Mittag–Leffler Stability and Decay-Rate Enhancement for a Nabla Discrete Fractional-Order Glucose–Insulin Regulatory System with a Nonsingular Mittag–Leffler Kernel

Grienggrai Rajchakit

Abstract

We study a reduced-order, discrete-time glucose–insulin regulatory model with an intended nonsingular Mittag-Leffler memory kernel. The aim is an interpretable control-theoretic benchmark, not a patient-specific artificial-pancreas model. For the bidirectionally coupled error system, optimisation of a scalar quadratic Young-inequality estimate gives \(\lambda^* = \tfrac{1}{2}\big[(a+c) - \sqrt{(a-c)^2 + \beta^2}\big]\), where \(\beta = bL_\phi + eL_\psi\); this algebraic quantity is positive exactly when \(\beta < 2\sqrt{ac}\). At \(a = 0.6\), \(c = 0.4\), and \(\beta = 0.5\), it increases the fixed-parameter value from \(0.150\) to \(0.231\). Replacing c by \(c + k_1\) with \(k_1 = 0.6\) gives \(0.480\). Strict gain monotonicity holds for \(\beta > 0\), while the formal \(\beta = 0\) extension is nondecreasing and saturates at a. Interpretation of these values as Mittag-Leffler decay certificates remains conditional on the operator definition and comparison lemmas flagged in this proof. The delayed estimate supplies a delay-independent algebraic condition, not a delay-uniform decay rate. The disturbance estimate, if its comparison properties hold, supplies an asymptotic residual radius. An explicit lagged predictor is reproducible and admits an endpoint refinement check, but this check does not establish consistency with the stated operator. In addition, the coupled code omits the equilibrium shifts of the raw model. These operator and model-alignment issues are retained as explicit editorial queries requiring resolution before publication. No clinical performance or safety claim is made.

