Effects of epidemic threshold definition on disease spread statistics

TL;DR

This paper analyzes how different definitions of epidemic thresholds affect the statistical predictions of disease spread in heterogeneous networks, using percolation theory and real network data.

Contribution

It introduces a framework to understand the impact of epidemic size thresholds on spread statistics and applies it to real-world network data.

Findings

$P_{}$ bounds the average epidemic size for large thresholds.

The threshold s_c influences the accuracy of percolation-based approximations.

Application to real networks shows threshold choice affects epidemic size predictions.

Abstract

We study the statistical properties of the SIR epidemics in heterogeneous networks, when an epidemic is defined as only those SIR propagations that reach or exceed a minimum size s_c. Using percolation theory to calculate the average fractional size <M_SIR> of an epidemic, we find that the strength of the spanning link percolation cluster $P_{\infty}$ is an upper bound to <M_SIR>. For small values of s_c, $P_{\infty}$ is no longer a good approximation, and the average fractional size has to be computed directly. The value of s_c for which $P_{\infty}$ is a good approximation is found to depend on the transmissibility T of the SIR. We also study Q, the probability that an SIR propagation reaches the epidemic mass s_c, and find that it is well characterized by percolation theory. We apply our results to real networks (DIMES and Tracerouter) to measure the consequences of the choice s_c on predictions of average outcome sizes of computer failure epidemics.