Skip to main content

Advertisement

SpringerLink
Log in

Infection propagator approach to compute epidemic thresholds on temporal networks: impact of immunity and of limited temporal resolution

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

Abstract

The epidemic threshold of a spreading process indicates the condition for the occurrence of the wide spreading regime, thus representing a predictor of the network vulnerability to the epidemic. Such threshold depends on the natural history of the disease and on the pattern of contacts of the network with its time variation. Based on the theoretical framework introduced in [E. Valdano, L. Ferreri, C. Poletto, V. Colizza, Phys. Rev. X 5, 21005 (2015)] for a susceptible-infectious-susceptible model, we formulate here an infection propagator approach to compute the epidemic threshold accounting for more realistic effects regarding a varying force of infection per contact, the presence of immunity, and a limited time resolution of the temporal network. We apply the approach to two temporal network models and an empirical dataset of school contacts. We find that permanent or temporary immunity do not affect the estimation of the epidemic threshold through the infection propagator approach. Comparisons with numerical results show the good agreement of the analytical predictions. Aggregating the temporal network rapidly deteriorates the predictions, except for slow diseases once the heterogeneity of the links is preserved. Weight-topology correlations are found to be the critical factor to be preserved to improve accuracy in the prediction.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R.M. Anderson, R.M. May, Infectious Diseases of Humans Dynamics and Control (Oxford University Press, 1992)

  2. M.J. Keeling, P. Rohani, Modeling Infectious Diseases in Humans and Animals (Princeton University Press, 2008)

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

    Article  ADS  Google Scholar 

  4. Y. Wang, D. Chakrabarti, C. Wang, C. Faloutsos, Epidemic spreading in real networks: an eigenvalue viewpoint, in Proceedings of 22nd International Symposium on Reliable Distributed Systems, 2003 (2003), pp. 25–34

  5. S. Gómez, A. Arenas, J. Borge-Holthoefer, S. Meloni, Y. Moreno, Europhys. Lett. 89, 38009 (2010)

    Article  ADS  Google Scholar 

  6. R. Cohen, K. Erez, D. ben Avraham, S. Havlin, Phys. Rev. Lett. 85, 4626 (2000)

    Article  ADS  Google Scholar 

  7. M.E.J. Newman, Phys. Rev. E 66, 016128 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  8. M. Boguñá, C. Castellano, R. Pastor-Satorras, Phys. Rev. Lett. 111, 068701 (2013)

    Article  ADS  Google Scholar 

  9. C. Castellano, R. Pastor-Satorras, Phys. Rev. Lett. 105, 218701 (2010)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  11. A. Barrat, M. Barthelemy, A. Vespignani, Dynamical Processes on Complex Networks (Cambridge University Press, 2008)

  12. S. Bansal, J. Read, B. Pourbohloul, L.A. Meyers, J. Biol. Dyn. 4, 478 (2010)

    Article  MathSciNet  Google Scholar 

  13. L. Isella, J. Stehlé, A. Barrat, C. Cattuto, J.F. Pinton, W. Van den Broeck, J. Theor. Biol. 271, 166 (2011)

  14. G. Miritello, E. Moro, R. Lara, Phys. Rev. E 83, 045102 (2011)

    Article  ADS  Google Scholar 

  15. L.E.C. Rocha, F. Liljeros, P. Holme, Proc. Natl. Acad. Sci. USA 107, 5706 (2010)

    Article  ADS  MATH  Google Scholar 

  16. P. Bajardi, A. Barrat, F. Natale, L. Savini, V. Colizza, PloS One 6, e19869 (2011)

    Article  ADS  Google Scholar 

  17. C.T. Butts, Science 325, 414 (2009)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  18. M. Karsai, M. Kivelä, R.K. Pan, K. Kaski, J. Kertész, A.L. Barabási, J. Saramäki, Phys. Rev. E 83, 025102 (2011)

    Article  ADS  Google Scholar 

  19. P. Holme, J. Saramäki, Phys. Rep. 519, 97 (2012)

    Article  ADS  Google Scholar 

  20. L.E.C. Rocha, V.D. Blondel, PLoS Comput. Biol. 9, e1002974 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  21. L. Ferreri, P. Bajardi, M. Giacobini, S. Perazzo, E. Venturino, Phys. Rev. E 90, 012812 (2014)

    Article  ADS  Google Scholar 

  22. J.L. Iribarren, E. Moro, Phys. Rev. Lett. 103, 038702 (2009)

    Article  ADS  Google Scholar 

  23. A. Vazquez, B. Rácz, A. Lukács, A.L. Barabási, Phys. Rev. Lett. 98, 158702 (2007)

    Article  ADS  Google Scholar 

  24. K.T. Eames, M.J. Keeling, Math. Biosc. 189, 115 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  25. N. Perra, B. Gonçalves, R. Pastor-Satorras, A. Vespignani, Sci. Rep. 2, 469 (2012)

