Neurofeedback control of rumor propagation dynamics with cognitive fatigue

Xiaoyu Miao; Haoran Song; Lipu Zhang; Xiaoyu Miao; Haoran Song; Lipu Zhang

doi:10.3934/math.2026728

AIMS Mathematics

2026, Volume 11, Issue 6: 17859-17879. doi: 10.3934/math.2026728

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Neurofeedback control of rumor propagation dynamics with cognitive fatigue

1.
School of Media Engineering, Communication University of Zhejiang, Hangzhou 310018, China
2.
Zhejiang Key Laboratory of Film and TV Media Technology, Hangzhou 310018, China

Received: 20 March 2026 Revised: 03 June 2026 Accepted: 09 June 2026 Published: 18 June 2026
MSC : 34H05, 37N20, 49J15

The persistent spread of rumors and misinformation on social media poses severe challenges to public safety. Traditional interventions overlook the idea that repeated debunking induces cognitive fatigue, thus causing diminishing efficiency over time. This paper proposes a novel SHIR-F (Susceptible-Hesitant-Infected-Recovered with Fatigue) dynamic model that incorporates a cognitive fatigue mechanism. By building upon the classical compartmental structure, the model introduces a fatigue state variable and characterizes the diminishing marginal effect of interventions through an exponentially decaying effective debunking efficiency, thereby achieving a quantitative description of public cognitive response mechanisms in information propagation. Regarding the control design, this work constructs a stage-weighted optimal control objective that balances the infection scale against the intervention costs, and develops a closed-loop feedback control strategy parameterized by a dual-branch neural network. This architecture achieves the end-to-end numerical approximation of continuous-time control problems through the collaborative operation of peak and regular branches combined with a fixed-weight fusion mechanism, thus circumventing the computational difficulties associated with solving high-dimensional adjoint equations in traditional Pontryagin methods. A rigorous theoretical analysis proves the positive invariance, global existence, and uniqueness of system solutions, derives explicit expressions for the basic reproduction number, and reveals the monotonic modulation relationship between the fatigue levels and propagation thresholds. Numerical experiments demonstrate that the proposed strategy reduces the infection peaks by 37.4% and the total costs by 48.8%, while ensuring bounded control inputs and fatigue variables, thus exhibiting favorable numerical robustness under parameter perturbation conditions.
- rumor propagation model,
- cognitive fatigue,
- SHIR-F system,
- neurofeedback control,
- optimal control
Citation: Xiaoyu Miao, Haoran Song, Lipu Zhang. Neurofeedback control of rumor propagation dynamics with cognitive fatigue[J]. AIMS Mathematics, 2026, 11(6): 17859-17879. doi: 10.3934/math.2026728

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Abstract

The persistent spread of rumors and misinformation on social media poses severe challenges to public safety. Traditional interventions overlook the idea that repeated debunking induces cognitive fatigue, thus causing diminishing efficiency over time. This paper proposes a novel SHIR-F (Susceptible-Hesitant-Infected-Recovered with Fatigue) dynamic model that incorporates a cognitive fatigue mechanism. By building upon the classical compartmental structure, the model introduces a fatigue state variable and characterizes the diminishing marginal effect of interventions through an exponentially decaying effective debunking efficiency, thereby achieving a quantitative description of public cognitive response mechanisms in information propagation. Regarding the control design, this work constructs a stage-weighted optimal control objective that balances the infection scale against the intervention costs, and develops a closed-loop feedback control strategy parameterized by a dual-branch neural network. This architecture achieves the end-to-end numerical approximation of continuous-time control problems through the collaborative operation of peak and regular branches combined with a fixed-weight fusion mechanism, thus circumventing the computational difficulties associated with solving high-dimensional adjoint equations in traditional Pontryagin methods. A rigorous theoretical analysis proves the positive invariance, global existence, and uniqueness of system solutions, derives explicit expressions for the basic reproduction number, and reveals the monotonic modulation relationship between the fatigue levels and propagation thresholds. Numerical experiments demonstrate that the proposed strategy reduces the infection peaks by 37.4% and the total costs by 48.8%, while ensuring bounded control inputs and fatigue variables, thus exhibiting favorable numerical robustness under parameter perturbation conditions.

