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Bounding the Size and Probability of Epidemics on Networks

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Joel C. Miller
Joel C. Miller*
Affiliation:
British Columbia Centre for Disease Control
*
∗Postal address: 655 W 12th Avenue, Vancouver, BC V5Z 4R4, Canada. Email address: [email protected]
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Abstract

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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 the 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 the 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.

Keywords

Epidemiologynetworkattack rateprobabilitytransmissibility

MSC classification

Secondary: 92D30: Epidemiology 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 2 , June 2008 , pp. 498 - 512
Copyright
Copyright © Applied Probability Trust 2008 

Footnotes

Much of this work was completed while at the Center for Nonlinear Studies and the Mathematical Modeling & Analysis Group, Los Alamos National Laboratory.

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