    Article  ADS  Google Scholar 

  26. E. Volz, L.A. Meyers, J. Roy. Soc. Int. 6, 233 (2009)

    Article  Google Scholar 

  27. T. Gross, C.J.D. D’Lima, B. Blasius, Phys. Rev. Lett. 96, 208701 (2006)

    Article  ADS  Google Scholar 

  28. M. Taylor, T.J. Taylor, I.Z. Kiss, Phys. Rev. E 85, 016103 (2012)

    Article  ADS  Google Scholar 

  29. Z. Zhao, J.P. Calderón, C. Xu, G. Zhao, D. Fenn, D. Sornette, R. Crane, P.M. Hui, N.F. Johnson, Phys. Rev. E 81, 56107 (2010)

    Article  ADS  Google Scholar 

  30. E. Valdano, L. Ferreri, C. Poletto, V. Colizza, Phys. Rev. X 5, 21005 (2015)

    Google Scholar 

  31. M. De Domenico, A. Solé-Ribalta, E. Cozzo, M. Kivelä, Y. Moreno, M.A. Porter, S. Gómez, A. Arenas, Phys. Rev. X 3, 041022 (2013)

    Google Scholar 

  32. M. Kivelä, A. Arenas, M. Barthelemy, J.P. Gleeson, Y. Moreno, M.A. Porter, J. Complex Networks 2, 203 (2014)

    Article  Google Scholar 

  33. P. Vanhems, A. Barrat, C. Cattuto, J.F. Pinton, N. Khanafer, C. Régis, B. Kim, B. Comte, N. Voirin, PLoS One 8, e73970 (2013)

    Article  ADS  Google Scholar 

  34. M. Konschake, H.H.K. Lentz, F.J. Conraths, P. Hövel, T. Selhorst, PLoS One 8, e55223 (2013)

    Article  ADS  Google Scholar 

  35. J. Fournet, A. Barrat, PLoS One 9, e107878 (2014)

    Article  ADS  Google Scholar 

  36. T. Obadia, R. Silhol, L. Opatowski, L. Temime, J. Legrand, A.C.M. Thiébaut, J.L. Herrmann, E. Fleury, D. Guillemot, P.Y. Boëlle, PLoS Comput. Biol. 11, e1004170 (2015)

    Article  ADS  Google Scholar 

  37. B. Ribeiro, N. Perra, A. Baronchelli, Sci. Rep. 3, 3006 (2013)

    Article  ADS  Google Scholar 

  38. P. Bajardi, A. Barrat, L. Savini, V. Colizza, J. Roy. Soc. Int. 9, 2814 (2012)

    Article  Google Scholar 

  39. A. Machens, F. Gesualdo, C. Rizzo, A. Tozzi, A. Barrat, C. Cattuto, BMC Infectious Diseases 13, 185 (2013)

    Article  Google Scholar 

  40. J.P. Gleeson, S. Melnik, J.A. Ward, M.A. Porter, P.J. Mucha, Phys. Rev. E 85, 026106 (2012)

    Article  ADS  Google Scholar 

  41. P. Van Mieghem, Computing 93, 147 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  42. P. Van Mieghem, R. van de Bovenkamp, Phys. Rev. Lett. 110, 108701 (2013)

    Article  ADS  Google Scholar 

  43. C. Li, H. Wang, P. Van Mieghem, Phys. Rev. E 88, 062802 (2013)

    Article  ADS  Google Scholar 

  44. K. Wehmuth, A. Ziviani, E. Fleury, arXiv:1402.3488 (2014)

  45. P.J. Mucha, T. Richardson, K. Macon, M.A. Porter, J.P. Onnela, Science 329, 277 (2010)

    MathSciNet  Google Scholar 

  46. I. Scholtes, N. Wider, R. Pfitzner, A. Garas, C.J. Tessone, F. Schweitzer, Nat. Commun. 5, 5024 (2014)

    Article  ADS  Google Scholar 

  47. E. Cozzo, R.A. Baños, S. Meloni, Y. Moreno, Phys. Rev. E 88, 050801 (2013)

    Article  ADS  Google Scholar 

  48. H. Wang, Q. Li, G. D’Agostino, S. Havlin, H.E. Stanley, P. Van Mieghem, Phys. Rev. E 88, 022801 (2013)

    Article  ADS  Google Scholar 

  49. C. Granell, S. Gómez, A. Arenas, Phys. Rev. Lett. 111, 128701 (2013)

    Article  ADS  Google Scholar 

  50. S. Elaydi, An Introduction to Difference Equations, 3rd edn. (Springer, New York, 2005)

  51. P.D. Powell, arXiv:1112.4379 (2011)

  52. H.K. Lentz, T. Selhorst, I.M. Sokolov, Phys. Rev. Lett. 110, 118701 (2013)

    Article  ADS  Google Scholar 

  53. P. Van Mieghem, F. Sahneh, C. Scoglio, An upper bound for the epidemic threshold in exact Markovian SIR and SIS epidemics on networks, in IEEE 53rd Annual Conference on Decision and Control (CDC), 2014 (2014), pp. 6228–6233

