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Markovian contact processes

Published online by Cambridge University Press:  01 July 2016

Denis Mollison
Affiliation:
Heriot-Watt University, Edinburgh

Abstract

Markovian contact processes (mcp's) include some of the commoner models for the spatial spread of population processes, such as the ‘simple’ and ‘general’ epidemics, percolation processes, and birth, death and migration processes. They are here set in a framework which relates them to each other, and to a particular basic model, the contact birth process (cbp); indeed they are defined as modifications of the cbp. The immediate advantage of this is that bounds for the velocity of the cbp, which can be obtained from the linear equation describing its expected numbers (provided the contact distribution has exponentially bounded tail) apply also to general mcp's. Existence of moments (and for some processes their time-derivatives) of St (θ), the distance to the furthest individual in an arbitrary direction θ, can also be deduced whenever the corresponding moment of the contact distribution exists.

An important subclass consists of population-monotone mcp's, for which the distributions of St can be shown to be subconvolutive, so that the work of Hammersley (1974) can be applied to obtain convergence theorems for St /t; and these can be extended to convergence of the convex hull of the set of inhabited points. These results are particularly valuable because they apply to some non-linear processes, e.g. the simple epidemic. Some special results on the cbp (Section 6) emphasize the differences between linear and non-linear spatial stochastic processes.

Although the paper is written throughout in terms of continuous-time processes, the results on subconvolutive distributions are actually more easily applied in the discrete-time case. (Conversely, the approach used in the accompanying paper by Biggins (pp. 62–84), which is written in terms of discrete-time processes, can be extended to deal also with continuous-time processes.)

Type
Research Article
Copyright
Copyright © Applied Probability Trust 1978 

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