Contact processes on random graphs with power law degree distributions have critical value 0

Abstract

If we consider the contact process with infection rate λ on a random graph on n vertices with power law degree distributions, mean field calculations suggest that the critical value λc of the infection rate is positive if the power α > 3. Physicists seem to regard this as an established fact, since the result has recently been generalized to bipartite graphs by Gómez-Gardeñes et al. [Proc. Natl. Acad. Sci. USA 105 (2008) 1399-1404. Here, we show that the critical value λc is zero for any value of α > 3, and the contact process starting from all vertices infected, with a probability tending to 1 as n → ∞, maintains a positive density of infected sites for time at least exp(n 1−δ ) for any δ > 0. Using the last result, together with the contact process duality, we can establish the existence of a quasi-stationary distribution in which a randomly chosen vertex is occupied with probability ρ(λ). It is expected that ρ(λ) ∼ Cλ β as λ → 0. Here we show that α − 1 ≤ β ≤ 2α − 3, and so β > 2 for α > 3. Thus even though the graph is locally tree-like, β does not take the mean field critical value β = 1.

Edge-based compartmental modelling for infectious disease spread

“…Thus if partnership duration is nonzero, our predictions may be significantly altered. This limitation is often not recognized but may lead to important failures of mean-field or MA models when applied to a population for which partnership duration is important [46].…”

Critical behaviour of the partner model

“…2 ≤ e −c(log t−log C) 2 which for t large enough is at most e −(c/2)(log t) 2 , so that if an event holds with good probability in n, then it holds with good probability in t.…”

Collective versus hub activation of epidemic phases on networks

“…Transmission of the disease is effected by a rate to transmit the disease through an edge connecting an infected to a susceptible node equal to a constant λ. After a considerable theoretical effort, it has been shown that the behavior of the SIS model in uncorrelated [2] SF networks is far from trivial [7][8][9][10][11][12][13][14][15][16]. Two competing theories were initially proposed to account for the SIS epidemic threshold.…”