Modeling and stability analysis of a memory-driven fractional labor dynamics system

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This article was retracted on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 investigates the existence and stability of a fractional-order labor dynamics model formulated using the fractional Nabla difference operator, specifically the Atangana-Baleanu fractional derivative in the Caputo sense. The model incorporates fractional dynamics to capture memory effects and the complex interactions associated with workforce layoffs. First, we present the mathematical formulation of the system and discuss its relevance to labor dynamics. Using the fixed-point theory, we establish the existence and uniqueness of solutions, demonstrating that the system is well posed. Furthermore, we examine the stability of the model in the Mittag-Leffler-Hyers-Ulam sense, providing insight into its long-term qualitative behavior. Numerical simulations support the theoretical findings and demonstrate that the fractional-order parameter significantly influences the system’s dynamics. Overall, this study offers a more general and flexible framework for modeling layoff processes using fractional calculus.

Keywords

fractional-order model / Atangana-Baleanu derivative / fractional Nabla difference operator / Mittag-Leffler-Hyers-Ulam 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This article has been retracted.Please see the Retraction Notice for more detail: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] J. D. Murray, Mathematical Biology: I. An Introduction , Springer, 2007.

[2] G. Awad, Introduction to labor market dynamics and employment inequality: theoretical frameworks, global and structural factors, employment patterns, and job quality, in Unveiling Developmental Disparities in the Middle East, (2025), 305-326. https://doi.org/10.4018/979-8-3693-7377-4.ch013

[3] X. Shi, Y. Hou, Y. Cai, Layoffs and enterprise value stability under major public health emergencies, J. Asia Pac. Econ., 2025 (2025), 1-31. https://doi.org/10.1080/13547860.2025.2479309

[4] M. B. Cetin, S. Gunduz, From boom to bust: unravelling the global tech layoffs phenomenon, in Economic Uncertainty in the Post-Pandemic Era, Routledge, (2024), 157-186. https://doi.org/10.4324/9781003461074-8

[5] S. J. Davis, P. M. Krolikowski, Sticky wages on the layoff margin, Am. Econ. Rev., 115 (2025), 491-524. https://doi.org/10.1257/aer.20240309

[6] M. Alikhani, H. Fazlollahtabar, A mathematical model for optimizing organizational learning capability, Adv. Oper. Res., 2014 (2014), 490210. https://doi.org/10.1155/2014/490210

[7] S. K. Pandey, K. Yadav, A mathematical model for viscous flow dynamics of tropical cyclones, Eur. J. Mech. B. Fluids, 111 (2025), 72-80. https://doi.org/10.1016/j.euromechflu.2024.12.003

[8] R. Kumar, D. Pamucar, A comprehensive and systematic review of multi-criteria decision-making (MCDM) methods to solve decision-making problems: two decades from 2004 to 2024, Spectrum Dec. Making Appl., 2 (2025), 178-197. https://doi.org/10.31181/sdmap21202524

[9] S. A. Zanib, T. Zubair, N. Abbas, W. Shatanawi, An advanced ABC finite difference approach for alcohol consumption dynamics, Model. Earth Syst. Environ., 11 (2025), 130. https://doi.org/10.1007/s40808-025-02307-0

[10] Y. Ordokhani, A. Hosseinian, P. Assari, Numerical simulation of a dynamic human capital model with demographic delays via the local discrete Galerkin method, Appl. Numer. Math., 217 (2025), 234-254. https://doi.org/10.1016/j.apnum.2025.06.007

[11] B. Ghosh, Fractional order modeling of ecological and epidemiological systems: ambiguities and challenges, J. Anal., 33 (2025), 341-366. https://doi.org/10.1007/s41478-024-00836-y

[12] Y. A. Madani, M. A. Almalahi, O. Osman, B. Muflih, K. Aldwoah, K. S. Mohamed, et al., Analysis of an acute diarrhea piecewise modified ABC fractional model: optimal control, stability and simulation, Fractal Fract., 9 (2025), 68. https://doi.org/10.3390/fractalfract9020068

[13] M. Revathy, A. Mohanapriya, R. Shenbagavalli, Optimizing Cancer Therapies with AI and Mathematical modeling, in 2025 IEEE International Conference on Compute, Control, Network & Photonics (ICCCNP) , (2025), 1-6. https://doi.org/10.1109/ICCCNP63914.2025.11233403

[14] K. S. Nisar, R. Chokkalingam, S. Sriramulu, S. Arunachalam, Mathematical study of Nabla fractional difference tech layoff model, Iran. J. Sci., 49 (2025), 345-356. https://doi.org/10.1007/s40995-024-01721-w

[15] J. N. Baron, M. T. Hannan, M. D. Burton, Labor pains: change in organizational models and employee turnover in young, high-tech firms, Am. J. Sociol., 106 (2001), 960-1012. https://doi.org/10.1086/320296

[16] T. Petaratip, P. Niamsup, Stability analysis of an unemployment model with time delay, AIMS Math., 6 (2021), 7421-7440. https://doi.org/10.3934/math.2021434

