Effect of initial infection size on a network susceptible-infected-recovered model

We consider the effect of a nonvanishing fraction of initially infected nodes (seeds) on the susceptible-infected-recovered epidemic model on random networks. This is relevant when the number of arriving infected individuals is large, or to the spread of ideas with publicity campaigns. This model is frequently studied by mapping to a bond percolation problem, in which edges are occupied with the probability p of eventual infection along an edge. This gives accurate measures of the final size of the infection and epidemic threshold in the limit of a vanishingly small seed fraction. We show, however, that when the initial infection occupies a nonvanishing fraction, f, of the network, this method yields ambiguous results, as the correspondence between edge occupation and contagion transmission no longer holds. We propose instead to measure the giant component of recovered individuals within the original contact network. We derive exact equations for the size of the epidemic and the epidemic threshold in the infinite size limit in heterogeneous sparse random networks, and we confirm them with numerical results. We observe that the epidemic threshold correctly depends on f, decreasing as f increases. When the seed fraction tends to zero, we recover the standard results.