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Two golden times in two-step contagion models: A nonlinear map approach

Wonjun Choi, Deokjae Lee, J. Kertész, and B. Kahng
Phys. Rev. E 98, 012311 – Published 19 July 2018

Abstract

The two-step contagion model is a simple toy model for understanding pandemic outbreaks that occur in the real world. The model takes into account that a susceptible person either gets immediately infected or weakened when getting into contact with an infectious one. As the number of weakened people increases, they eventually can become infected in a short time period and a pandemic outbreak occurs. The time required to reach such a pandemic outbreak allows for intervention and is often called golden time. Understanding the size-dependence of the golden time is useful for controlling pandemic outbreak. Using an approach based on a nonlinear mapping, here we find that there exist two types of golden times in the two-step contagion model, which scale as O(N1/3) and O(Nζ) with the system size N on Erdős-Rényi networks, where the measured ζ is slightly larger than 1/4. They are distinguished by the initial number of infected nodes, o(N) and O(N), respectively. While the exponent 1/3 of the N-dependence of the golden time is universal even in other models showing discontinuous transitions induced by cascading dynamics, the measured ζ exponents are all close to 1/4 but show model-dependence. It remains open whether or not ζ reduces to 1/4 in the asymptotically large-N limit. Our method can be applied to several models showing a hybrid percolation transition and gives insight into the origin of the two golden times.

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  • Received 27 June 2017
  • Revised 26 March 2018

DOI:https://doi.org/10.1103/PhysRevE.98.012311

©2018 American Physical Society

Physics Subject Headings (PhySH)

NetworksStatistical Physics & ThermodynamicsInterdisciplinary PhysicsCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Wonjun Choi1, Deokjae Lee1, J. Kertész2,3, and B. Kahng1,*

  • 1CCSS, CTP and Department of Physics and Astronomy, Seoul National University, Seoul 08826, Korea
  • 2Center for Network Science, Central European University, H-1051, Budapest, Hungary
  • 3Department of Theoretical Physics, Budapest University of Technology and Economics, H-1111, Budapest, Hungary

  • *bkahng@snu.ac.kr

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Issue

Vol. 98, Iss. 1 — July 2018

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