Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Critical regimes driven by recurrent mobility patterns of reaction–diffusion processes in networks


Reaction–diffusion processes1 have been widely used to study dynamical processes in epidemics2,3,4 and ecology5 in networked metapopulations. In the context of epidemics6, reaction processes are understood as contagions within each subpopulation (patch), while diffusion represents the mobility of individuals between patches. Recently, the characteristics of human mobility7, such as its recurrent nature, have been proven crucial to understand the phase transition to endemic epidemic states8,9. Here, by developing a framework able to cope with the elementary epidemic processes, the spatial distribution of populations and the commuting mobility patterns, we discover three different critical regimes of the epidemic incidence as a function of these parameters. Interestingly, we reveal a regime of the reaction–diffussion process in which, counter-intuitively, mobility is detrimental to the spread of disease. We analytically determine the precise conditions for the emergence of any of the three possible critical regimes in real and synthetic networks.

This is a preview of subscription content, access via your institution

Relevant articles

Open Access articles citing this article.

Access options

Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Fig. 1: Criticality regimes in the spread of an epidemic.
Fig. 2: Prediction of epidemics in the city of Cali (Colombia) from empirical mobility flows.


  1. Marro, J. & Dickman, R. Nonequilibrium Phase Transitions in Lattice Models (Cambridge Univ. Press, Cambridge, 2005).

    MATH  Google Scholar 

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

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

  4. 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 

  5. Hanski, I. Metapopulation dynamics. Nature 396, 41–50 (1998).

    Article  ADS  Google Scholar 

  6. Pastor-Satorras, R., Castellano, C., Van Mieghem, P. & Vespignani, A. Epidemic processes in complex networks. Rev. Mod. Phys. 87, 925–979 (2015).

    Article  ADS  MathSciNet  Google Scholar 

  7. Gonzalez, M. C., Hidalgo, C. A. & Barabasi, A.-L. Understanding individual human mobility patterns. Nature 453, 779–782 (2008).

    Article  ADS  Google Scholar 

  8. 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 

  9. Balcan, D. & Vespignani, A. Phase transitions in contagion processes mediated by recurrent mobility patterns. Nat. Phys. 7, 581–586 (2011).

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  11. Grenfell, B. & Harwood, J. (Meta)population dynamics of infectious diseases. Trends Ecol. Evol. 12, 395–399 (1997).

    Article  Google Scholar 

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

    MATH  Google Scholar 

  13. Hanski, I. & Gaggiotti, O. E. Ecology, Genetics, and Evolution of Metapopulations (Princeton Univ. Press, Princeton, NJ, 2004).

    Google Scholar 

  14. Bajardi, P., Barrat, A., Natale, F., Savini, L. & Colizza, V. Dynamical patterns of cattle trade movements. PLoS ONE 6, e19869 (2011).

    Article  ADS  Google Scholar 

  15. Ball, F. et al. Seven challenges for metapopulation models of epidemics, including households models. Epidemics 10, 63–67 (2015).

    Article  Google Scholar 

  16. Funk, S. et al. Nine challenges in incorporating the dynamics of behaviour in infectious diseases models. Epidemics 10, 21–25 (2015).

    Article  Google Scholar 

  17. Eubank, S. et al. Modelling disease outbreaks in realistic urban social networks. Nature 429, 180–184 (2004).

    Article  ADS  Google Scholar 

  18. Ferguson, N. M. et al. Strategies for mitigating an influenza pandemic. Nature 442, 448–452 (2006).

    Article  ADS  Google Scholar 

  19. Van den Broeck, W. et al. The GLEaMviz computational tool, a publicly available software to explore realistic epidemic spreading scenarios at the global scale. BMC Infect. Dis. 11, 37 (2011).

    Article  Google Scholar 

  20. Lofgren, E. T. et al. Opinion: mathematical models: a key tool for outbreak response. Proc. Natl Acad. Sci. USA 111, 18095–18096 (2014).

    Article  ADS  Google Scholar 

  21. Scarpino, S. V., Allard, A. & Hebert-Dufresne, L. The effect of a prudent adaptive behaviour on disease transmission. Nat. Phys. 12, 1042–1046 (2016).

    Article  Google Scholar 

  22. Marcus, R. A. Brief comments on perturbation theory of a nonsymmetric matrix: the GF matrix. J. Phys. Chem. A 105, 2612–2616 (2001).

    Article  Google Scholar 

  23. Estrada, E., Meloni, S., Sheerin, M. & Moreno, Y. Epidemic spreading in random rectangular networks. Phys. Rev. E 94, 052316 (2016).

    Article  ADS  Google Scholar 

  24. Gross, T., D’Lima, C. J. D. & Blasius, B. Epidemic dynamics on an adaptive network. Phys. Rev. Lett. 96, 208701 (2006).

    Article  ADS  Google Scholar 

  25. Schwarzkopf, Y., Rákos, A. & Mukamel, D. Epidemic spreading in evolving networks. Phys. Rev. E 82, 036112 (2010).

    Article  ADS  MathSciNet  Google Scholar 

  26. Shaw, L. B. & Schwartz, I. B. Fluctuating epidemics on adaptive networks. Phys. Rev. E 77, 066101 (2008).

    Article  ADS  MathSciNet  Google Scholar 

  27. Barton, N. H. The probability of fixation of a favoured allele in a subdivided population. Genet. Res. 62, 149–157 (1993).

    Article  Google Scholar 

  28. Whitlock, M. C. Fixation probability and time in subdivided populations. Genetics 164, 767–779 (2003).

    Google Scholar 

  29. Colizza, V., Barrat, A., Barthelemy, M., Valleron, A.-J. & Vespignani, A. Modeling the worldwide spread of pandemic influenza: Baseline case and containment interventions. PLoS Med. 4, 0095–0110 (2007).

    Article  Google Scholar 

  30. Kitsak, M. et al. Identification of influential spreaders in complex networks. Nat. Phys. 6, 888–893 (2010).

    Article  Google Scholar 

  31. Escobar-Morales, G. Cali en Cifras 2013 (Departamento Administrativo de Planeación, Santiago de Cali, 2013).

Download references


We acknowledge S. Meloni for the very many discussions and useful suggestions. J.G.-G. and A.A. acknowledge financial support from MINECO (projects FIS2015-71582-C2 and FIS2014-55867-P) and from the Departamento de Industria e Innovación del Gobierno de Aragón y Fondo Social Europeo (FENOL group E-19). A.A. acknowledges also financial support from the ICREA Academia, the James S. McDonnell Foundation.

Author information

Authors and Affiliations



All of the authors wrote the paper and contributed equally to the production of the manuscript.

Corresponding authors

Correspondence to J. Gómez-Gardeñes or A. Arenas.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Additional information

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary notes 1–8, Supplementary figures 1–7, Supplementary Notes

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Gómez-Gardeñes, J., Soriano-Paños, D. & Arenas, A. Critical regimes driven by recurrent mobility patterns of reaction–diffusion processes in networks. Nature Phys 14, 391–395 (2018).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing