Dynamical analysis of impulsive insulin delivery in a glucose-insulin system with insulin-degrading enzyme effects

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

This paper investigated impulsive insulin therapy within an extended glucose-insulin regulatory system that included insulin-degrading enzyme dynamics. Two treatment modes were examined: periodic delivery and state-dependent feedback. For the periodic model, we established solution positivity and boundedness, proved the global asymptotic stability of a unique positive periodic solution for type I diabetes, and derived permanence bounds along with a safe dosing region for type II diabetes. For the state-dependent model, under mild conditions, we demonstrated the existence and global orbital asymptotic stability of a unique order-1 periodic orbit within a physiologically relevant glucose range. Numerical simulations supported the theoretical results and illustrated how the choice of dosing period, bolus size, and threshold level influenced long-term glucose control. The results provided a unified theoretical framework for comparing and optimizing impulsive insulin therapies.

Keywords

glucose-insulin model / impulsive injection / periodic solution / stability

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] C. Chen, H. Tsai, Modeling the physiological glucose-insulin system on normal and diabetic subjects, Comput. Methods Programs Biomed., 97 (2010), 130-140. https://doi.org/10.1016/j.cmpb.2009.06.005

[2] A. Boutayeb, A. Chetouani, A critical review of mathematical models and data used in diabetology, Biomed. Eng. Online, 5 (2006), 43. https://doi.org/10.1186/1475-925X-5-43

[3] P. Palumbo, S. Ditlevsen, A. Bertuzzi, A. Gaetano, Mathematical modeling of the glucose-insulin system: A review, Math. Biosci., 244 (2013), 69-81. https://doi.org/10.1016/j.mbs.2013.05.006

[4] F. A. Rihan, Delay Differential Equations and Applications to Biology , 1^st edition, Springer, 2021. https://doi.org/10.1007/978-981-16-0626-7

[5] F. A. Rihan, K. Udhayakumar, Optimal control of glucose-insulin dynamics via delay differential model with fractional-order, Alexandria Eng. J., 114 (2025), 243-255. https://doi.org/10.1016/j.aej.2024.11.071

[6] S. Liu, G. Lv, Almost sure polynomial stability and stabilization of stochastic differential systems with impulsive effects, Stat. Probab. Lett., 206 (2024), 109980. https://doi.org/10.1016/j.spl.2023.109980

[7] F. A. Rihan, K. Udhayakumar, S. F. Nasrin, C. Rajivganthi, A stochastic delay differential model for glucose-insulin interactions, Ain Shams Eng. J., 16 (2025), 103668. https://doi.org/10.1016/j.asej.2025.103668

[8] 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

[9] 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

[10] C. T. Li, Y. T. Liu, X. Z. Feng, Y. Z. Wang, Dynamic analysis of a class of insulin-glucocorticoid model with nonlinear pulse, Nonlinear Anal. Real World Appl., 85 (2025), 104352. https://doi.org/10.1016/j.nonrwa.2025.104352

[11] M. Koschorreck, E. D. Gilles, Mathematical modeling and analysis of insulin clearance in vivo, BMC Syst. Biol., 2 (2008), 43. https://doi.org/10.1186/1752-0509-2-43

[12] H. Y. Wang, J. X. 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

[13] C. T. Li, Y. T. Liu, Y. Z. Wang, X. Z. Feng, Dynamic modeling of the glucose-insulin system with inhibitors impulsive control, Math. Methods Appl. Sci., 2024 (2024), 20907267. https://doi.org/10.1002/mma.10266

[14] S. Y. Tang, Y. N. Xiao, One-compartment model with Michaelis-Menten elimination kinetics and therapeutic window: An analytical approach, J. Pharmacokinet. Pharmacodyn., 34 (2007), 807-827. https://doi.org/10.1007/s10928-007-9070-4

[15] S. M. Alzahrani, Statistical insights into stochastic glucose-insulin dynamics: Modeling insulin degradation with the Michaelis-Menten function, Eur. J. Pure Appl. Math., 18 (2025), 5404. https://doi.org/10.29020/nybg.ejpam.v18i1.5404

[16] W. Farris, S. Mansourian, Y. Chang, L. Lindsley, E. A. Eckman, M. P. Frosch, et al., Insulin-degrading enzyme regulates the levels of insulin, amyloid β-protein, and the β-amyloid precursor protein intracellular domain in vivo, Proc. Natl. Acad. Sci. U.S.A., 100 (2003), 4162-4167. https://doi.org/10.1073/pnas.0230450100

