Dynamics and optimal control of a fractional-order propagation model with time-varying intervention costs

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This article was retracted on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

The growing role of artificial intelligence (AI) in information dissemination has made algorithmic recommendation a significant pathway for rumor spread. To address this, we developed a dual-path rumor propagation model that integrates interpersonal communication with AI-driven algorithmic diffusion. The model employs Caputo fractional derivatives to capture memory effects and historical dependencies in the spreading process. For intervention, we proposed a coordinated control framework combining traditional measures (e.g., legal regulation and science education) with dynamic AI algorithmic control. A key innovation was the design of a time-varying cost function and the introduction of a time-sensitive factor to model the realistic constraint of rising AI control costs over time, forming a novel fractional-order optimal control problem. The trade-off between control cost and governance effectiveness was analyzed via Pareto frontier optimization. Theoretical analysis confirmed that the equilibrium and its global stability are governed by the basic reproduction number. Numerical simulations revealed the existence of a critical regime for the time-sensitive factor (ρ = 0.8), in which the optimal strategy exhibits a distinct staged control behavior: intensive AI-based intervention is emphasized during the early phase, followed by a gradual transition toward traditional measures for sustained long-term governance. This strategy effectively reduces the peak and overall scale of rumor propagation, demonstrates robustness across parameters, and achieves a balanced trade-off between control cost and social benefit. The study provides a dynamic modeling framework for rumor governance in the AI era, highlighting the importance of integrating algorithmic and traditional controls while accounting for time-varying costs.

Keywords

fractional-order optimal control / time-varying cost / time-sensitive factor / Pareto frontier / rumor propagation control

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] Z. Sha, S. Zhu, D. Jian, D Xiong, Overcoming language barriers via machine translation with sparse Mixture-of-Experts fusion of large language models, Inf. Process. Manag., 62(2025), 104078. https://doi.org/10.1016/j.ipm.2025.104078

[2] K. Shu, A. Sliva, S. Wang, J. Tang, H. Liu, Fake news detection on social media: A data mining perspective, ACM SIGKDD Explor. Newsl., 19(2017), 22-36. https://doi.org/10.1145/3137597.3137600

[3] S. Vosoughi, D. Roy, S. Aral, The spread of true and false news online, Science, 359(2018), 1146-1151. https://doi.org/10.1126/science.aap9559

[4] D. J. Daley, D. G. Kendall, Stochastic rumours, IMA J. Appl. Math., 1(1965), 42-55. https://doi.org/10.1093/imamat/1.1.42

[5] D. H. Zanette, Dynamics of rumor propagation on small-world networks, Phys. Rev. E, 65(2002), 041908. https://doi.org/10.1103/PhysRevE.65.041908

[6] M. Nekovee, Y. Moreno, G. Bianconi, M. Marsili, Theory of rumour spreading in complex social networks, Phys. A, 374(2007), 457-470. https://doi.org/10.1016/j.physa.2006.07.017

[7] D. J. Watts, S. H. Strogatz, Collective dynamics of ’small-world’networks, Nature, 393(1998), 440-442. https://doi.org/10.1038/30918

[8] A. L. Barabási, R. Albert, Emergence of scaling in random networks, Science, 286(1999), 509-512. https://doi.org/10.1126/science.286.5439.509

[9] X. Chen, N. Wang, Rumor spreading model considering rumor credibility, correlation and crowd classification based on personality, Sci. Rep., 10(2020), 5887. https://doi.org/10.1038/s41598-020-62585-9

[10] M. Kivelä, A. Arenas, M. Barthelemy, J. P. Gleeson, Y. Moreno, M. A. Porter, Multilayer networks, J. Complex Netw., 2(2014), 203-271. https://doi.org/10.1093/comnet/cnu016

[11] N. Hamdan, A. Kilicman, A fractional order SIR epidemic model for dengue transmission, Chaos Solitons Fractals, 114(2018), 55-62. https://doi.org/10.1016/j.chaos.2018.06.031

[12] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, North-Holland Mathematics Studies , Elsevier, 204(2006). Available from: https://www.sciencedirect.com/bookseries/north-holland-mathematics-studies/vol/204/suppl/C.

[13] K. Diethelm, The Analysis of Fractional Differential Equations , Springer, Berlin, Heidelberg, 2010. https://doi.org/10.1007/978-3-642-14574-2

[14] K. Diethelm, N. J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl., 265(2002), 229-248. https://doi.org/10.1006/jmaa.2000.7194

[15] V. Lakshmikantham, A. S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal., 69(2008), 2677-2682. https://doi.org/10.1016/j.na.2007.08.042

[16] R. P. Agarwal, M. Benchohra, S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math., 109(2010), 973-1033. https://doi.org/10.1007/s10440-008-9356-6

[17] H. G. Sun, Y. Zhang, D. Baleanu, W. Chen, Y. Chen, A new collection of real world applications of fractional calculus in science and engineering, Commun. Nonlinear Sci. Numer. Simul., 64(2018), 213-231. https://doi.org/10.1016/j.cnsns.2018.04.019

[18] W. Ding, S. Patnaik, S. Sidhardh, F. Semperlotti, Applications of distributed-order fractional operators: A review, Entropy, 23(2021), 110. https://doi.org/10.3390/e23010110

[19] P. Malik, Deepika, Stability analysis of fractional order modelling of social media addiction, Math. Found. Comput., 6(2023), 670-690. https://doi.org/10.3934/mfc.2022040

