SIR epidemics with long-range infection in one dimension

Grassberger, Peter

doi:10.1088/1742-5468/2013/04/P04004

Journal Article

FZJ-2014-01354

SIR epidemics with long-range infection in one dimension

Grassberger, P. (Corresponding author)FZJ*

2013
IOP Publ. Bristol

Journal of statistical mechanics: theory and experiment 2013(04), P04004 - (2013) [10.1088/1742-5468/2013/04/P04004]

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Please use a persistent id in citations: doi:10.1088/1742-5468/2013/04/P04004

Abstract: We study epidemic processes with immunization on very large 1-dimensional lattices, where at least some of the infections are non-local, with rates decaying as power laws p(x) ~ x−σ−1 for large distances x. When starting with a single infected site, the cluster of infected sites stays always bounded if σ > 1 (and dies with probability 1, if its size is allowed to fluctuate down to zero), but the process can lead to an infinite epidemic for σ < 1. For σ < 0 the behavior is essentially of mean-field type, but for 0 < σ ≤ 1 the behavior is non-trivial, both for the critical and for supercritical cases. For critical epidemics we confirm a previous prediction that the critical exponents controlling the correlation time and the correlation length are simply related to each other, and we verify detailed field theoretic predictions for σ drarr 1/3. For σ = 1 we find generic power laws with continuously varying exponents even in the supercritical case, and confirm in detail the predicted Kosterlitz–Thouless nature of the transition. Finally, the mass N(t) of supercritical clusters grows for 0 < σ < 1 like a stretched exponential. This implies that networks embedded in 1-d space with power-behaved link distributions have infinite intrinsic dimension (based on the graph distance), but are not small world.

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