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Asymptotic Stability of a Rumor Spreading Model with Three Time Delays and Two Saturation Functions

Author

Listed:
  • Teng Sheng

    (Department of Computer Science, Sichuan University Jinjiang College, Meishan 620860, China)

  • Chunlong Fu

    (Department of Computer Science, Sichuan University Jinjiang College, Meishan 620860, China)

  • Xiaofan Yang

    (Department of Computer Science, Sichuan University Jinjiang College, Meishan 620860, China
    School of Big Data and Software Engineering, Chongqing University, Chongqing 400044, China)

  • Yang Qin

    (School of Big Data and Software Engineering, Chongqing University, Chongqing 400044, China)

  • Luxing Yang

    (School of Information Technology, Deakin University, Melbourne, VIC 3125, Australia)

Abstract

Time delays and saturation effects are critical elements describing complex rumor spreading behaviors. In this article, a rumor spreading model with three time delays and two saturation functions is proposed. The basic properties of the model are reported. The structure of the rumor-endemic equilibria is deduced. A criterion for the global asymptotic stability of the rumor-free equilibrium is derived. In the presence of very small delays, a criterion for the local asymptotic stability of a rumor-endemic equilibrium is provided. The influence of the delays and the saturation effects on the dynamics of the model is made clear through simulation experiments. In particular, it is found that (a) extended time delays lead to slower change in the number of spreaders or stiflers and (b) lifted saturation coefficients lead to slower change in the number of spreaders or stiflers. This work helps to deepen the understanding of complex rumor spreading phenomenon and develop effective rumor-containing schemes.

Suggested Citation

  • Teng Sheng & Chunlong Fu & Xiaofan Yang & Yang Qin & Luxing Yang, 2025. "Asymptotic Stability of a Rumor Spreading Model with Three Time Delays and Two Saturation Functions," Mathematics, MDPI, vol. 13(12), pages 1-25, June.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:12:p:2015-:d:1682105
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    References listed on IDEAS

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    1. Cheng, Yingying & Huo, Liang'an & Zhao, Laijun, 2022. "Stability analysis and optimal control of rumor spreading model under media coverage considering time delay and pulse vaccination," Chaos, Solitons & Fractals, Elsevier, vol. 157(C).
    2. Huo, Liang’an & Chen, Sijing, 2020. "Rumor propagation model with consideration of scientific knowledge level and social reinforcement in heterogeneous network," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 559(C).
    3. Liang’an Huo & Fan Ding & Chen Liu & Yingying Cheng, 2018. "Dynamical Analysis of Rumor Spreading Model considering Node Activity in Complex Networks," Complexity, Hindawi, vol. 2018, pages 1-10, November.
    4. Linhe Zhu & Hongyong Zhao, 2017. "Dynamical behaviours and control measures of rumour-spreading model with consideration of network topology," International Journal of Systems Science, Taylor & Francis Journals, vol. 48(10), pages 2064-2078, July.
    5. Xuefeng Yue & Liangan Huo, 2022. "Analysis of the Stability and Optimal Control Strategy for an ISCR Rumor Propagation Model with Saturated Incidence and Time Delay on a Scale-Free Network," Mathematics, MDPI, vol. 10(20), pages 1-20, October.
    6. Wang, Jinling & Jiang, Haijun & Hu, Cheng & Yu, Zhiyong & Li, Jiarong, 2021. "Stability and Hopf bifurcation analysis of multi-lingual rumor spreading model with nonlinear inhibition mechanism," Chaos, Solitons & Fractals, Elsevier, vol. 153(P2).
    7. Zhu, Linhe & Guan, Gui, 2019. "Dynamical analysis of a rumor spreading model with self-discrimination and time delay in complex networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 533(C).
    8. Chunfeng Wei & Chunlong Fu & Xiaofan Yang & Yang Qin & Luxing Yang, 2025. "Stability Analysis of a Rumor-Spreading Model with Two Time Delays and Saturation Effect," Mathematics, MDPI, vol. 13(11), pages 1-27, May.
    9. Zhao, Laijun & Wang, Jiajia & Chen, Yucheng & Wang, Qin & Cheng, Jingjing & Cui, Hongxin, 2012. "SIHR rumor spreading model in social networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(7), pages 2444-2453.
    10. El Fatini, Mohamed & Sekkak, Idriss & Laaribi, Aziz, 2019. "A threshold of a delayed stochastic epidemic model with Crowly–Martin functional response and vaccination," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 520(C), pages 151-160.
    11. Xupeng Luo & Haijun Jiang & Jiarong Li & Shanshan Chen & Yang Xia, 2024. "Modeling and controlling delayed rumor propagation with general incidence in heterogeneous networks," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 35(02), pages 1-29, February.
    12. Zhao, Juan & Yang, Lu-Xing & Zhong, Xiang & Yang, Xiaofan & Wu, Yingbo & Tang, Yuan Yan, 2019. "Minimizing the impact of a rumor via isolation and conversion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 526(C).
    13. Chunru Li & Zujun Ma, 2022. "Dynamics Analysis and Optimal Control for a Delayed Rumor-Spreading Model," Mathematics, MDPI, vol. 10(19), pages 1-25, September.
    14. Shuzhen Yu & Zhiyong Yu & Haijun Jiang, 2022. "Stability, Hopf Bifurcation and Optimal Control of Multilingual Rumor-Spreading Model with Isolation Mechanism," Mathematics, MDPI, vol. 10(23), pages 1-29, December.
    15. Liang’an Huo & Sijing Chen & Xiaoxiao Xie & Huiyuan Liu & Jianjia He & Xuzhen Zhu, 2021. "Optimal Control of ISTR Rumor Propagation Model with Social Reinforcement in Heterogeneous Network," Complexity, Hindawi, vol. 2021, pages 1-15, December.
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    1. Wei Zhang & Xiaofan Yang & Luxing Yang, 2025. "A Delayed Malware Propagation Model Under a Distributed Patching Mechanism: Stability Analysis," Mathematics, MDPI, vol. 13(14), pages 1-31, July.

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