Skip to main content
Log in

Sudden spreading of infections in an epidemic model with a finite seed fraction

  • Regular Article
  • Published:
The European Physical Journal B Aims and scope Submit manuscript

Abstract

We study a simple case of the susceptible-weakened-infected-removed model in regular random graphs in a situation where an epidemic starts from a finite fraction of initially infected nodes (seeds). Previous studies have shown that, assuming a single seed, this model exhibits a kind of discontinuous transition at a certain value of infection rate. Performing Monte Carlo simulations and evaluating approximate master equations, we find that the present model has two critical infection rates for the case with a finite seed fraction. At the first critical rate the system shows a percolation transition of clusters composed of removed nodes, and at the second critical rate, which is larger than the first one, a giant cluster suddenly grows and the order parameter jumps even though it has been already rising. Numerical evaluation of the master equations shows that such sudden epidemic spreading does occur if the degree of the underlying network is large and the seed fraction is small.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. R. Albert, A.-L. Barabási, Rev. Mod. Phys. 74, 47 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  2. M.E.J. Newman, SIAM Rev. 45, 167 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  3. A. Barrat, M. Barthélemy, A. Vespignani, Dynamical processes on complex networks (Cambridge University Press, Cambridge, 2008)

  4. S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, Rev. Mod. Phys. 80, 1275 (2008)

    Article  ADS  Google Scholar 

  5. R. Pastor-Satorras, A. Vespignani, Phys. Rev. Lett. 86, 3200 (2001)

    Article  ADS  Google Scholar 

  6. Y. Moreno, R. Pastor-Satorras, A. Vespignani, Eur. Phys. J. B 26, 521 (2002)

    ADS  Google Scholar 

  7. W.O. Kermack, A.G. McKendrick, Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci. 115, 700 (1927)

    Article  ADS  Google Scholar 

  8. R.M. Anderson, R.M. May, Infectious diseases of humans: dynamics and control (Oxford University Press, New York, 1992)

  9. R. Pastor-Satorras, C. Castellano, P. Van Mieghem, A. Vespignani, Rev. Mod. Phys. 87, 925 (2015)

    Article  ADS  Google Scholar 

  10. J.C. Miller, A.C. Slim, E.M. Volz, J. R. Soc. Interface 9, 890 (2012)

    Article  Google Scholar 

  11. J.P. Gleeson, Phys. Rev. Lett. 107, 068701 (2011)

    Article  ADS  Google Scholar 

  12. J. Lindquist, J. Ma, P. Van den Driessche, F.H. Willeboordse, J. Math. Biol. 62, 143 (2011)

    Article  MathSciNet  Google Scholar 

  13. J.P. Gleeson, Phys. Rev. X 3, 021004 (2013)

    Google Scholar 

  14. A.V. Goltsev, S.N. Dorogovtsev, J.G. Oliveira, J.F.F. Mendes, Phys. Rev. Lett. 109, 128702 (2012)

    Article  ADS  Google Scholar 

  15. G. Ódor, Phys. Rev. E 90, 032110 (2014)

    Article  ADS  Google Scholar 

  16. P. Moretti, M.A. Muñoz, Nat. Commun. 4 (2013)

  17. G. Ódor, R. Dickman, G. Ódor, Sci. Rep. 5 (2015)

  18. W. Cota, S.C. Ferreira, G. Ódor, Phys. Rev. E 93, 032322 (2016)

    Article  ADS  Google Scholar 

  19. C. Castellano, S. Fortunato, V. Loreto, Rev. Mod. Phys. 81, 591 (2009)

    Article  ADS  Google Scholar 

  20. P.S. Dodds, D.J. Watts, Phys. Rev. Lett. 92, 218701 (2004)

    Article  ADS  Google Scholar 

  21. P.S. Dodds, D.J. Watts, J. Theor. Biol. 232, 587 (2005)

    Article  Google Scholar 

  22. D. Centola, V.M. Eguíluz, M.W. Macy, Physica A 374, 449 (2007)

    Article  ADS  Google Scholar 

  23. D. Centola, Science 329, 1194 (2010)

    Article  ADS  Google Scholar 

  24. P.L. Krapivsky, S. Redner, D. Volovik, J. Stat. Mech. Theory Exp. 2011, P12003 (2011)

    Article  Google Scholar 

  25. M. Zheng, L. Lü, M. Zhao, Phys. Rev. E 88, 012818 (2013)

    Article  ADS  Google Scholar 

  26. E. Campbell, M. Salathé, Sci. Rep. 3 (2013)

  27. S. Melnik, J.A. Ward, J.P. Gleeson, M.A. Porter, Chaos 23, 013124 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  28. T. Hasegawa, K. Nemoto, J. Stat. Mech. Theory Exp. 2014, P11024 (2014)

