Long-range epidemic spreading in a random environment

TL;DR

This paper investigates the long-range epidemic spreading in disordered environments using a combination of renormalization group methods and simulations, revealing critical behaviors and Griffiths phases in the model.

Contribution

It introduces a detailed analysis of a long-range contact process with decaying infection rates, highlighting the emergence of Griffiths phases and critical scaling behaviors.

Findings

Survival probability decays as a power law with time.

A Griffiths phase with a continuously varying dynamical exponent.

Growth of infected cluster size follows a power law at the threshold.

Abstract

Modeling long-range epidemic spreading in a random environment, we consider a quenched disordered, $d$-dimensional contact process with infection rates decaying with the distance as $1/r^{d+\sigma}$. We study the dynamical behavior of the model at and below the epidemic threshold by a variant of the strong-disorder renormalization group method and by Monte Carlo simulations in one and two spatial dimensions. Starting from a single infected site, the average survival probability is found to decay as $P(t) \sim t^{-d/z}$ up to multiplicative logarithmic corrections. Below the epidemic threshold, a Griffiths phase emerges, where the dynamical exponent $z$ varies continuously with the control parameter and tends to $z_c=d+\sigma$ as the threshold is approached. At the threshold, the spatial extension of the infected cluster (in surviving trials) is found to grow as $R(t) \sim t^{1/z_c}$ with a multiplicative logarithmic correction, and the average number of infected sites in surviving trials is found to increase as $N_s(t) \sim (\ln t)^{\chi}$ with $\chi=2$ in one dimension.