Qualitative Behavior of Solutions to a Mathematical Model of Memes Transmission

  • Authors

    • Reem Al-Amoudi King Abdulaziz University
    • Sarah Al-Sheikh King Abdulaziz University
    • Salma Al-Tuwairqi King Abdulaziz University
    2014-01-24
    https://doi.org/10.14419/ijamr.v3i1.1642

  • Abstract

    Researchers have applied epidemiological models to studythe dynamics of social and behavioral processes, based on the fact that both biological diseases and social behavioral are a result from interactions between individuals. The main feature of the paper is to understand the dynamics of spreading a meme on a large scale in a short time through a chain of communications. In this paper we study a meme transmission model, which is an extension of the deterministic Daley-Kendall model and we analyze it by using stability theory of nonlinear differential equations. The model is based on dividing the population into three disjoint classes of individuals according to their reaction to the meme. We examine the existence of equilibria of the model and investigate their stability using linearization methods, Lyapunov method and Hopf bifurcation analysis. One of the significantresultsinthispaperisfinding conditions that will lead to persistent of memes. Also numerical simulations are used to support the results.

    Keywords: Basic reproduction number, Global stability, Hopf bifurcation, Local stability, Lyapunov function, Memes transmission model.

  • References

    1. K. Dietz, Epidemics and Rumors: A survey, Journal of the Royal Statistical Society, Series A (General), Vol. 130, No.4, (1967) 505-528.
    2. V. Cane, A Note on the size of epidemics and the number of people hearing a rumor, Journal of the Royal Statistical Society, Series B (Methodological), Vol. 28, No. 3, (1966) 487-490.
    3. K. Thompson, R. Estrada, D. Daugherty, and A. Cintron-Arias, A deterministic approach to the spread of rumors, Working paper, Washington, DC, USA, (2003).
    4. L. Bettencourt, A. Cintron-Arias, D. Kaiser, and C. Castillo-Chavez, The power of a good idea: Quantitative modeling of the spread of ideas from epidemiological models, Physica A, 364 (2006) 513-536.
    5. K. Kawachi, Deterministic models for rumors transmission, Nonlinear analysis: Real world applications, 9 (2008) 1989-2028.
    6. K. Kawachi, M. Seki, H. Yoshida, Y. Otake, k. Warashina and H. Ueda, A rumor transmission model with various contact interactions, Journal of theoretical biology, 253 (2008) 55-60.
    7. J. Piqueira, Rumor propagation model: an equilibrium study, Hindawi Publishing Corporation Mathematical Problems in Engineering, Vol. (2010), Article ID 631357, doi:10.1155/2010/631357.
    8. W. Huang, On rumor spreading with skepticism and denial, Working paper, (2011).
    9. L. Wang, B. Wood, An epidemiological approach to model the viral propagation of memes, Applied Mathematical Modelling 35 (2011) 5442—5447.
    10. L. an Huo, P.Huang, and C.X. Guo, Analyzing the dynamics of a rumor transmission model with incubation, Hindawi Publishing Corporation Discrete Dynamics in Nature and Society, Vol. (2012), Article ID:328151, doi:10.1155/2012/328151.
    11. L. Zaho, Q. Wang, J. Cheng, D. Zhang, T. Ma, Y. Chen, and J. Wang, The impact of authorities media and rumor dissemination on the evolution of emergency, Physica A: Statistical Mechanics and its Applications, Volume 391, Issue 15 (2012) 3978-3987.
    12. F.Brauer, P.van den Driessche , and J.Wu, Mathematical Epidemiology, Springer-Verlag, (2008).
    13. L.Perko, Differential Equations and Dynamic Systems, Springer Verlag,(1991).
    14. L.Edelstein- Keshet, Mathematical Models in Biology, SIAM, (2005).
    15. M.W.Hirsch, S.Smale, and R.L.Devaney, Differential equations, dynamical systems and an introduction to chaos, Elsevier Academic press,(1974).

  • Downloads

  • How to Cite

    Al-Amoudi, R., Al-Sheikh, S., & Al-Tuwairqi, S. (2014). Qualitative Behavior of Solutions to a Mathematical Model of Memes Transmission. International Journal of Applied Mathematical Research, 3(1), 36-44. https://doi.org/10.14419/ijamr.v3i1.1642

    Received date: 2013-12-18

    Accepted date: 2014-01-17

    Published date: 2014-01-24