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A deterministic mathematical model for the spread of two rumors


In this paper we propose a deterministic mathematical model that attempts to explain the propagation of a rumor using SIRS type epidemiological models with temporary immunity and nonlinear incidence rate. In particular, we speculate about the dissemination of information when the so-called “complex networks” are used. The effect of introducing a second rumor, inspired by a vaccination model, in the same population of individuals, which will try to counteract the effect of the original rumor, is studied. That is a situation that occurs frequently in communities, when a rumor is counteracted by a contrary information or news, which behaves in the same way as a rumor. Furthermore, qualitative analysis and numerical experimentation of the dynamic model are performed. We corroborate that the dynamics of spreading rumors show similar behavior to that found in the dynamics of an infectious disease.

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  1. Allen, L., Lahondny Jr., G.: Extinction thresholds in deterministic and stochastic epidemic models. J. Biol. Dyn. 6(2), 590–611 (2012)

    Article  Google Scholar 

  2. Brauer, F.: Backward bifurcations in simple vaccination models. J. Math. Anal. Appl. 298, 418–431 (2004)

    MathSciNet  Article  Google Scholar 

  3. Brauer, F.: Backward bifurcations in simple vaccination/treatment models. J. Biol. Dyn. 5(5), 410–418 (2011)

    MathSciNet  Article  Google Scholar 

  4. Brauer, F., Castillo-Chavez, C.: Mathematical Models in Population Biology and Epidemiology, 2nd edn. Springer, New York (2012)

    Book  Google Scholar 

  5. Brauer, F., Castillo-Chavez, C.: Mathematical Models for Communicable Deseases. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2013)

    MATH  Google Scholar 

  6. Cheng, J.J., Liu, Y., Shen, B., Yuan, W.G.: An epidemic model of rumor diffusion in online social networks. Eur. Phys. J. B 86, 29 (2013)

    MathSciNet  Article  Google Scholar 

  7. Chierichetti, F., Lattanzi, S., Panconesi, A.: Rumor spreading in social networks. Theor. Comput Sci. 412(24), 2602–2610 (2011).[Selected Papers from 36th International Colloquium on Automata, Languages and Programming (ICALP 2009)]

    MathSciNet  Article  Google Scholar 

  8. Daley, D., Gani, J.: Epidemic Modelling: An Introduction. Cambridge University Press, Cambridge (2001)

    MATH  Google Scholar 

  9. Daley, D., Kendall, D.: Stochastic rumors. J. Inst. Math. Appl. 1, 42–55 (1965)

    MathSciNet  Article  Google Scholar 

  10. van den Driessche, P., Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180(1–2), 29–48 (2002)

    MathSciNet  Article  Google Scholar 

  11. Edelstein-Keshet, L.: Mathematical Models in Biology. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2005)

    Book  Google Scholar 

  12. Erciyes, K.: Complex Networks, An Algorithmic Perspective. CRC Press, Boca Raton (2015)

    MATH  Google Scholar 

  13. Hu, Y., Pan, Q., Hou, W., He, M.: Rumor spreading model with the different attitudes towards rumors. Phys. A Stat. Mech. Appl. 502, 331–344 (2018).

    MathSciNet  Article  Google Scholar 

  14. Kermack, W., McKendrick, A.: A contributions to the mathematical theory epidemics. In: Proceedings of the Royal Society of London, Series A, vol. 115, no. 772, pp. 700–721. The Royal Society, London (1927)

  15. Liu, X., Li, T., Tian, M.: Rumor spreading of a seir model in complex social networks with hesitating mechanism. Adv. Differ. Equations 2018(1), 391 (2018).

    MathSciNet  Article  MATH  Google Scholar 

  16. Maki, D., Thompson, M.: Mathematical Models and Applications, With Emphasis on Social, Life, and Management Sciences. Prentice-Hall, Englewood Cliffs (1973)

    Google Scholar 

  17. Moreno, Y., Nekovee, M., Pacheco, A.: Dynamics of rumor spreading in complex networks. Phys. Rev. E 69, 066130 (2004)

    Article  Google Scholar 

  18. Nekovee, M., Moreno, Y., Bianconi, G., Marsili, M.: Theory of rumour spreading in complex social networks. Phys. A 374, 457–470 (2007)

    Article  Google Scholar 

  19. Pearce, J.: Spreading rumors (2016). Accessed on 04 Apr 2016

  20. The Math Works: Constrained-nonlinear-optimization algorithms (2017). Accessed on 07 Jul 2017

  21. The Math Works: fmincon (2017). Accessed on 07 Apr 2017

  22. The Math Works: fminsearch (2017). Accessed on 07 Apr 2017

  23. The World Bank: Fixed broadband subscriptions. Indicators (2016). Accessed on 7 Apr 2016

  24. The World Bank: Mobile cellular subscriptions. Indicators (2016). Accessed on 05 Apr 2016

  25. Thompson, K., Estrada, R.C., Daugherty, D., Cintrón-Arias, A.: A deterministic approach to the spread of rumors. Tech. Rep. BU-1642-M, University of Manchester, Cornell University, Dept. of Biological Statistics and Computational Biology (2003)

  26. Turenne, N.: The rumour spectrum. PLoS One 13(1), 1–27 (2018).

    Article  Google Scholar 

  27. Wang, C., Tan, Z.X., Ye, Y., Wang, L., Cheong, K.H., Xie, N.g.: A rumor spreading model based on information entropy. Sci. Rep. 7(1), 9615 (2017).

  28. Zanette, D.: Critical behavior of propagation on small-world networks. Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 64, 050901 (2001)

    Article  Google Scholar 

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This research was partially supported by the Decanato de Investigación y Desarrollo (DID) at USB.

Special thanks to the reviewers for their valuable work.

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Correspondence to Marco Odehnal.

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Escalante, R., Odehnal, M. A deterministic mathematical model for the spread of two rumors. Afr. Mat. 31, 315–331 (2020).

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  • Mathematical model
  • Spreading rumor
  • SIRS model
  • Vaccination model
  • Epidemic

Mathematics Subject Classification

  • 92D50
  • 92D30
  • 92D25
  • 92D99