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Bounding basic characteristics of spatial epidemics with a new percolation model

Part of: Special processes

Published online by Cambridge University Press:  01 July 2016

Ronald Meester  and
Pieter Trapman
Ronald Meester*
Affiliation:
VU University Amsterdam
Pieter Trapman*
Affiliation:
VU University Amsterdam and University Medical Center Utrecht
*
∗ Postal address: Department of Mathematics, VU University Amsterdam, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands.
∗∗ Current address: Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden. Email address: [email protected]
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Abstract

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We introduce a new 1-dependent percolation model to describe and analyze the spread of an epidemic on a general directed and locally finite graph. We assign a two-dimensional random weight vector to each vertex of the graph in such a way that the weights of different vertices are independent and identically distributed, but the two entries of the vector assigned to a vertex need not be independent. The probability for an edge to be open depends on the weights of its end vertices, but, conditionally on the weights, the states of the edges are independent of each other. In an epidemiological setting, the vertices of a graph represent the individuals in a (social) network and the edges represent the connections in the network. The weights assigned to an individual denote its (random) infectivity and susceptibility, respectively. We show that one can bound the percolation probability and the expected size of the cluster of vertices that can be reached by an open path starting at a given vertex from above by the corresponding quantities for independent bond percolation with a certain density; this generalizes a result of Kuulasmaa (1982). Many models in the literature are special cases of our general model.

Keywords

Dependent percolationepidemics

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 92D30: Epidemiology
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 43 , Issue 2 , June 2011 , pp. 335 - 347
Copyright
Copyright © Applied Probability Trust 2011 

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