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Phase transitions in contagion processes mediated by recurrent mobility patterns

Nature Physics volume 7, pages 581–586 (2011)Cite this article

Abstract

Human mobility and activity patterns mediate contagion on many levels, including the spatial spread of infectious diseases, diffusion of rumours, and emergence of consensus. These patterns however are often dominated by specific locations and recurrent flows and poorly modelled by the random diffusive dynamics generally used to study them. Here we develop a theoretical framework to analyse contagion within a network of locations where individuals recall their geographic origins. We find a phase transition between a regime in which the contagion affects a large fraction of the system and one in which only a small fraction is affected. This transition cannot be uncovered by continuous deterministic models because of the stochastic features of the contagion process and defines an invasion threshold that depends on mobility parameters, providing guidance for controlling contagion spread by constraining mobility processes. We recover the threshold behaviour by analysing diffusion processes mediated by real human commuting data.

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Figure 1: Statistical properties of commuting networks in the United States and France.
Figure 2: Illustration of the subpopulation invasion dynamics.
Figure 3: Phase diagrams separating the global invasion regime from the extinction regime.
Figure 4: Dynamical behaviour of an SIR epidemic on the real US commuting network data.

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References

  1. Marro, J. & Dickman, R. Nonequilibrium Phase Transitions in Lattice Models (Cambridge Univ. Press, 1999).

    Book  Google Scholar 

  2. van Kampen, N. G. Stochastic Processes in Physics and Chemistry (North-Holland, 1981).

    MATH  Google Scholar 

  3. May, R. M. & Anderson, R. M. Spatial heterogeneity and the design of immunization programs. Math. Biosci. 72, 83–111 (1984).

    Article  MathSciNet  Google Scholar 

  4. Bolker, B. M. & Grenfell, T. Chaos and biological complexity in measles dynamics. Proc. R. Soc. Lond. B 251, 75–81 (1993).

    Article  ADS  Google Scholar 

  5. Bolker, B. M. & Grenfell, T. Space persistence and dynamics of measles epidemics. Phil. Trans. R. Soc. Lond. B 348, 309–320 (1995).

    Article  ADS  Google Scholar 

  6. Sattenspiel, L. & Dietz, K. A structured epidemic model incorporating geographic mobility among regions. Math. Biosci. 128, 71–91 (1995).

    Article  Google Scholar 

  7. Lloyd, A. L. & May, R. M. Spatial heterogeneity in epidemic models. J. Theor. Biol. 179, 1–11 (1996).

    Article  Google Scholar 

  8. Keeling, M. J. & Rohani, P. Estimating spatial coupling in epidemiological systems: A mechanistic approach. Ecol. Lett. 5, 20–29 (2002).

    Article  Google Scholar 

  9. Watts, D., Muhamad, R., Medina, D. C. & Dodds, P. S. Multiscale resurgent epidemics in a hierarchical metapopulation model. Proc. Natl Acad. Sci. USA 102, 11157–11162 (2005).

    Article  ADS  Google Scholar 

  10. Rapoport, A. Spread of information through a population with socio-structural bias: I. Assumption of transitivity. Bull. Math. Biol. 15, 523–533 (1953).

    MathSciNet  Google Scholar 

  11. Goffman, W. & Newill, V. A. Generalization of epidemic theory: An application to the transmission of ideas. Nature 204, 225–228 (1964).

    Article  ADS  Google Scholar 

  12. Goffman, W. Mathematical approach to the spread of scientific ideas—the history of mast cell research. Nature 212, 449–452 (1966).

    Article  ADS  Google Scholar 

  13. Dietz, K. Epidemics and rumours: A survey. J. R. Stat. Soc. A 130, 505–528 (1967).

    Article  MathSciNet  Google Scholar 

  14. Tabah, A. N. Literature dynamics: Studies on growth, diffusion, and epidemics. Annu. Rev. Inform. Sci. Technol. 34, 249–286 (1999).

    Google Scholar 

  15. Daley, D. J. & Gani, J. Epidemic Modeling: An Introduction (Cambridge Univ. Press, 2000).

    MATH  Google Scholar 

  16. Rvachev, L. A. & Longini, I. M. A mathematical model for the global spread of influenza. Math. Biosci. 75, 3–22 (1985).

