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Avalanche outbreaks emerging in cooperative contagions

Nature Physics volume 11, pages 936–940 (2015)Cite this article

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Abstract

The spreading of contagions can exhibit a percolation transition, which separates transitory prevalence from outbreaks that reach a finite fraction of the population1,2. Such transitions are commonly believed to be continuous, but empirical studies have shown more violent spreading modes when the participating agents are not limited to one type. Striking examples include the co-epidemic of the Spanish flu and pneumonia that occurred in 1918 (refs 3, 4), and, more recently, the concurrent prevalence of HIV/AIDS and a host of diseases5,6,7. It remains unclear to what extent an outbreak in the presence of interacting pathogens differs from that due to an ordinary single-agent process. Here we study a mechanistic model for understanding contagion processes involving inter-agent cooperation. Our stochastic simulations reveal the possible emergence of a massive avalanche-like outbreak right at the threshold, which is manifested as a discontinuous phase transition. Such an abrupt change arises only if the underlying network topology supports a bottleneck for cascaded mutual infections. Surprisingly, all these discontinuous transitions are accompanied by non-trivial critical behaviours, presenting a rare case of hybrid transition8. The findings may imply the origin of catastrophic occurrences in many realistic systems, from co-epidemics to financial contagions9.

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Figure 1: Illustration of co-epidemic spreading process on a complex network.
Figure 2: Hybrid phase transition on Erdős–Rényi networks with mean degree 〈k〉 = 4.
Figure 3: Dependence of transition order on dimensionality.
Figure 4: Critical dimension for co-epidemics on lattices.

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Acknowledgements

We thank D. Brockmann and W. Nadler for discussions.

Author information

Author notes

  1. Weiran Cai and Li Chen: These authors contributed equally to this work.

Authors and Affiliations

  1. Medical Faculty Carl Gustav Carus, Technische Universität Dresden, 01307 Dresden, Germany

    Weiran Cai

  2. Department of Electrical and Information Engineering, Technische Universität Dresden, 01069 Dresden, Germany

    Weiran Cai

  3. Robert Koch Institute, 13353 Berlin, Germany

    Li Chen & Fakhteh Ghanbarnejad

  4. Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany

    Li Chen, Fakhteh Ghanbarnejad & Peter Grassberger

  5. Jülich Supercomputing Centre, Forschungszentrum Jülich, 52425 Jülich, Germany

    Peter Grassberger

Authors
  1. Weiran Cai
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  2. Li Chen
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  3. Fakhteh Ghanbarnejad
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  4. Peter Grassberger
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Contributions

All authors conceived and designed the research, carried out numerical experiments, analysed the data, worked out the mechanism and wrote the manuscript.

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Correspondence to Weiran Cai.

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The authors declare no competing financial interests.

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Cai, W., Chen, L., Ghanbarnejad, F. et al. Avalanche outbreaks emerging in cooperative contagions. Nature Phys 11, 936–940 (2015). https://doi.org/10.1038/nphys3457

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