Dynamical Analysis of a Rumor Propagation Model With Saturated Recruitment and Incorporated Control Strategies

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-24-2026-308
Published: Oct 5, 2026
Cite as:
Lukman Hakim, Fatmawati, Ahmadin, Chinwendu E. Madubueze, Dynamical Analysis of a Rumor Propagation Model With Saturated Recruitment and Incorporated Control Strategies, Int. J. Anal. Appl., 24 (2026), 308.

Main Article Content

Lukman Hakim, Fatmawati, Ahmadin, Chinwendu E. Madubueze

Abstract

Unverified information can spread quickly and influence the public. In this work, we suggest and analyze a nonlinear rumor propagation model incorporating saturated incidence and optimal control. In this model, the qualitative properties of the model are rigorously analyzed. The positivity and boundedness of the solutions are established to ensure biological feasibility. The threshold dynamics is governed, equilibrium points are analyzed and their stability. Bifurcation analysis shows that forward bifurcations occur at a reproduction number equal to one. Sensitivity analysis reveals that rumor propagation is strongly influenced by contact rate with rumor, the direct probability of rumor, and the transition from rumor spreaders to lurkers. To design mitigation strategies, an optimal control is produced with two controls: educational control and truth-defender control. The optimality conditions are derived using Pontryagin’s and numerically utilizing the forward-backward sweep method. Numerical simulations demonstrate that truth dissemination is the dominant intervention in reducing rumor spreaders and lurkers, while contact reduction provides strong support in the early stage. The combined control strategy produces the most robust outcome by simultaneously minimizing the rumor and strengthening the corrective spread of information.