References

1.
Bergman, R.N.; Phillips, L.S.; Cobelli, C. Physiologic evaluation of factors controlling glucose tolerance in man: Measurement of insulin sensitivity and beta-cell glucose sensitivity from the response to intravenous glucose. J. Clin. Invest. 1981, 68, 1456–1467. https://doi.org/10.1172/jci110398.
2.
Ahmed, K.I.A.; Mirgani, S.M.; Seadawy, A.; et al. 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. 2025, 19, 2460280. https://doi.org/10.1080/16583655.2025.2460280.
3.
Alhazmi, M.; Mirgani, S.M.; Aljohani, A.F.; et al. Numerical simulation of a fractional glucose-insulin model via successive approximation and ABM schemes. AIMS Math. 2025, 10, 22817–22849. https://doi.org/10.3934/math.20251014.
4.
Rafique, M.; Ur Rehamn, M.A.; Rafiq, M.; et al. Time delayed fractional diabetes mellitus model and consistent numerical algorithm. Sci. Rep. 2024, 14, 23871. https://doi.org/10.1038/s41598-024-74767-w.
5.
AL-Mekhlafi, S.M.; Abou Hasan, M.M.; Al-Ali, H.A.; et al. A Crossover Optimal Control Framework for a Time-Delayed Fractional-Order Diabetes Model: Stability, Bifurcation, and Numerical Analysis. Math. Methods Appl. Sci. 2025, 48, 15723–15741. https://doi.org/10.1002/mma.70046.
6.
Alhazmi, M.; Mirgani, S.M.; Saber, S. Hybrid Euler-Lagrange approach for fractional-order modeling of glucose-insulin dynamics. Axioms 2025, 14, 800. https://doi.org/10.3390/axioms14110800.
7.
Saber, S.; Solouma, E. Advanced fractional modeling of diabetes: Bifurcation analysis, chaos control, and a comparative study of numerical methods Adams-Bashforth-Moulton and Laplace-Adomian-Pad´e method. Indian J. Phys. 2025, 99, 5151–5169. https://doi.org/10.1007/s12648-025-03712-y.
8.
Saber, S.; Alahmari, A.; Shokeralla, A.A.; et al. Numerical Techniques for Solving Fractional Glucose-Insulin Regulatory Systems. Lobachevskii J. Math. 2025, 46, 5471–5483. https://doi.org/10.1134/s1995080225605405.
9.
Saber, S.; Solouma, E.; Aljohani, A.F.; et al. A hybrid approach to diabetes modeling: Fractional derivatives and stochastic analysis. AIMS Math. 2026, 11, 5029–5061. https://doi.org/10.3934/math.2026206.
10.
Shenbagam, M.; Kanagaraj, G.; Giri, J.; et al. Optimizing blood glucose regulation in type 1 diabetes: A fractional order controller approach. AIP Adv. 2024, 14, 045331. https://doi.org/10.1063/5.0199980.
11.
Farman, M.; Hasan, A.; Xu, C.; et al. Computational techniques to monitoring fractional order type-1 diabetes mellitus model for feedback design of artificial pancreas. Comput. Methods Programs Biomed. 2024, 257, 108420. https://doi.org/10.1016/j.cmpb.2024.108420.
12.
Atangana, A.; Baleanu, D. New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model. Therm. Sci. 2016, 20, 763–769. https://doi.org/10.2298/tsci160111018a.
13.
Abdeljawad, T.; Baleanu, D. Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels. Adv. Differ. Equ. 2016, 2016, 232. https://doi.org/10.1186/s13662-016-0949-5.
14.
Abdeljawad, T. Different type kernel h-fractional differences and their fractional h-sums. Chaos Solitons Fractals 2018, 116, 146–156. https://doi.org/10.1016/j.chaos.2018.09.022.
15.
Abdeljawad, T. On Riemann and Caputo fractional differences. Comput. Math. Appl. 2011, 62, 1602–1611. https://doi.org/10.1016/j.camwa.2011.03.036.
16.
Syed Ali, M.; Narayanan, G.; Shekher, V.; et al. Global Mittag-Leffler stability analysis of impulsive fractional-order complex-valued BAM neural networks with time varying delays. Commun. Nonlinear Sci. Numer. Simul. 2020, 83, 105088. https://doi.org/10.1016/j.cnsns.2019.105088.
17.
Narayanan, G.; Syed Ali, M.; Karthikeyan, R.; et al. Global Mittag-Leffler boundedness of nabla discrete-time fractional-order fuzzy complex-valued molecular models of mRNA and protein in regulatory mechanisms. Commun. Nonlinear Sci. Numer. Simul. 2024, 129, 107669. https://doi.org/10.1016/j.cnsns.2023.107669.
18.
Kchaou, M.; Narayanan, G.; Syed Ali, M.; et al. Finite-time Mittag-Leffler synchronization of delayed fractional-order discrete-time complex-valued genetic regulatory networks: Decomposition and direct approaches. Inf. Sci. 2024, 664, 120337. https://doi.org/10.1016/j.ins.2024.120337.
19.
Wei, Y.; Wei, Y.; Chen, Y.; et al. Mittag-Leffler stability of nabla discrete fractional-order dynamic systems. Nonlinear Dyn. 2020, 101, 407–417. https://doi.org/10.1007/s11071-020-05776-3.
20.
Wei, Y.; Chen, Y.; Wei, Y.; et al. Lyapunov stability analysis for nonlinear nabla tempered fractional order systems. Asian J. Control. 2023, 25, 3057–3066. https://doi.org/10.1002/asjc.3003.
21.
Wei, Y.; Zhao, L.;Wei, Y.; et al. Lyapunov theorem for stability analysis of nonlinear nabla fractional order systems. Commun. Nonlinear Sci. Numer. Simul. 2023, 126, 107443. https://doi.org/10.1016/j.cnsns.2023.107443.
22.
Wu, X.; Yang, X.; Liu, D.Y.; et al. Wirtinger-based fractional summation inequality for stability analysis of nabla discrete fractional-order time-delay systems. Nonlinear Dyn. 2024, 112, 17055–17068. https://doi.org/10.1007/s11071-024-09916-x.
23.
Mustapha, E.Z.; Abdellatif, E.O.; Karim, E.M.; et al. Fractional Caputo Operator and Takagi-Sugeno Fuzzy Modeling to Diabetes Analysis. Symmetry 2024, 16, 1395. https://doi.org/10.3390/sym16101395.
24.
Narayanan, G.; Syed Ali, M.; Rajchakit, G.; et al. Stability analysis for nabla discrete fractional-order of Glucose-Insulin Regulatory System on diabetes mellitus with Mittag-Leffler kernel. Biomed. Signal Process. Control. 2023, 80, 104295. https://doi.org/10.1016/j.bspc.2022.104295.
25.
Abdeljawad, T.; Al-Mdallal, Q.M. Discrete Mittag-Leffler kernel type fractional difference initial value problems and Gronwall’s inequality. J. Comput. Appl. Math. 2018, 339, 218–230. https://doi.org/10.1016/j.cam.2017.10.021.
26.
Chen, Y.; Li, X.; Liu, S. Finite-time stability of ABC type fractional delay difference equations. Chaos Solitons Fractals 2021, 152, 111430. https://doi.org/10.1016/j.chaos.2021.111430.

Scilight Press

Author Information

Abstract

Keywords

References

About Scilight

Journals

Publishing Policies

Contact Us