Effects of epidemic threshold definition on disease spread statistics

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 sc. Using percolation theory to calculate the average fractional size 〈MSIR〉 of an epidemic, we find that the strength of the spanning link percolation cluster P∞ is an upper bound to 〈MSIR〉. For small values of sc, P∞ is no longer a good approximation, and the average fractional size has to be computed directly. The value of sc for which P∞ 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 sc, 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 sc on predictions of average outcome sizes of computer failure epidemics.