  54. P. Shu, W. Wang, M. Tang, Y. Do, Chaos 25, 063104 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  55. M. Salathé, M. Kazandjieva, J.W. Lee, P. Levis, M.W. Feldman, J.H. Jones, Proc. Natl. Acad. Sci USA 107, 22020 (2010)

    Article  ADS  Google Scholar 

  56. S.C. Ferreira, C. Castellano, R. Pastor-Satorras, Phys. Rev. E 86, 041125 (2012)

    Article  ADS  Google Scholar 

  57. P. Crépey, F.P. Alvarez, M. Barthélemy, Phys. Rev. E 73, 046131 (2006)

    Article  ADS  Google Scholar 

  58. S.C. Ferreira, R.S. Ferreira, C. Castellano, R. Pastor-Satorras, Phys. Rev. E 84, 066102 (2011)

    Article  ADS  Google Scholar 

  59. P. Holme, PLoS Comput Biol 9, e1003142 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  60. J. Stehlé, N. Voirin, A. Barrat, C. Cattuto, V. Colizza, L. Isella et al., BMC Medicine 9, 87 (2011)

    Article  Google Scholar 

  61. P. Holme, Phys. Rev. E 71, 046119 (2005)

    Article  ADS  Google Scholar 

  62. R.K. Pan, J. Saramaki, Phys. Rev. E 84, 016105 (2011)

    Article  ADS  Google Scholar 

  63. M.S. Granovetter, Am. J. Sociol. 78, 1360 (1973)

    Article  Google Scholar 

  64. Y. Gang, Z. Tao, W. Jie, F. Zhong-Qian, W. Bing-Hong, Chinese Phys. Lett. 22, 510 (2005)

    Article  ADS  Google Scholar 

  65. J.P. Onnela, J. Saramäki, J. Hyvönen, G. Szabó, D. Lazer, K. Kaski, J. Kertész, A.L. Barabási, Proc. Natl. Acad. Sci. 104, 7332 (2007)

    Article  ADS  Google Scholar 

  66. R. Lambiotte, J.C. Delvenne, M. Barahona, arXiv:0812.1770 (2009)

  67. R. Toivonen, X. Castelló, V.M. Eguíluz, J. Saramäki, K. Kaski, M. San Miguel, Phys. Rev. E 79, 016109 (2009)

    Article  ADS  Google Scholar 

  68. K.T. Eames, J.M. Read, W.J. Edmunds, Epidemics 1, 70 (2009)

    Article  Google Scholar 

  69. M. Salathé, J.H. Jones, PLoS Comput. Biol. 6, e1000736 (2010)

    Article  ADS  Google Scholar 

  70. Z. Yang, T. Zhou, Phys. Rev. E 85, 056106 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  71. C. Kamp, M. Moslonka-Lefebvre, S. Alizon, PLoS Comput. Biol. 9, e1003352 (2013)

    Article  ADS  Google Scholar 

  72. M. Deijfen, Math. Biosc. 232, 57 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  73. T. Britton, M. Deijfen, F. Liljeros, J. Stat. Phys. 145, 1368 (2011)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  74. T. Britton, D. Lindenstrand, Math. Biosc. 240, 124 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  75. G. Miritello, R. Lara, M. Cebrian, E. Moro, Sci. Rep. 3, 1950 (2013)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Sorbonne Universités, UPMC Univ. Paris 06, INSERM, Institut Pierre Louis d’Épidémiologie et de Santé Publique (IPLESP, UMRS 1136), 75012, Paris, France

    Eugenio Valdano, Chiara Poletto & Vittoria Colizza

  2. ISI Foundation, 10126, Torino, Italy

    Vittoria Colizza

Authors
  1. Eugenio Valdano
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chiara Poletto
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Vittoria Colizza
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Valdano, E., Poletto, C. & Colizza, V. Infection propagator approach to compute epidemic thresholds on temporal networks: impact of immunity and of limited temporal resolution. Eur. Phys. J. B 88, 341 (2015). https://doi.org/10.1140/epjb/e2015-60620-5

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1140/epjb/e2015-60620-5

Keywords

Search

Navigation