Effect of initial infection size on network SIR model

TL;DR

This paper investigates how a non-negligible initial infection size affects the SIR epidemic model on networks, revealing that traditional percolation methods become ambiguous and proposing a new approach based on the recovered giant component.

Contribution

It introduces a novel method to analyze epidemic size and threshold when initial infection fraction is significant, correcting limitations of existing percolation-based approaches.

Findings

Epidemic threshold decreases as initial infection fraction increases.

Standard percolation methods become ambiguous for large seed fractions.

Exact equations for epidemic size and threshold are derived for large networks.

Abstract

We consider the effect of a nonvanishing fraction of initially infected nodes (seeds) on the SIR epidemic model on random networks. This is relevant when, for example, the number of arriving infected individuals is large, but also to the modeling of a large number of infected individuals, but also to more general situations such as the spread of ideas in the presence of publicity campaigns. This model is frequently studied by mapping to a bond percolation problem, in which edges in the network are occupied with the probability, $p$, of eventual infection along an edge connecting an infected individual to a susceptible neighbor. This approach allows one to calculate the total 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. This has an unambiguous interpretation and correctly captures the dependence of the epidemic size on $f$. We give exact equations for the size of the epidemic and the epidemic threshold in the infinite size limit. We observe a second order phase transition as in the original formulation, however with an epidemic threshold which decreases with increasing $f$. When the seed fraction $f$ tends to zero we recover the standard results.