[20] X. Yue, W. Zhu, The dynamics and control of an ISCRM fractional-order rumor propagation model containing media reports, AIMS Math., 9(2024), 9721-9745. https://doi.org/10.3934/math.2024476

[21] W. Zhu, Dynamic fractional-order ISDR rumor propagation model incorporating refutation mechanism in complex networks, Sci. Rep., 15(2025), 31137. https://doi.org/10.1038/s41598-025-16369-8

[22] S. Dai, L. Zhu, L. He, Y. Dong, Dynamics and control of infectious diseases on complex networks: A theoretical and empirical study, J. Math. Anal. Appl., 557(2026), 130334. https://doi.org/10.1016/j.jmaa.2025.130334

[23] O. P. Agrawal, A general formulation and solution scheme for fractional optimal control problems, Nonlinear Dyn., 38(2004), 323-337. https://doi.org/10.1007/s11071-004-3764-6

[24] D. F. M. Torres, A. B. Malinowska, Introduction to the Fractional Calculus of Variations, Imperial College Press, London, 2012. Available from: https://archive.org/details/introductiontofr0000mali.

[25] S. A. Rakhshan, S. Effati, Fractional optimal control problems with time-varying delay: A new delay fractional Euler-Lagrange equations, J. Franklin Inst., 357(2020), 5954-5988. https://doi.org/10.1016/j.jfranklin.2020.03.038

[26] C. Dineshkumar, V. Vijayakumar, R. Udhayakumar, K. S. Nisar, A. Shukla, Results on approximate controllability results for fractional stochastic delay differential systems of order r ∈ (1, 2) , Stoch. Dyn., 23(2023), 2350047. https://doi.org/10.1142/S0219493723500478

[27] C. Dineshkumar, Y. Hoon Joo, A note concerning to approximate controllability of Atangana-Baleanu fractional neutral stochastic integro-differential system with infinite delay, Math. Methods Appl. Sci., 46(2023), 9922-9941. https://doi.org/10.1002/mma.9093

[28] H. M. Ali, S. Z. Rida, A. M. Yousef, A. S. Zaki, Dynamical behavior of fractional-order rumor model based on the activity of spreaders, Prog. Fract. Differ. Appl., 9(2023), 419-428. https://doi.org/10.18576/pfda/090308

[29] Y. Li, Y. Liu, J. Zhang, Dynamic analysis of a fractional-order SINPR rumor propagation model with emotional mechanisms, Fractal Fract., 9(2025), 546. https://doi.org/10.3390/fractalfract9080546

[30] L. Huo, L. Wang, X. Zhao, Stability analysis and optimal control of a rumor spreading model with media report, Phys. A, 517(2019), 551-562. https://doi.org/10.1016/j.physa.2018.11.047

[31] D. Li, J. Ma, How the government’s punishment and individual’s sensitivity affect the rumor spreading in online social networks, Phys. A, 469(2017), 284-292. https://doi.org/10.1016/j.physa.2016.11.033

[32] Z. Tufekci, Engineering the public: Big data, surveillance, and computational politics, First Monday, 19(2014). https://doi.org/10.5210/fm.v19i7.4901

[33] S. P. Sethi, G. L. Thompson, Optimal Control Theory: Applications to Management Science and Economics , Springer, Cham, 2000.

[34] M. Caputo, Linear models of dissipation whose Q is almost frequency independent-II, Geophys. J. Int., 13(1967), 529-539. https://doi.org/10.1111/j.1365-246X.1967.tb02303.x

[35] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications , Elsevier, 1998.

[36] H. L. Li, L. Zhang, C. Hu, Y. L. 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

[37] A. M. Lyapunov, The general problem of the stability of motion, Int. J. Control, 55(1992), 531-534. https://doi.org/10.1080/00207179208934253

[38] E. X. DeJesus, C. Kaufman, Routh-Hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations, Phys. Rev. A, 35(1987), 5288-5290. https://doi.org/10.1103/PhysRevA.35.5288

[39] R. M. Anderson, R. M. May, Infectious Diseases of Humans: Dynamics and Control , Oxford University Press, 1991. https://doi.org/10.1093/oso/9780198545996.001.0001

[40] Y. Ding, Z. Wang, H. Ye, Optimal control of a fractional-order HIV-immune system with memory, IEEE Trans. Control Syst. Technol., 20(2012), 763-769. https://doi.org/10.1109/TCST.2011.2153203

[41] J. P. LaSalle, The Stability of Dynamical Systems, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, 1976. https://doi.org/10.1137/1.9781611970432

[42] Y. Li, Y. Q. Chen, I. Podlubny, Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability, Comput. Math. Appl., 59(2010), 1810-1821. https://doi.org/10.1016/j.camwa.2009.08.019

[43] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishchenko, The Mathematical Theory of Optimal Processes , John Wiley and Sons, New York, 1962. Available from: https://archive.org/details/mathematicaltheo0000pont.

[44] Z. Yue, L. Wang, Lurkers, propagator classification and clarifiers in rumor propagation: A study of dynamics analysis and optimal control strategies, Int. J. Dyn. Control, 13(2025), 363. https://doi.org/10.1007/s40435-025-01877-2

[45] D. Matignon, Stability results for fractional differential equations with applications to control processing, in Computational Engineering in Systems Applications, 2(1996), 963-968. Available from: https://www.researchgate.net/publication/2581881.

[46] M. S. Tavazoei, A note on fractional-order derivatives of periodic functions, Automatica, 46(2010), 945-948. https://doi.org/10.1016/j.automatica.2010.02.023