Effects of epidemic threshold definition on disease spread statistics

We study the statistical properties of SIR epidemics in random networks, when an epidemic is defined as only those SIR propagations that reach or exceed a minimum size scsc. 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∞P∞ is an upper bound to 〈MSIR〉. For small values of scsc, P∞P∞ is no longer a good approximation, and the average fractional size has to be computed directly. We find that the choice of scsc is generally (but not always) guided by the network structure and the value of TT of the disease in question. If the goal is to always obtain P∞P∞ as the average epidemic size, one should choose scsc to be the typical size of the largest percolation cluster at the critical percolation threshold for the transmissibility. We also study QQ, the probability that an SIR propagation reaches the epidemic mass scsc, 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 scsc on predictions of average outcome sizes of computer failure epidemics.