References

[1]	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 doi: 10.1093/imamat/1.1.42
[2]	D. P. Maki, M. Thompson, Mathematical models and applications: with emphasis on the social, life, and management sciences, Englewood Cliffs: Prentice-Hall, 1973.
[3]	D. J. Daley, J. Gani, Rumours: modelling spread and its cessation, In: Epidemic modelling: an introduction, Cambridge Studies in Mathematical Biology, Cambridge University Press, 1999,133–153.
[4]	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 doi: 10.1103/PhysRevE.65.041908
[5]	Y. Moreno, M. Nekovee, A. F. Pacheco, Dynamics of rumor spreading in complex networks, Phys. Rev. E, 69 (2004), 066130. https://doi.org/10.1103/PhysRevE.69.066130 doi: 10.1103/PhysRevE.69.066130
[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 doi: 10.1016/j.physa.2006.07.017
[7]	L. Zhao, H. Cui, X. Qiu, X. Wang, J. Wang, SIR rumor spreading model in the new media age, Phys. A, 392 (2013), 995–1003. https://doi.org/10.1016/j.physa.2012.09.030 doi: 10.1016/j.physa.2012.09.030
[8]	Y. Zan, J. Wu, P. Li, Q. Yu, SICR rumor spreading model in complex networks: counterattack and self-resistance, Phys. A, 405 (2014), 159–170. https://doi.org/10.1016/j.physa.2014.03.021 doi: 10.1016/j.physa.2014.03.021
[9]	S. Ahmed, M. E. Rasul, Examining the association between social media fatigue, cognitive ability, narcissism and misinformation sharing: cross-national evidence from eight countries, Sci. Rep., 13 (2023), 1541614. https://doi.org/10.1038/s41598-023-42614-z doi: 10.1038/s41598-023-42614-z
[10]	Y. Hwang, J. So, S. H. Jeong, Does COVID-19 message fatigue lead to misinformation acceptance? An extension of the risk information seeking and processing model, Health Commun., 38 (2023), 2742–2749. https://doi.org/10.1080/10410236.2022.2111636 doi: 10.1080/10410236.2022.2111636
[11]	A. K. M. N. Islam, S. Laato, S. Talukder, E. Sutinen, Misinformation sharing and social media fatigue during COVID-19: an affordance and cognitive load perspective, Technol. Forecast. Soc. Change, 159 (2020), 120201. https://doi.org/10.1016/j.techfore.2020.120201 doi: 10.1016/j.techfore.2020.120201
[12]	E. C. Tandoc, H. K. Kim, Avoiding real news, believing in fake news? Investigating pathways from information overload to misbelief, Journalism, 24 (2023), 1174–1192. https://doi.org/10.1177/14648849221090744 doi: 10.1177/14648849221090744
[13]	O. D. Apuke, B. Omar, E. A. Tunca, C. V. Gever, Information overload and misinformation sharing behaviour of social media users: testing the moderating role of cognitive ability, J. Inf. Sci., 50 (2024), 1371–1381. https://doi.org/10.1177/01655515221121942 doi: 10.1177/01655515221121942
[14]	L. Zhao, J. Wang, Y. Chen, Q. Wang, J. Cheng, H. Cui, SIHR rumor spreading model in social networks, Phys. A, 391 (2012), 2444–2453. https://doi.org/10.1016/j.physa.2011.12.008 doi: 10.1016/j.physa.2011.12.008
[15]	H. Zhu, J. Ma, Analysis of SHIR rumor propagation in random heterogeneous networks with dynamic friendships, Phys. A, 513 (2019), 257–271. https://doi.org/10.1016/j.physa.2018.09.015 doi: 10.1016/j.physa.2018.09.015
[16]	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 doi: 10.1038/s41598-020-62585-9
[17]	W. Liu, J. Wang, Y. Ouyang, Rumor transmission in online social networks under Nash equilibrium of a psychological decision game, Netw. Spat. Econ., 22 (2022), 831–854. https://doi.org/10.1007/s11067-022-09574-9 doi: 10.1007/s11067-022-09574-9
[18]	J. Gu, W. Li, X. Cai, The effect of the forget-remember mechanism on spreading, Eur. Phys. J. B, 62 (2008), 247–255. https://doi.org/10.1140/epjb/e2008-00139-4 doi: 10.1140/epjb/e2008-00139-4