Article Details

References

  1. 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.
  2. W. Li, H. Shen, Z. Huang, H. Yang, Research on the Dynamical Behavior of Public Opinion Triggered by Rumor Based on a Nonlinear Oscillator Model, Entropy 25 (2023), 1614. https://doi.org/10.3390/e25121614.
  3. F. Yin, X. Jiang, X. Qian, X. Xia, Y. Pan, et al., Modeling and Quantifying the Influence of Rumor and Counter-Rumor on Information Propagation Dynamics, Chaos Solitons Fractals 162 (2022), 112392. https://doi.org/10.1016/j.chaos.2022.112392.
  4. S. Yu, Z. Yu, H. Jiang, J. Li, Dynamical Study and Event-Triggered Impulsive Control of Rumor Propagation Model on Heterogeneous Social Network Incorporating Delay, Chaos Solitons Fractals 145 (2021), 110806. https://doi.org/10.1016/j.chaos.2021.110806.
  5. M. Ghosh, P. Das, P. Das, A Comparative Study of Deterministic and Stochastic Dynamics of Rumor Propagation Model with Counter-Rumor Spreader, Nonlinear Dyn. 111 (2023), 16875–16894. https://doi.org/10.1007/s11071-023-08768-1.
  6. M. Jiang, Q. Gao, J. Zhuang, Reciprocal Spreading and Debunking Processes of Online Misinformation: A New Rumor Spreading–Debunking Model with a Case Study, Phys. A 565 (2021), 125572. https://doi.org/10.1016/j.physa.2020.125572.
  7. J. Song, X. Zhu, Research on Public Opinion Guidance of Converging Media Based on AHP and Transmission Dynamics, Math. Biosci. Eng. 18 (2021), 6857–6886. https://doi.org/10.3934/mbe.2021341.
  8. M. Wang, Y. Hu, Dynamic Analysis and Control Measures of Rumor Propagation Model Considering True Individuals and Public Opinion Individuals, Inf. Sci. 676 (2024), 120779. https://doi.org/10.1016/j.ins.2024.120779.
  9. W. Pan, W. Yan, Y. Hu, R. He, L. Wu, Dynamic Analysis of a SIDRW Rumor Propagation Model Considering the Effect of Media Reports and Rumor Refuters, Nonlinear Dyn. 111 (2023), 3925–3936. https://doi.org/10.1007/s11071-022-07947-w.
  10. S. Yu, Z. Yu, H. Jiang, S. Yang, The Dynamics and Control of 2I2SR Rumor Spreading Models in Multilingual Online Social Networks, Inf. Sci. 581 (2021), 18–41. https://doi.org/10.1016/j.ins.2021.08.096.
  11. D.J. Daley, D.G. Kendall, Epidemics and Rumours, Nature 204 (1964), 1118. https://doi.org/10.1038/2041118a0.
  12. D.P. Maki, M. Thompson, Mathematical Models and Applications: With Emphasis on the Social, Life, and Management Sciences, Prentice-Hall, 1973.
  13. L. Huo, P. Huang, C.-X. Guo, Analyzing the Dynamics of a Rumor Transmission Model with Incubation, Discret. Dyn. Nat. Soc. 2012 (2012), 328151. https://doi.org/10.1155/2012/328151.
  14. J. Dhar, A. Jain, V.K. Gupta, A Mathematical Model of News Propagation on Online Social Network and a Control Strategy for Rumor Spreading, Soc. Netw. Anal. Min. 6 (2016), 57. https://doi.org/10.1007/s13278-016-0366-5.
  15. R. Isea, K.E. Lonngren, A New Variant of the SEIZ Model to Describe the Spreading of a Rumor, Int. J. Data Sci. Anal. 3 (2017), 28–33. https://doi.org/10.11648/j.ijdsa.20170304.12.
  16. K. Afassinou, Analysis of the Impact of Education Rate on the Rumor Spreading Mechanism, Phys. A 414 (2014), 43–52. https://doi.org/10.1016/j.physa.2014.07.041.
  17. R. Jie, J. Qiao, G. Xu, Y. Meng, A Study on the Interaction between Two Rumors in Homogeneous Complex Networks under Symmetric Conditions, Phys. A 454 (2016), 129–142. https://doi.org/10.1016/j.physa.2016.02.048.
  18. Y. Hu, Q. Pan, W. Hou, M. He, Rumor Spreading Model Considering the Proportion of Wisemen in the Crowd, Phys. A 505 (2018), 1084–1094. https://doi.org/10.1016/j.physa.2018.04.056.
  19. R. Escalante, M. Odehnal, A Deterministic Mathematical Model for the Spread of Two Rumors, Afrika Mat. 31 (2020), 315–331. https://doi.org/10.1007/s13370-019-00726-8.
  20. Y. Liu, C. Zeng, Y. Luo, Dynamics of a New Rumor Propagation Model with the Spread of Truth, Appl. Math. 9 (2018), 536–549. https://doi.org/10.4236/am.2018.95038.
  21. L. Zhu, G. Guan, Dynamical Analysis of a Rumor Spreading Model with Self-Discrimination and Time Delay in Complex Networks, Phys. A 533 (2019), 121953. https://doi.org/10.1016/j.physa.2019.121953.
  22. J. Wang, H. Jiang, C. Hu, Z. Yu, J. Li, Stability and Hopf Bifurcation Analysis of Multi-Lingual Rumor Spreading Model with Nonlinear Inhibition Mechanism, Chaos Solitons Fractals 153 (2021), 111464. https://doi.org/10.1016/j.chaos.2021.111464.
  23. S. Chen, H. Jiang, L. Li, J. Li, Dynamical Behaviors and Optimal Control of Rumor Propagation Model with Saturation Incidence on Heterogeneous Networks, Chaos Solitons Fractals 140 (2020), 110206. https://doi.org/10.1016/j.chaos.2020.110206.
  24. L. Zhu, B. Wang, Stability Analysis of a SAIR Rumor Spreading Model with Control Strategies in Online Social Networks, Inf. Sci. 526 (2020), 1–19. https://doi.org/10.1016/j.ins.2020.03.076.
  25. Y. Cheng, L. Huo, L. Zhao, Dynamical Behaviors and Control Measures of Rumor-Spreading Model in Consideration of the Infected Media and Time Delay, Inf. Sci. 564 (2021), 237–253. https://doi.org/10.1016/j.ins.2021.02.047.
