Multiple transitions of the susceptible-infected-susceptible epidemic model on complex networks

Abstract

The epidemic threshold of the susceptible-infected-susceptible (SIS) dynamics on random networks having a power law degree distribution with exponent γ > 3 has been investigated using different mean-field approaches, which predict different outcomes. We performed extensive simulations in the quasistationary state for a comparison with these mean-field theories. We observed concomitant multiple transitions in individual networks presenting large gaps in the degree distribution and the obtained multiple epidemic thresholds are well described by different mean-field theories. We observed that the transitions involving thresholds which vanishes at the thermodynamic limit involve localized states, in which a vanishing fraction of the network effectively contribute to epidemic activity, whereas an endemic state, with a finite density of infected vertices, occurs at a finite threshold. The multiple transitions are related to the activations of distinct sub-domains of the network, which are not directly connected.

Super compact pairwise model for SIS epidemic on heterogeneous networks

“…Referring back to our numerical tests, we point out that we did not explicitly consider the 'close' to threshold regime, since the super compact PW model is highly coarse grained and thus unlikely to produce as good as or better agreement than more detailed or sophisticated models. This is supported by past and recent research which confirms that agreement between mean-field and simulation models close to the threshold remains difficult to obtain and often requires more sophisticated models, see [15,16]. The issue of the threshold's dependency on model and the precise value of the threshold for SIS dynamics on networks have recently been subject to a vigorous debate.…”

“…In [16], the authors reinforce and show that different mean-field approaches lead to different outcomes in terms of the threshold. Similarly, in [17], the authors show that the heterogenous mean-field theory [3], with closures at the level of pairs, fails to correctly capture the transition point and finite-size scalings close to the threshold when a contact process dynamic is considered.…”

“…Here, this ratio is chosen to be 3, its actual value has only a minor influence on the results, generally this need to be greater than 1 to have an epidemic. We note that τ has been chosen to avoid the 'close to threshold' regime, where mean-field models generally fail to accurately predict simulation results [15,16].…”

Scale-free networks of the earth’s surface

“…Since the surprising discovery of the scale-free networks with power-law degree distributions proposed by Barabási and Albert (BA), 8 most of real networks have been proved displaying power-law-shaped degree distribution P (k) ∼ k −γ (where k is the degree which is defined as the number of edges connected to a vertex, P (k) is the degree distribution and γ is the power exponent), with exponents varying in the range 2 < γ < 3, such as the transportation networks, the power networks, the Internet, the earthquake networks, etc. 4,[9][10][11][12][13] The term "scale-free" refers to any functional form f (x) that remains unchanged within a multiplicative factor under a rescaling of the independent variable x. In effect, this means power-law forms, since these are the only solutions to f (ax) = bf (x), and hence "power-law" and "scale-free" are, for our purposes, synonymous.…”

Survival time of the susceptible-infected-susceptible infection process on a graph

“…, where D is the degree of a randomly chosen node in G. For power-law degree graphs, there is evidence of multiple phase transitions [11] in the SIS process.…”