[19]	L. Zhao, X. Qiu, X. Wang, J. Wang, Rumor spreading model considering forgetting and remembering mechanisms in inhomogeneous networks, Phys. A, 392 (2013), 987–994. https://doi.org/10.1016/j.physa.2012.10.031 doi: 10.1016/j.physa.2012.10.031
[20]	J. Chen, H. Ma, S. Yang, SEIOR rumor propagation model considering hesitating mechanism and different rumor-refuting ways in complex networks, Mathematics, 11 (2023), 283. https://doi.org/10.3390/math11020283 doi: 10.3390/math11020283
[21]	A. El Bhih, R. Ghazzali, S. Ben Rhila, M. Rachik, A. El Alami Laaroussi, A discrete mathematical modeling and optimal control of the rumor propagation in online social network, Discrete Dyn. Nat. Soc., 2020 (2020), 4386476. https://doi.org/10.1155/2020/4386476 doi: 10.1155/2020/4386476
[22]	N. Ding, G. Guan, S. Shen, L. Zhu, Dynamical behaviors and optimal control of delayed S$^2$IS rumor propagation model with saturated conversion function over complex networks, Commun. Nonlinear Sci. Numer. Simul., 128 (2024), 107603. https://doi.org/10.1016/j.cnsns.2023.107603 doi: 10.1016/j.cnsns.2023.107603
[23]	H. Guo, X. Yan, J. Zhang, Modeling and simulation of rumor propagation and optimal control strategy based on social positive reinforcement, Eur. Phys. J. Plus, 140 (2025), 49. https://doi.org/10.1140/epjp/s13360-024-05953-y doi: 10.1140/epjp/s13360-024-05953-y
[24]	D. Sharma, A. Shah, C. Gopalappa, A multi-agent reinforcement learning framework for public health decision analysis, Healthcare Anal., 8 (2025), 100436. https://doi.org/10.1016/j.health.2025.100436 doi: 10.1016/j.health.2025.100436
[25]	G. Liu, H. Li, L. Xiong, Y. Chen, A. Wang, D. Shen, Reinforcement learning for mitigating malware propagation in wireless radar sensor networks with channel modeling, Mathematics, 13 (2025), 1397. https://doi.org/10.3390/math13091397 doi: 10.3390/math13091397
[26]	S. Shen, X. Hao, Y. Shen, H. Xu, J. Dong, Z. Fang, Deep-reinforcement-learning-based botnet propagation control in the social internet of things, IEEE Int. Things J., 12 (2025), 27481–27495. https://doi.org/10.1109/JIOT.2025.3562583 doi: 10.1109/JIOT.2025.3562583
[27]	H. Lin, C. Tian, L. Chen, D. Liao, Y. Wang, Y. Hua, Model-based reinforcement learning for containing malware propagation in wireless radar sensor networks, Actuators, 14 (2025), 434. https://doi.org/10.3390/act14090434 doi: 10.3390/act14090434
[28]	D. Zhan, L. F. Toso, J. Anderson, Coreset-based task selection for sample-efficient meta-reinforcement learning, 2025 IEEE 64th Conference on Decision and Control (CDC), Rio de Janeiro, Brazil, 2025. https://doi.org/10.1109/CDC57313.2025.11312186
[29]	M. Raissi, P. Perdikaris, G. E. Karniadakis, Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys., 378 (2019), 686–707. https://doi.org/10.1016/j.jcp.2018.10.045 doi: 10.1016/j.jcp.2018.10.045
[30]	M. Kim, Y. Kim, Y. Kim, Physics-informed neural networks for optimal vaccination plan in SIR epidemic models, Math. Biosci. Eng., 22 (2025), 1598–1633. https://doi.org/10.3934/mbe.2025059 doi: 10.3934/mbe.2025059
[31]	A. Zhong, B. She, P. E. Paré, A physics-informed neural networks-based model predictive control framework for SIR epidemics, IEEE Open J. Control Syst., 5 (2025), 31–48. https://doi.org/10.1109/OJCSYS.2025.3641369 doi: 10.1109/OJCSYS.2025.3641369
[32]	G. C. Calafiore, C. Novara, C. Possieri, A time-varying SIRD model for the COVID-19 contagion in Italy, Annu. Rev. Control, 50 (2020), 361–372. https://doi.org/10.1016/j.arcontrol.2020.10.005 doi: 10.1016/j.arcontrol.2020.10.005
[33]	P. van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29–48. https://doi.org/10.1016/S0025-5564(02)00108-6 doi: 10.1016/S0025-5564(02)00108-6

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