaction, Int. J. Nonlin. Sci. Num., 24 (2023), 1123–1140. https://doi.org/10.1515/ijnsns-2020-0201 doi: 10.1515/ijnsns-2020-0201
[10]	S. Saber, A. Alalyani, Stability analysis and numerical simulations of IVGTT glucose-insulin interaction models with two time delays, Math. Model. Anal., 27 (2022), 383–407. https://doi.org/10.3846/mma.2022.14007 doi: 10.3846/mma.2022.14007
[11]	K. I. A. Ahmed, H. D. S. Adam, M. Y. Youssif, S. Saber, Different strategies for diabetes by mathematical modeling: Modified minimal model, Alex. Eng. J., 80 (2023), 74–87. https://doi.org/10.1016/j.aej.2023.07.050 doi: 10.1016/j.aej.2023.07.050
[12]	K. I. A. Ahmed, D. S. A. Adam, M. Y. Youssif, S. Saber, Different strategies for diabetes by mathematical modeling: Applications of fractal-fractional derivatives in the sense of Atangana-Baleanu, Results Phys., 52 (2023), 106892. https://doi.org/10.1016/j.rinp.2023.106892 doi: 10.1016/j.rinp.2023.106892
[13]	R. Caponetto, G. Dongola, L. Fortuna, I. Petras, Fractional order systems: Modelling and control applications, World Scientific, 2010. https://doi.org/10.1142/7709
[14]	S. B. Minucci, R. L. Heise, A. M. Reynolds, Review of mathematical modeling of the inflammatory response in lung infections and injuries, Front. Appl. Math. Stat. 6 (2020), 1–25. https://doi.org/10.3389/fams.2020.00036 doi: 10.3389/fams.2020.00036
[15]	M. A. Dokuyucu, E. Çelik, H. Bulut, H. M. Baskonus, Cancer treatment model with the Caputo-Fabrizio fractional derivative, Eur. Phys. J. Plus, 133 (2018), 92. https://doi.org/10.1140/epjp/i2018-11950-y doi: 10.1140/epjp/i2018-11950-y
[16]	M. F. Faraloya, S. Shafie, F. M. Siam, R. Mahmud, S. O. Ajadi, Numerical simulation and optimization of radiotherapy cancer treatments using the Caputo fractional derivative, Malays. J. Math. Sci. 15 (2021), 161–187.
[17]	A. Alalyani, S. Saber, Stability analysis and numerical simulations of the fractional COVID-19 pandemic model, Int. J. Nonlin. Sci. Num. 24 (2023), 989–1002. https://doi.org/10.1515/ijnsns-2021-0042 doi: 10.1515/ijnsns-2021-0042
[18]	S. Saber, A. M. Alghamdi, G. A. Ahmed, K. M. Alshehri, Mathematical modelling and optimal control of pneumonia disease in sheep and goats in Al-Baha region with cost-effective strategies, AIMS Math., 7 (2022), 12011–12049. https://doi.org/10.3934/math.2022669 doi: 10.3934/math.2022669
[19]	S. M. Al-Zahrani, F. E. I. Elsmih, K. S. Al-Zahrani, S. Saber, A fractional order SITR model for forecasting of transmission of COVID-19: Sensitivity statistical analysis, Malays. J. Math. Sci., 16 (2022), 517–536. https://doi.org/10.47836/mjms.16.3.08 doi: 10.47836/mjms.16.3.08
[20]	S. Saber, Solution to $\bar{\partial}$-problem with support conditions in weakly $q$-convex domains, Commun. Korean Math. Soc., 33 (2018), 409–421. https://doi.org/10.4134/CKMS.c170022 doi: 10.4134/CKMS.c170022
[21]	S. Saber, Solvability of the tangential Cauchy-Riemann equations on boundaries of strictly $q$-convex domains, Lobachevskii J. Math., 32 (2011), 189–193. https://doi.org/10.1134/S1995080211030115 doi: 10.1134/S1995080211030115
[22]	M. Althubyani, H. D. S. Adam, A. Alalyani, N. E. Taha, K. O. Taha, R. A. Alharbi, S. Saber, Understanding zoonotic disease spread with a fractional order epidemic model, Sci. Rep., 15 (2025), 13921. https://doi.org/10.1038/s41598-025-95943-6 doi: 10.1038/s41598-025-95943-6
[23]	H. D. S. Adam, M. Althubyani, S. M. Mirgani, S. Saber, An application of Newton's interpolation polynomials to the zoonotic disease transmission between humans and baboons system based on a time-fractal fractional derivative with a power-law kernel, AIP Adv., 15 (2025), 045217. https://doi.org/10.1063/5.0253869 doi: 10.1063/5.0253869
[24]	M. Alhazmi, S. Saber, Glucose-insulin regulatory system: Chaos control and stability analysis via Atangana-Baleanu fractal-fractional derivatives, Alex. Eng. J., 122 (2025), 77–90. https://doi.org/10.1016/j.aej.2025.02.066 doi: 10.1016/j.aej.2025.02.066
[25]	S. Saber, $L^{2}$ estimates and existence theorems for $\bar{\partial}_b$ on Lipschitz boundaries of $Q$-pseudoconvex domains, C. R. Math., 358 (2020), 435–458. https://doi.org/10.5802/crmath.43 doi: 10.5802/crmath.43
