Table of Contents Author Guidelines Submit a Manuscript
Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 328151, 21 pages
http://dx.doi.org/10.1155/2012/328151
Research Article

Analyzing the Dynamics of a Rumor Transmission Model with Incubation

1Antai College of Economics and Management, Shanghai Jiao Tong University, Shanghai 200052, China
2Business School of Sichuan University, Chengdu 610065, China

Received 9 November 2011; Accepted 3 January 2012

Academic Editor: Hassan A. El-Morshedy

Copyright © 2012 Liang'an Huo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. H. Hayakawa, Sociology of Rumor-Approach from Formal Sociology, Seikyusya, Tokyo, Japan, 2002.
  2. T. Shibutani, Improvised News: A Sociological Study of Rumor, Bobbs-Merrill, Indianapolis, Ind, USA, 1966.
  3. M. Kosfeld, “Rumours and markets,” Journal of Mathematical Economics, vol. 41, no. 6, pp. 646–664, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. A. Rapoport, “Spread of information through a population with socio-structural bias. I. Assumption of transitivity,” vol. 15, pp. 523–533, 1953. View at Google Scholar
  5. A. Rapoport, “Spread of information through a population with socio-structural bias. II. Various models with partial transitivity,” vol. 15, pp. 535–546, 1953. View at Google Scholar
  6. A. Rapoport and L. I. Rebhun, “On the mathematical theory of rumor spread,” The Bulletin of Mathematical Biophysics, vol. 14, pp. 375–383, 1952. View at Google Scholar
  7. K. Dietz, “Epidemics and rumours: a survey,” Journal of the Royal Statistical Society A, vol. 130, no. 4, pp. 505–528, 1967. View at Publisher · View at Google Scholar
  8. D. J. Daley and J. Gani, Epidemic modelling: An Introduction, vol. 15 of Cambridge Studies in Mathematical Biology, Cambridge University Press, Cambridge, 1999. View at Publisher · View at Google Scholar
  9. L. M. A. Bettencourt, A. Cintrón-Arias, D. I. Kaiser, and C. Castillo-Chávez, “The power of a good idea: quantitative modeling of the spread of ideas from epidemiological models,” Physica A, vol. 364, pp. 513–536, 2006. View at Publisher · View at Google Scholar · View at Scopus
  10. D. J. Daley and D. G. Kendall, “Stochastic rumours,” Journal of the Institute of Mathematics and its Applications, vol. 1, pp. 42–55, 1965. View at Publisher · View at Google Scholar
  11. D. P. Maki and M. Thompson, Mathematical Models and Applications, with Emphasis on Social, Life, and Management Sciences, Prentice-Hall, Englewood Cliffs, NJ, USA, 1973.
  12. D. J. Daley, J. Gani et al., Epidemic Modelling, Cambridge University Press, Cambridge, UK, 2000.
  13. S. Belen, E. Kropa, and G. W. Weber, “Rumours within time dependent Maki-Thompson Model,” Working paper, 2008. View at Google Scholar
  14. W. Huang, “On Rumour Spreading with Skepticism and Denia,” Working paper, 2011. View at Google Scholar
  15. C. Lefèvre and P. Picard, “Distribution of the final extent of a rumour process,” Journal of Applied Probability, vol. 31, no. 1, pp. 244–249, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. B. Pittel, “On a Daley-Kendall model of random rumours,” Journal of Applied Probability, vol. 27, no. 1, pp. 14–27, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. K. Thompson, R. Castro Estrada, D. Daugherty, and A. Cintrn Arias, “A deterministic approach to the spread of rumors,” Working paper, Washington, DC, USA, 2003. View at Google Scholar
  18. K. Kawachi, “Deterministic models for rumor transmission,” Nonlinear Analysis, vol. 9, no. 5, pp. 1989–2028, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. 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, vol. 253, no. 1, pp. 55–60, 2008. View at Publisher · View at Google Scholar · View at Scopus
  20. E. Lebensztayn and F. P. Machado, “Limit theorems for a general stochastic rumour model,” ARXIV, 2010. View at Google Scholar
  21. C. E. M. Pearce, “The exact solution of the general stochastic rumour,” Mathematical and Computer Modelling, vol. 31, no. 10–12, pp. 289–298, 2000, Stochastic models in engineering, technology, and management (Gold Coast, 1996). View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. J. Gani, “The Maki-Thompson rumour model: a detailed analysis,” Environmental Modelling and Software, vol. 15, no. 8, pp. 721–725, 2000. View at Publisher · View at Google Scholar · View at Scopus