[17] O. Pivovarova, A. Höhn, T Grune, A. Pfeiffer, N. Rudovich, Insulin-degrading enzyme: New therapeutic target for diabetes and Alzheimer’s disease?, Ann. Med., 48 (2016), 614-624. https://doi.org/10.1080/07853890.2016.1197416

[18] M. A. Leissring, C. M. González-Casimiro, B. Merino, C. N. Suire, G. Perdomo, Targeting insulin-degrading enzyme in insulin clearance, Int. J. Mol. Sci., 22 (2021), 2235. https://doi.org/10.3390/ijms22052235

[19] G. Tundo, D. Sbardella, C. Ciaccio, G. Grasso, M. Gioia, A. Coletta, et al., Multiple functions of insulin-degrading enzyme: A metabolic crosslight?, Crit. Rev. Biochem. Mol. Biol., 52 (2017), 554-582. https://doi.org/10.1080/10409238.2017.1337707

[20] A. Mari, A. Tura, E. Grespan, R. Bizzotto, Mathematical modeling for the physiological and clinical investigation of glucose homeostasis and diabetes, Front. Physiol., 11 (2020), 575789. https://doi.org/10.3389/fphys.2020.575789

[21] F. Rao, Z. L. Zhang, J. X. Li, Dynamical analysis of a glucose-insulin regulatory system with insulin-degrading enzyme and multiple delays, J. Math. Biol., 87 (2023), 73. https://doi.org/10.1007/s00285-023-02003-6

[22] C. M. González-Casimiro, B. Merino, E. Casanueva-Álvarez, T. Postigo-Casado, P. Cámara-Torres, C. M. Fernández-Daz, et al., Modulation of insulin sensitivity by insulin-degrading enzyme, Biomedicines, 9 (2021), 86. https://doi.org/10.3390/biomedicines9010086

[23] J. Yang, S. Y. Tang, R. A. Cheke, The regulatory system for diabetes mellitus: Modeling rates of glucose infusions and insulin injections, Commun. Nonlinear Sci. Numer. Simul., 37 (2016), 305-325. https://doi.org/10.1016/j.cnsns.2016.02.001

[24] J. Yang, S. Y. Tang, R. A. Cheke, Modelling the regulatory system for diabetes mellitus with a threshold window, Commun. Nonlinear Sci. Numer. Simul., 22 (2015), 478-491. https://doi.org/10.1016/j.cnsns.2014.08.012

[25] C. T. Li, J. Tian, X. Z. Feng, Y. T. Liu, Study on dynamical behavior of a type of diabetes mellitus model with nonlinear impulsive injection of insulin, Math. Methods Appl. Sci., 48 (2025), 16765-16777. https://doi.org/10.1002/mma.70125

[26] J. Jia, Z. Zhao, J. G. Yang, A. Zeb, Parameter estimation and global sensitivity analysis of a bacterial-plasmid model with impulsive drug treatment, Chaos, Solitons Fractals, 183 (2024), 114901. https://doi.org/10.1016/j.chaos.2024.114901

[27] Q. Li, B. W. Liu, Traveling wave fronts to a Mackey-Glass model involving distinct delays in diffusion term and birth function, Math. Methods Appl. Sci., 48 (2025), 15549-15558. https://doi.org/10.1002/mma.11109

[28] D. D. Bainov, P. S. Simeonov, System with Impulse Effect: Stability, Theory and Applications , Ellis Horwood, Chichester, 1989.

[29] X. D. Li, M. Bohner, C. K. Wang, Impulsive differential equations: periodic solutions and applications, Automatica, 52 (2015), 173-178. https://doi.org/10.1016/j.automatica.2014.11.009

[30] G. Yang, L. G. Yao, Positive almost periodic solutions for an epidemic model with saturated treatment, Taiwanese J. Math., 20 (2016), 1377-1392. https://doi.org/10.11650/tjm.20.2016.7639

[31] G. Yang, Dynamical behaviors on a delay differential neoclassical growth model with patch structure, Math. Methods Appl. Sci., 41 (2018), 3856-3867. https://doi.org/10.1002/mma.4872