    Article  MathSciNet  Google Scholar 

  17. Grais, R. F., Hugh Ellis, J. & Glass, G. E. Assessing the impact of airline travel on the geographic spread of pandemic influenza. Eur. J. Epidemiol. 18, 1065–1072 (2003).

    Article  Google Scholar 

  18. Hufnagel, L., Brockmann, D. & Geisel, T. Forecast and control of epidemics in a globalized world. Proc. Natl Acad. Sci. USA 101, 15124–15129 (2004).

    Article  ADS  Google Scholar 

  19. Colizza, V., Barrat, A., Barthélemy, M. & Vespignani, A. The role of the airline transportation network in the prediction and predictability of global epidemics. Proc. Natl Acad. Sci. USA 103, 2015–2020 (2006).

    Article  ADS  Google Scholar 

  20. Balcan, D. et al. Modeling the spatial spread of infectious diseases: The GLobal Epidemic and Mobility computational model. J. Comput. Sci. 1, 132–145 (2010).

    Article  Google Scholar 

  21. Chowell, G., Hyman, J. M., Eubank, S. & Castillo-Chavez, C. Scaling laws for the movement of people between locations in a large city. Phys. Rev. E 68, 066102 (2003).

    Article  ADS  Google Scholar 

  22. Barrat, A., Barthélemy, M., Pastor-Satorras, R. & Vespignani, A. The architecture of complex weighted networks. Proc. Natl Acad. Sci. USA 101, 3747–3752 (2004).

    Article  ADS  Google Scholar 

  23. Guimerá, R., Mossa, S., Turtschi, A. & Amaral, L. A. N. The worldwide air transportation network: Anomalous centrality, community structure, and cities’ global roles. Proc. Natl Acad. Sci. USA 102, 7794–7799 (2005).

    Article  ADS  MathSciNet  Google Scholar 

  24. Brockmann, D., Hufnagel, L. & Geisel, T. The scaling laws of human travel. Nature 439, 462–465 (2006).

    Article  ADS  Google Scholar 

  25. Patuelli, R., Reggiani, R., Gorman, S. P., Nijkamp, P. & Bade, F-J. Network analysis of commuting flows: A comparative static approach to German data. Netw. Spatial Econ. 7, 315–331 (2007).

    Article  Google Scholar 

  26. González, M. C., Hidalgo, C. A. & Barabási, A-L. Understanding individual human mobility patterns. Nature 453, 779–782 (2008).

    Article  ADS  Google Scholar 

  27. Wang, P. & González, M. C. Understanding spatial connectivity of individuals with non-uniform population density. Phil. Trans. R. Soc. A 367, 3321–3329 (2009).

    Article  ADS  Google Scholar 

  28. Song, C., Qu, Z., Blumm, N. & Barabási, A-L. Limits of predictability in human mobility. Science 327, 1018–1021 (2010).

    Article  ADS  MathSciNet  Google Scholar 

  29. Song, C., Koren, T., Wang, P. & Barabási, A-L. Modelling the scaling properties of human mobility. Nature Phys. 6, 818–823 (2010).

    Article  ADS  Google Scholar 

  30. Balcan, D. et al. Multiscale mobility networks and the spatial spreading of infectious diseases. Proc. Natl Acad. Sci. USA 106, 21484–21489 (2009).

    Article  ADS  Google Scholar 

  31. Colizza, V., Pastor-Satorras, R. & Vespignani, A. Reaction-diffusion processes and metapopulation models in heterogeneous networks. Nature Phys. 3, 276–282 (2007).

    Article  ADS  Google Scholar 

  32. Colizza, V. & Vespignani, A. Invasion threshold in heterogeneous metapopulation networks. Phys. Rev. Lett. 99, 148701 (2007).

    Article  ADS  Google Scholar 

  33. Colizza, V. & Vespignani, A. Epidemic modelling in metapopulation systems with heterogeneous coupling pattern: Theory and simulations. J. Theor. Biol. 251, 450–467 (2008).

    Article  Google Scholar 

  34. Barthélemy, M., Godrèche, C. & Luck, J-M. Fluctuation effects in metapopulation models: percolation and pandemic threshold. J. Theor. Biol. 267, 554–564 (2010).

    Article  MathSciNet  Google Scholar 

  35. Ni, S. & Weng, W. Impact of travel patterns on epidemic dynamics in heterogeneous spatial metapopulation networks. Phys. Rev. E 79, 016111 (2009).