  23. R. E. Dickinson and C. E. M. Pearce, “Rumours, epidemics, and processes of mass action: synthesis and analysis,” Mathematical and Computer Modelling, vol. 38, no. 11–13, pp. 1157–1167, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. D. H. Zanette, “Dynamics of rumor propagation on small-world networks,” Physical Review E, vol. 65, no. 4, Article ID 041908, 2002. View at Publisher · View at Google Scholar · View at Scopus
  25. M. Nekovee, Y. Moreno, G. Bianconi, and M. Marsili, “Theory of rumour spreading in complex social networks,” Physica A, vol. 374, no. 1, pp. 457–470, 2007. View at Publisher · View at Google Scholar · View at Scopus
  26. V. Isham, S. Harden, and M. Nekovee, “Stochastic epidemics and rumours on finite random networks,” Physica A, vol. 389, no. 3, pp. 561–576, 2010. View at Publisher · View at Google Scholar · View at Scopus
  27. A. Sudbury, “The proportion of the population never hearing a rumour,” Journal of Applied Probability, vol. 22, no. 2, pp. 443–446, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. D. H. Zanette, “Critical behavior of propagation on small-world networks,” Physical Review E, vol. 64, no. 5, Article ID 050901, 2001. View at Google Scholar · View at Scopus
  29. P. Berenbrink, R. Elsässer, and T. Sauerwald, “Communication complexity of quasirandom rumor spreading,” in Algorithms—ESA 2010, M. de Berg, U. Meyer et al., Eds., pp. 134–145, Springer, Berlin, Germany, 2010. View at Publisher · View at Google Scholar
  30. T. Sauerwald, A. Stauery et al., “Rumor spreading and vertex expansion on regular graphs,” in Proceedings of the 22nd ACM-SIAM Symposium, pp. 462–475, San Francisco, Calif, USA, January 2011.
  31. L. Buzna, K. Peters, and D. Helbing, “Modelling the dynamics of disaster spreading in networks,” Physica A, vol. 363, no. 1, pp. 132–140, 2006. View at Publisher · View at Google Scholar · View at Scopus
  32. Z. Liu, Y. C. Lai, and N. Ye, “Propagation and immunization of infection on general networks with both homogeneous and heterogeneous components,” Physical Review E, vol. 67, no. 3, Article ID 031911, 2003. View at Google Scholar · View at Scopus
  33. J. Zhou, Z. Liu, and B. Li, “Influence of network structure on rumor propagation,” Physics Letters A, vol. 368, no. 6, pp. 458–463, 2007. View at Publisher · View at Google Scholar · View at Scopus
  34. Y. Moreno, M. Nekovee, and A. F. Pacheco, “Dynamics of rumor spreading in complex networks,” Physical Review E, vol. 69, no. 6, Article ID 066130, 2004. View at Publisher · View at Google Scholar · View at Scopus
  35. L. Zhao, Q. Wang, J. Cheng, Y. Chen, J. Wang, and W. Huang, “Rumor spreading model with consideration of forgetting mechanism: a case of online blogging LiveJournal,” Physica A, vol. 390, no. 13, pp. 2619–2625, 2011. View at Publisher · View at Google Scholar
  36. A. Cintron-Arias, Modeling and parameter estimation of contact processes, Ph.D. thesis, Cornell University, 2006.
  37. K. Mischaikow, H. Smith, and H. R. Thieme, “Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions,” Transactions of the American Mathematical Society, vol. 347, no. 5, pp. 1669–1685, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  38. C. Castillo-Chavez and B. Song, “Dynamical models of tuberculosis and their applications,” Mathematical Biosciences and Engineering, vol. 1, no. 2, pp. 361–404, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  39. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, vol. 42, Springer, Berlin, Germany, 1983.
  40. C. Castillo-Chavez, Z. Feng, and W. Huang, Mathematical Approaches for Emerging and Re-Emerging Infectious Diseases: An Introduction, vol. 125, Springer, New York, NY, USA, 2002. View at Zentralblatt MATH
  41. M. Y. Li and J. S. Muldowney, “A geometric approach to global-stability problems,” SIAM Journal on Mathematical Analysis, vol. 27, no. 4, pp. 1070–1083, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH