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 individualsis large, or to the spread of ideas with publicity campaigns. This model is frequently studied by mapping to abond percolation problem, in which edges are occupied with the probabilitypof eventual infection along an edge.This gives accurate measures of the final size of the infection and epidemic threshold in the limit of a vanishinglysmall seed fraction. We show, however, that when the initial infection occupies a nonvanishing fraction,f,ofthenetwork, this method yields ambiguous results, as the correspondence between edge occupation and contagiontransmission no longer holds. We propose instead to measure the giant component of recovered individualswithin the original contact network. We derive exact equations for the size of the epidemic and the epidemicthreshold in the infinite size limit in heterogeneous sparse random networks, and we confirm them with numericalresults. We observe that the epidemic threshold correctly depends onf, decreasing asfincreases. When the seedfraction tends to zero, we recover the standard results.

Automated summary

We consider the impact of a nonvanishing fraction of initially infected nodes on the susceptible-infected-recovered epidemic model on random networks. We propose a new method to measure the size of the epidemic and epidemic threshold.