  26. Y. Dong, L. Huo, L. Zhao, An Improved Two-Layer Model for Rumor Propagation Considering Time Delay and Event-Triggered Impulsive Control Strategy, Chaos Solitons Fractals 164 (2022), 112711. https://doi.org/10.1016/j.chaos.2022.112711.
  27. Y. Xia, H. Jiang, Z. Yu, S. Yu, X. Luo, Dynamic Analysis and Optimal Control of a Reaction-Diffusion Rumor Propagation Model in Multi-Lingual Environments, J. Math. Anal. Appl. 521 (2023), 126967. https://doi.org/10.1016/j.jmaa.2022.126967.
  28. N. Ding, G. Guan, S. Shen, L. Zhu, Dynamical Behaviors and Optimal Control of Delayed S2IS 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.
  29. Z. Yu, S. Wu, Y. Zhang, X. Cong, K. Wu, IRLTS Rumor Propagation Model Considering the Influence of Lurking Psychology and the Truth of Rumors, Nonlinear Dyn. 112 (2024), 18593–18609. https://doi.org/10.1007/s11071-024-09741-2.
  30. H. Castillo-Alvino, M. Marvá, The Competition Model with Holling Type II Competitive Response to Interfering Time, J. Biol. Dyn. 14 (2020), 222–244. https://doi.org/10.1080/17513758.2020.1742392.
  31. M. Gümüş, K. Türk, Dynamical Behavior of a Hepatitis B Epidemic Model and Its NSFD Scheme, J. Appl. Math. Comput. 70 (2024), 3767–3788. https://doi.org/10.1007/s12190-024-02103-6.
  32. F. Jarad, T. Abdeljawad, Z. Hammouch, On a Class of Ordinary Differential Equations in the Frame of Atangana–Baleanu Fractional Derivative, Chaos Solitons Fractals 117 (2018), 16–20. https://doi.org/10.1016/j.chaos.2018.10.006.
  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.
  34. M. Haque, N. Ali, S. Chakravarty, Study of a Tri-Trophic Prey-Dependent Food Chain Model of Interacting Populations, Math. Biosci. 246 (2013), 55–71. https://doi.org/10.1016/j.mbs.2013.07.021.
  35. M. Margaliot, G. Langholz, The Routh–Hurwitz Array and Realization of Characteristic Polynomials, IEEE Trans. Autom. Control 45 (2000), 2424–2427. https://doi.org/10.1109/9.895587.
  36. J.P. LaSalle, Stability Theory for Ordinary Differential Equations, J. Differ. Equ. 4 (1968), 57–65. https://doi.org/10.1016/0022-0396(68)90048-X.
  37. S. Sangsawang, U.W. Humphries, A. Khan, P. Pongsumpun, Sensitivity Analysis of Cassava Mosaic Disease with Saturation Incidence Rate Model, AIMS Math. 8 (2023), 6233–6254. https://doi.org/10.3934/math.2023315.
  38. X. Zhu, H. Liu, X. Lin, Q. Zhang, Y. Wei, Global Stability and Optimal Vaccination Control of SVIR Models, AIMS Math. 9 (2024), 3453–3482. https://doi.org/10.3934/math.2024170.
  39. C. Castillo-Chavez, B. Song, Dynamical Models of Tuberculosis and Their Applications, Math. Biosci. Eng. 1 (2004), 361–404. https://doi.org/10.3934/mbe.2004.1.361.
  40. L. Hakim, A Pontryagin Principle and Optimal Control of Spreading COVID-19 with Vaccination and Quarantine Subtype, Commun. Math. Biol. Neurosci. 2023 (2023), 94. https://doi.org/10.28919/cmbn/8157.
  41. C.E. Madubueze, Z. Chazuka, C.P. Ogbogbo, F. Fatmawati, Mathematical Analysis and Optimal Control of Lassa Fever Transmission, Int. J. Appl. Comput. Math. 11 (2025), 207. https://doi.org/10.1007/s40819-025-02030-5.
  42. T. Jiang, X. Lv, D. Fan, Q. Li, J. Yang, Higher-Order Dynamics and Optimal Control of SEIR Rumor Propagation Models in Homogeneous and Heterogeneous Networks, Math. Model. Control 6 (2026), 72–87. https://doi.org/10.3934/mmc.2026006.
  43. L. Huo, T. Lin, C. Fan, C. Liu, J. Zhao, Optimal Control of a Rumor Propagation Model with Latent Period in Emergency Event, Adv. Differ. Equ. 2015 (2015), 54. https://doi.org/10.1186/s13662-015-0394-x.
  44. G. Zhang, H. Dong, H. Li, C. Xu, H.R. Karimi, et al., Dynamics Mechanism of Time-Delay Reaction–Diffusion Rumor-Propagation Model with Saturation Control, Adv. Contin. Discrete Models 2025 (2025), 116. https://doi.org/10.1186/s13662-025-03974-0.
  45. M. Naim, F. Lahmidi, A. Namir, Dynamics of a Delayed Rumor Propagation Model with Consideration of Psychological Factors and Forgetting Mechanism, Appl. Math. Inf. Sci. 14 (2020), 597–604. https://doi.org/10.18576/amis/140408.
  46. 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.
  47. A. Yang, X. Huang, X. Cai, X. Zhu, L. Lu, ILSR Rumor Spreading Model with Degree in Complex Network, Phys. A 531 (2019), 121807. https://doi.org/10.1016/j.physa.2019.121807.
  48. X. Liu, T. Li, M. Tian, Rumor Spreading of a SEIR Model in Complex Social Networks with Hesitating Mechanism, Adv. Differ. Equ. 2018 (2018), 391. https://doi.org/10.1186/s13662-018-1852-z.
  49. L. Zhao, X. Wang, J. Wang, X. Qiu, W. Xie, Rumor-Propagation Model with Consideration of Refutation Mechanism in Homogeneous Social Networks, Discret. Dyn. Nat. Soc. 2014 (2014), 659273. https://doi.org/10.1155/2014/659273.
  50. R. Li, Y. Li, Z. Meng, Y. Song, G. Jiang, Rumor Spreading Model Considering Individual Activity and Refutation Mechanism Simultaneously, IEEE Access 8 (2020), 63065–63076. https://doi.org/10.1109/ACCESS.2020.2983249.
  51. J. Liao, J. Wang, J. Li, X. Jiang, The Dynamics and Control of a Multi-Lingual Rumor Propagation Model with Non-Smooth Inhibition Mechanism, Math. Biosci. Eng. 21 (2024), 5068–5091. https://doi.org/10.3934/mbe.2024224.