Bounding the Size and Probability of Epidemics on Networks

TL;DR

This paper derives bounds on the size and probability of epidemics on networks with clustering and heterogeneity, identifying conditions under which epidemics are most or least likely or large.

Contribution

It introduces a general framework for bounding epidemic outcomes based on transmissibility distributions, applicable to complex networks with clustering.

Findings

Bounds on epidemic size and probability depend on transmissibility distributions.

Homogeneous populations maximize epidemic size and probability.

Maximal variance in transmissibility minimizes epidemic outcomes.

Abstract

We consider an infectious disease spreading along the edges of a network which may have significant clustering. The individuals in the population have heterogeneous infectiousness and/or susceptibility. We define the out-transmissibility of a node to be the marginal probability that it would infect a randomly chosen neighbor given its infectiousness and the distribution of susceptibility. For a given distribution of out-transmissibility, we find the conditions which give the upper [or lower] bounds on size and probability of an epidemic, under weak assumptions on the transmission properties, but very general assumptions on the network. We find similar bounds for a given distribution of in-transmissibility (the marginal probability of being infected by a neighbor). We also find conditions giving global upper bounds on size and probability. The distributions leading to these bounds are network-independent. In the special case of networks with high girth (locally tree-like), we are able to prove stronger results. In general the probability and size of epidemics are maximal when the population is homogeneous and minimal when the variance of in- or out-transmissibility is maximal.