    Article  Google Scholar 

  29. W. Wang, M. Tang, H.-F. Zhang, Y.-C. Lai, Phys. Rev. E 92, 012820 (2015)

    Article  ADS  Google Scholar 

  30. D.J. O’Sullivan, G.J. O’Keeffe, P.G. Fennell, J.P. Gleeson, Front. Phys. 3, 71 (2015)

    Google Scholar 

  31. J.C. Miller, J. Complex Netw. 4, 201 (2016)

    Article  MathSciNet  Google Scholar 

  32. W. Wang, M. Tang, P. Shu, Z. Wang, New J. Phys. 18, 013029 (2016)

    Article  ADS  Google Scholar 

  33. D. Lee, W. Choi, J. Kertész, B. Kahng, Sci. Rep. 7, 5723 (2017)

    Article  ADS  Google Scholar 

  34. W. Choi, D. Lee, B. Kahng, Phys. Rev. E 95, 022304 (2017)

    Article  ADS  Google Scholar 

  35. H.-K. Janssen, M. Müller, O. Stenull, Phys. Rev. E 70, 026114 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  36. G. Bizhani, M. Paczuski, P. Grassberger, Phys. Rev. E 86, 011128 (2012)

    Article  ADS  Google Scholar 

  37. K. Chung, Y. Baek, D. Kim, M. Ha, H. Jeong, Phys. Rev. E 89, 052811 (2014)

    Article  ADS  Google Scholar 

  38. H.-K. Janssen, O. Stenull, Europhys. Lett. 113, 26005 (2016)

    Article  ADS  Google Scholar 

  39. G.J. Baxter, S.N. Dorogovtsev, A.V. Goltsev, J.F. Mendes, Phys. Rev. E 83, 051134 (2011)

    Article  ADS  Google Scholar 

  40. J.C. Miller, PLoS ONE 9, e101421 (2014)

    Article  ADS  Google Scholar 

  41. Z.-L. Hu, J.-G. Liu, G.-Y. Yang, Z.-M. Ren, Europhys. Lett. 106, 18002 (2014)

    Article  ADS  Google Scholar 

  42. S. Ji, L. Lu, C.H. Yeung, Y. Hu, arXiv:1508.04294 (2015)

  43. T. Hasegawa, K. Nemoto, Phys. Rev. E 93, 032324 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  44. T. Tomé, R.M. Ziff, Phys. Rev. E 82, 051921 (2010)

    Article  ADS  Google Scholar 

  45. D.R. de Souza, T. Tomé, R.M. Ziff, J. Stat. Mech. Theory Exp. 2011, P03006 (2011)

    Article  Google Scholar 

  46. D. Stauffer, A. Aharony, Introduction to percolation theory (Taylor and Francis, London, 1994)

  47. T. Hasegawa, T. Nogawa, K. Nemoto, Europhys. Lett. 104, 16006 (2013)

    Article  ADS  Google Scholar 

  48. T. Hasegawa, T. Nogawa, K. Nemoto, Discontin. Nonlinearity Complex. 3, 319 (2014)

    Article  Google Scholar 

  49. W. Choi, D. Lee, B. Kahng, Phys. Rev. E 95, 062115 (2017)

    Article  ADS  Google Scholar 

  50. L. Chen, F. Ghanbarnejad, W. Cai, P. Grassberger, Europhys. Lett. 104, 50001 (2013)

    Article  ADS  Google Scholar 

  51. W. Cai, L. Chen, F. Ghanbarnejad, P. Grassberger, Nat. Phys. 11, 936 (2015)

    Article  Google Scholar 

  52. L. Hébert-Dufresne, B.M. Althouse, Proc. Natl. Acad. Sci. USA 112, 10551 (2015)

    Article  MathSciNet  Google Scholar 

  53. P. Grassberger, L. Chen, F. Ghanbarnejad, W. Cai, Phys. Rev. E 93, 042316 (2016)

    Article  ADS  Google Scholar 

  54. N. Azimi-Tafreshi, Phys. Rev. E 93, 042303 (2016)

    Article  ADS  Google Scholar 

  55. R. Juhász, G. Ódor, C. Castellano, M.A. Muñoz, Phys. Rev. E 85, 066125 (2012)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Takehisa Hasegawa.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hasegawa, T., Nemoto, K. Sudden spreading of infections in an epidemic model with a finite seed fraction. Eur. Phys. J. B 91, 58 (2018). https://doi.org/10.1140/epjb/e2018-80343-3

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1140/epjb/e2018-80343-3

Keywords

Navigation