TY - JOUR
T1 - Two golden times in two-step contagion models
T2 - A nonlinear map approach
AU - Choi, Wonjun
AU - Lee, Deokjae
AU - Kertész, J.
AU - Kahng, B.
N1 - Publisher Copyright:
© 2018 American Physical Society.
PY - 2018/7/19
Y1 - 2018/7/19
N2 - 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 Erdos-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.
AB - 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 Erdos-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.
UR - http://www.scopus.com/inward/record.url?scp=85050459202&partnerID=8YFLogxK
U2 - 10.1103/PhysRevE.98.012311
DO - 10.1103/PhysRevE.98.012311
M3 - Article
C2 - 30110730
AN - SCOPUS:85050459202
SN - 2470-0045
VL - 98
JO - Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
JF - Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
IS - 1
M1 - 012311
ER -