[17] W. O. Kermack, A. G. McKendrick, Contributions to the mathematical theory of epidemics-III. Further studies of the problem of endemicity, Bull. Math. Biol., 53 (1991), 89-118. https://doi.org/10.1016/S0092-8240(05)80042-4

[18] P. Prakash, V. Sakthivel, Layoffs analysis and prediction using machine learning algorithms, in Proceedings of the 6th International Conference on Communications and Cyber Physical Engineering, (2024), 535-543. https://doi.org/10.1007/978-981-99-7137-4_53

[19] G. Rak, Careers: these tech jobs are in demand: despite recent layoffs, big data and AI roles are growing, IEEE Spectr., 62 (2025), 15-16. https://doi.org/10.1109/MSPEC.2025.10918570

[20] S. Abrahams, An analysis of the gender layoff gap implied by a gender gap in wage bargaining, Econ. Lett., 234 (2023), 111505. https://doi.org/10.1016/j.econlet.2023.111505

[21] A. Selvam, S. Boulaaras, S. Sabarinathan, T. Radwan, Nonlinear fractional order financial system: chaotic behavior and Ulam-Hyers stability, Fractals, 33 (2025), 2540082. https://doi.org/10.1142/S0218348X25400821

[22] Z. Feng, Z. Xiang, Finite-time stability of fractional-order nonlinear systems, Chaos, 34 (2024), 023105. https://doi.org/10.1063/5.0170419

[23] S. Y. Yang, G. C. Wu, Discrete Gronwall’s inequality for Ulam stability of delay fractional difference equations, Math. Model. Anal., 30 (2025), 169-185. https://doi.org/10.3846/mma.2025.20017

[24] C. Liu, G. C. Wu, H. S. Hou, Asymptotic stability of fractional linear discrete-time equations with arbitrary time delays, Eur. Phys. J. Spec. Top., 234 (2025), 2881-2898. https://doi.org/10.1140/epjs/s11734-024-01442-6

[25] A. Mohanapriya, V. Sivakumar, P. Periasamy, A generalized approach of fractional Fourier transform to stability of fractional differential equation, Korean J. Math., 29 (2021), 749-763. https://doi.org/10.11568/kjm.2021.29.4.749

[26] N. I. Mahmudov, M. Aydin, Qualitative analysis of Caputo fractional delayed difference system: a novel delayed discrete fractional sine and cosine-type function, J. Nonlinear Sci. Appl., 18 (2025), 43-63. https://dx.doi.org/10.22436/jnsa.018.01.05

[27] R. Shah, N. Irshad, Ulam-Hyers-Mittag-Leffler stability for a class of nonlinear fractional reaction-diffusion equations with delay, Int. J. Theor. Phys., 64 (2025), 20. https://doi.org/10.1007/s10773-025-05884-z

[28] S. M. Ulam, Problems in Modern Mathematics , Wiley, 1964.

[29] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A., 27 (1941), 222-224. https://doi.org/10.1073/pnas.27.4.222

[30] T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Am. Math. Soc., 72 (1978), 297-300. https://doi.org/10.1090/S0002-9939-1978-0507327-1

[31] M. Shankar, R. Metzler, C. Li, A survey on the Ulam-Hyers stability of fractional-order differential equations, J. Phys. A: Math. Theor., 58 (2025), 453001. https://doi.org/10.1088/1751-8121/ae189e

[32] R. Debbar, H. Boulares, A. Moumen, T. Alraqad, H. Saber, Existence and uniqueness of neutral functional differential equations with sequential fractional operators, PLoS One, 19 (2024), e0304575. https://doi.org/10.1371/journal.pone.0304575

[33] C. Promsakon, I. Ansari, M. Wetsah, A. Kumar, K. Karthikeyan, T. Sitthiwirattham, Existence and uniqueness of solutions for fractional-differential equation with boundary condition using nonlinear multi-fractional derivatives, Math. Probl. Eng., 2024 (2024), 6844686. https://doi.org/10.1155/2024/6844686

[34] T. Abdeljawad, D. Baleanu, Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels, Adv. Differ. Equations, 2016 (2016), 232. https://doi.org/10.1186/s13662-016-0949-5

[35] A. I. Mueller, D. Osterwalder, J. Zweimüller, A. Kettemann, Vacancy durations and entry wages: Evidence from linked vacancy-employer-employee data, Rev. Econ. Stud., 91 (2024), 1807-1841. https://doi.org/10.1093/restud/rdad051

[36] M. B. Harari, A. L. Rubenstein, K. M. McCombs, S. Dennett, A meta-analysis of perceived job alternatives and employee turnover: addressing the availability versus quality distinction, J. Managerial Psychol., 40 (2025), 337-350. https://doi.org/10.1108/JMP-12-2023-0775

[37] W. F. Cascio, R. Montealegre, How technology is changing work and organizations, Annu. Rev. Organ. Psychol. Organ. Behav., 3 (2016), 349-375. https://doi.org/10.1146/annurev-orgpsych-041015-062352

[38] F. De Stefano, S. Bagdadli, A. Camuffo, The HR role in corporate social responsibility and sustainability: A boundary-shifting literature review, Hum. Resour. Manage., 57 (2018), 549-566. https://doi.org/10.1002/hrm.21870