[26]	N. Almutairi, S. Saber, Application of a time-fractal fractional derivative with a power-law kernel to the Burke-Shaw system based on Newton's interpolation polynomials, MethodsX, 12 (2024), 102510. https://doi.org/10.1016/j.mex.2023.102510 doi: 10.1016/j.mex.2023.102510
[27]	S. Saber, Control of chaos in the Burke-Shaw system of fractal-fractional order in the sense of Caputo-Fabrizio, J. Appl. Math. Comput. Mech., 23 (2024), 83–96. https://doi.org/10.17512/jamcm.2024.1.07 doi: 10.17512/jamcm.2024.1.07
[28]	N. Almutairi, S. Saber, Existence of chaos and the approximate solution of the Lorenz-Lü-Chen system with the Caputo fractional operator, AIP Adv., 14 (2024), 015112. https://doi.org/10.1063/5.0185906 doi: 10.1063/5.0185906
[29]	K. I. A. Ahmed, H. D. S. Adam, N. Almutairi, S. Saber, Analytical solutions for a class of variable-order fractional Liu system under time-dependent variable coefficients, Results Phys., 56 (2024), 107311. https://doi.org/10.1016/j.rinp.2023.107311 doi: 10.1016/j.rinp.2023.107311
[30]	N. Almutairi, S. Saber, On chaos control of nonlinear fractional Newton-Leipnik system via fractional Caputo-Fabrizio derivatives, Sci. Rep., 13 (2023), 22726. https://doi.org/10.1038/s41598-023-49541-z doi: 10.1038/s41598-023-49541-z
[31]	N. Almutairi, S. Saber, H. Ahmad, The fractal-fractional Atangana-Baleanu operator for pneumonia disease: Stability, statistical and numerical analyses, AIMS Math., 8 (2023), 29382–29410. https://doi.org/10.3934/math.20231504 doi: 10.3934/math.20231504
[32]	N. Almutairi, S. Saber, Chaos control and numerical solution of time-varying fractional Newton-Leipnik system using fractional Atangana–Baleanu derivatives, AIMS Math., 8 (2023), 25863–25887. https://doi.org/10.3934/math.20231319 doi: 10.3934/math.20231319
[33]	S. K. Choi, B. Kang, N. Koo, Stability for Caputo fractional differential systems, Abstr. Appl. Anal., 2014 (2014), 631419. https://doi.org/10.1155/2014/631419 doi: 10.1155/2014/631419
[34]	A. Boukhouima, K. Hattaf, N. Yousfi, Dynamics of a fractional order HIV infection model with specific functional response and cure rate, Int. J. Differ. Equ., 2017 (2017), 8372140. https://doi.org/10.1155/2017/8372140 doi: 10.1155/2017/8372140
[35]	J. P. La Salle, The stability of dynamical systems, Society for Industrial and Applied Mathematics, 1976.
[36]	H. L. Li, L. Zhang, C. Hu, Y. Jiang, Z. Teng, Dynamical analysis of a fractional-order predator-prey model incorporating a prey refuge, J. Appl. Math. Comput., 54 (2017), 435–449. https://doi.org/10.1007/s12190-016-1017-8 doi: 10.1007/s12190-016-1017-8
[37]	I. Podlubny, Fractional differential equations, 1999.
[38]	X. Mao, Stochastic differential equations and applications, 1997.
[39]	H. D. S. Adam, K. I. Adam, S. Saber, G. Farid, Existence theorems for the $\bar{\partial}$ equation and Sobolev estimates on $q$-convex domains, AIMS Math., 8 (2023), 31141–31157. https://doi.org/10.3934/math.20231594 doi: 10.3934/math.20231594
[40]	X. Zhong, S. Guo, M. Peng, Stability of stochastic SIRS epidemic models with saturated incidence rates and delay, Stoch. Anal. Appl., 35 (2017), 1–26. https://doi.org/10.1080/07362994.2016.1244644 doi: 10.1080/07362994.2016.1244644
[41]	T. Khan, I. H. Jung, G. Zaman, A stochastic model for the transmission dynamics of hepatitis B virus, J. Biol. Dyn., 13 (2019), 328–344. https://doi.org/10.1080/17513758.2019.1600750 doi: 10.1080/17513758.2019.1600750
[42]	A. Din, A. Khan, D. Baleanu, Stationary distribution and extinction of stochastic coronavirus (COVID-19) epidemic model, Chaos Soliton. Fract., 139 (2020), 110036. https://doi.org/10.1016/j.chaos.2020.110036 doi: 10.1016/j.chaos.2020.110036
[43]	M. Otero, H. G. Solari, Stochastic eco-epidemiological model of dengue disease transmission by Aedes aegypti mosquito, Math. Biosci., 223 (2010), 32–46. https://doi.org/10.1016/j.mbs.2009.10.005 doi: 10.1016/j.mbs.2009.10.005
[44]	H. Woodall, B. Adams, Stochastic modelling for age-structured dengue epidemiology with and without seasonal variation, In: 9th European Conference on Mathematical and Theoretical Biology, 2014.
