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

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(N^{1/3}) and O(N^{ζ}) 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.

Hartree theory calculations of quasiparticle properties in twisted bilayer graphene

“…At the magic angle, the band structure of undoped tBLG is qualitatively different as compared to the other twist angles [24,64]. In particular, the two valence bands at Γ are pushed up and are now higher in energy than the states at K and K .…”

Interplay between competitive and cooperative interactions in a three-player pathogen system

“…This phenomenon, however, is sensitive to the network topology, with continuous, discontinuous and hybrid, i.e. continuous in the outbreak probability and discontinuous in the prevalence, transitions observed according to the topology of the network [40,43,39,42,53,54,55]. Here we found rich dynamics as the impact of stochastic effects.…”

The dynamics of two-stage contagion

“…Of prime interest here is the fact that, in contrast with the one-stage epidemic model, in the two-stage model one has, under certain conditions, a discontinuous ('first order' in the language of statistical physics) dependence of the total size of the epidemic on the contact parameter. This has led to interest in the statistical physics community, and several works investigate the two-stage contagion process on lattices and random graphs [4,10,11,13,22].…”

“…To incorporate the idea of stages of contagion into mathematical models, one can divide the population into classes, each of which consists of individuals at a certain stage in the adoption process, with movement among the classes due to contact with adopters. Models of this type have been proposed studied in several works [4,10,11,13,14,22,27,28,33], and in section 1.2 we will briefly describe and compare these with the model studied here.…”