    Article  ADS  Google Scholar 

  36. Ben-Zion, Y., Cohena, Y. & Shnerba, N.M. Modeling epidemics dynamics on heterogenous networks. J. Theor. Biol. 264, 197–204 (2010).

    Article  MathSciNet  Google Scholar 

  37. Pastor-Satorras, R. & Vespignani, A. Epidemic spreading in scale-free networks. Phys. Rev. Lett. 86, 3200–3203 (2001).

    Article  ADS  Google Scholar 

  38. Lloyd, A. L. & May, R. M. How viruses spread among computers and people. Science 292, 1316–1317 (2001).

    Article  Google Scholar 

  39. Cohen, R., Havlin, S. & ben-Avraham, D. Efficient immunization strategies for computer networks and populations. Phys. Rev. Lett. 91, 247901 (2003).

    Article  ADS  Google Scholar 

  40. Barrat, A., Barthélemy, M. & Vespignani, A. Dynamical Processes on Complex Networks (Cambridge Univ. Press, 2008).

    Book  Google Scholar 

  41. Keeling, M. J. & Rohani, P. Modeling Infectious Diseases in Humans and Animals (Princeton Univ. Press, 2008).

    MATH  Google Scholar 

  42. Ball, F., Mollison, D. & Scalia-Tomba, G. Epidemics with two levels of mixing. Ann. Appl. Probab. 7, 46–89 (1997).

    Article  MathSciNet  Google Scholar 

  43. Cross, P., Lloyd-Smith, J. O., Johnson, P. L. F. & Wayne, M. G. Duelling timescales of host movement and disease recovery determine invasion of disease in structured populations. Ecol. Lett. 8, 587–595 (2005).

    Article  Google Scholar 

  44. Cross, P., Johnson, P. L. F., Lloyd-Smith, J. O. & Wayne, M. G. Utility of R0 as a predictor of disease invasion in structured populations. J. R. Soc. Interface 4, 315–324 (2007).

    Article  Google Scholar 

  45. Harris, T. E. The Theory of Branching Processes (Dover, 1989).

    Google Scholar 

  46. Vázquez, A. Polynomial growth in age-dependent branching processes with diverging reproductive number. Phys. Rev. Lett. 96, 038702 (2006).

    Article  ADS  Google Scholar 

  47. Molloy, M. & Reed, B. The size of the largest component of a random graph on a fixed degree sequence. Comb. Probab. Comput. 7, 295–306 (1998).

    Article  Google Scholar 

  48. Catanzaro, M., Bogunã, M. & Pastor-Satorras, R. Generation of uncorrelated random scale-free networks. Phys. Rev. E 71, 027103 (2005).

    Article  ADS  Google Scholar 

  49. US Census Bureau http://www.census.gov/.

  50. Bailey, N. T. The Mathematical Theory of Infectious Diseases (Macmillan, 1975).

    MATH  Google Scholar 

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Acknowledgements

We would like to thank to C. Poletto and V. Colizza for interesting discussions during the preparation of this manuscript. This work was partially funded by the NIH R21-DA024259 award and the DTRA-1-0910039 award to A.V.; the work was also partly sponsored by the Army Research Laboratory and was accomplished under Cooperative Agreement Number W911NF-09-2-0053. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the US Government.

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Authors and Affiliations

  1. Center for Complex Networks and Systems Research (CNetS), School of Informatics and Computing, Indiana University, Bloomington, Indiana 47408, USA

    Duygu Balcan & Alessandro Vespignani

  2. Pervasive Technology Institute, Indiana University, Bloomington, Indiana 47406, USA

    Duygu Balcan & Alessandro Vespignani

  3. Institute for Scientific Interchange (ISI), Torino 10133, Italy

    Alessandro Vespignani

Authors
  1. Duygu Balcan
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Contributions

D.B. and A.V. conceived and executed the study, performed the analytical calculations and drafted the manuscript. D.B. performed the numerical simulations.

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Correspondence to Alessandro Vespignani.

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Balcan, D., Vespignani, A. Phase transitions in contagion processes mediated by recurrent mobility patterns. Nature Phys 7, 581–586 (2011). https://doi.org/10.1038/nphys1944

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