[45]	T. Khan, A. Khan, G. Zaman, The extinction and persistence of the stochastic hepatitis B epidemic model, Chaos Soliton. Fract., 108 (2018), 123–128. https://doi.org/10.1016/j.chaos.2018.01.036 doi: 10.1016/j.chaos.2018.01.036
[46]	A. Atangana, B. Dumitru, 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
[47]	M. Sivashankar, S. Sabarinathan, K. S. Nisar, C. Ravichandran, B. V. Senthil Kumar, Some properties and stability of Helmholtz model involved with nonlinear fractional difference equations and its relevance with quadcopter, Chaos Soliton. Fract., 168 (2023), 113161. https://doi.org/10.1016/j.chaos.2023.113161 doi: 10.1016/j.chaos.2023.113161
[48]	N. I. Okposo, E. Addai, J. S. Apanapudor, J. F. Gómez-Aguilar, A study on a monkeypox transmission model within the scope of fractal-fractional derivative with power-law kernel, Eur. Phys. J. Plus, 138 (2023), 684. https://doi.org/10.1140/epjp/s13360-023-04334-1 doi: 10.1140/epjp/s13360-023-04334-1
[49]	J. Alzabut, R. Dhineshbabu, A. G. M. Selvam, J. F. Gómez-Aguilar, H. Khan, Existence, uniqueness and synchronization of a fractional tumor growth model in discrete time with numerical results, Results Phys., 54 (2023), 107030. https://doi.org/10.1016/j.rinp.2023.107030 doi: 10.1016/j.rinp.2023.107030
[50]	X. J. Yang, Local fractional functional analysis and its applications, Hong Kong: Asian Academic Publisher, 2011.
[51]	M. Mohammad, M. Saadaoui, A new fractional derivative extending classical concepts: Theory and applications, Part. Differ. Equ. Appl. Math., 11 (2024), 100889. https://doi.org/10.1016/j.padiff.2024.100889 doi: 10.1016/j.padiff.2024.100889
[52]	K. M. Saad, New fractional derivative with non-singular kernel for deriving Legendre spectral collocation method, Alex. Eng. J., 59 (2020), 1909–1917. https://doi.org/10.1016/j.aej.2019.11.017 doi: 10.1016/j.aej.2019.11.017
[53]	J. V. da C. Sousa, E. Capelas de Oliveira, A new truncated $M$-fractional derivative type unifying some fractional derivative types with classical properties, Int. J. Anal. Appl., 16 (2018), 83–96. https://doi.org/10.28924/2291-8639-16-2018-83 doi: 10.28924/2291-8639-16-2018-83
[54]	R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65–70. https://doi.org/10.1016/j.cam.2014.01.002 doi: 10.1016/j.cam.2014.01.002
[55]	R. Almeida, A. B. Malinowska, M. T. T. Monteiro, Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications, Math. Method. Appl. Sci., 41 (2018), 336–352. https://doi.org/10.1002/mma.4617 doi: 10.1002/mma.4617
[56]	Y. H. Liang, K. J. Wang, Dynamics of new exact wave solutions to the local fractional Vakhnenko-Parkes equation, Fractals, 34 (2026), 2550102. https://doi.org/10.1142/S0218348X25501026 doi: 10.1142/S0218348X25501026
[57]	K. J. Wang, The fractal active low-pass filter within the local fractional derivative on the Cantor set, COMPEL, 42 (2023), 1396–1407. https://doi.org/10.1108/COMPEL-09-2022-0326 doi: 10.1108/COMPEL-09-2022-0326
[58]	W. Liu, Q. Zuo, C. Xu, Finite-time and global Mittag-Leffler stability of fractional-order neural networks with S-type distributed delays, AIMS Math., 9 (2024), 8339–8352. https://doi.org/10.3934/math.2024405 doi: 10.3934/math.2024405
[59]	D. Xu, W. Liu, Stochastic asymptotic stability for stochastic inertial Cohen-Grossberg neural networks with time-varying delay, J. Comput. Methods Sci., 23 (2023), 921–931. https://doi.org/10.3233/JCM-226480 doi: 10.3233/JCM-226480
[60]	Y. Zhang, Z. Li, W. Jiang, W. Liu, Exponential stability of stochastic inertial Cohen-Grossberg neural networks, Int. J. Pattern Recogn., 37 (2023), 2259032. https://doi.org/10.1142/S0218001422590327 doi: 10.1142/S0218001422590327
[61]	S. M. Ulam, A collection of mathematical problems, 1960.
[62]	S. M. Ulam, Problems in modern mathematics, Courier Corporation, 2004.
[63]	K. Diethelm, N. J. Ford, A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dyn., 29 (2002), 3–22. https://doi.org/10.1023/A:1016592219341 doi: 10.1023/A:1016592219341
[64]	P. E. Kloeden, E. Platen, Numerical solution of stochastic differential equations, Berlin: Springer, 1999. https://doi.org/10.1007/978-3-662-12616-5

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)