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Random walks on a complete graph: a model for infection

Part of: Stochastic processes Applications

Published online by Cambridge University Press:  14 July 2016

Nilanjana Datta  and
Tony C. Dorlas
Nilanjana Datta*
Affiliation:
University of Cambridge
Tony C. Dorlas*
Affiliation:
Dublin Institute for Advanced Studies
*
∗ Postal address: Statistical Laboratory, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK. Email address: n.datta@statslab.cam.ac.uk
∗∗ Postal address: School of Theoretical Physics, Dublin Institute for Advanced Studies, 10 Burlington Road, Dublin 4, Ireland. Email address: dorlas@stp.dias.ie
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Abstract

We introduce a new model for the infection of one or more subjects by a single agent, and calculate the probability of infection after a fixed length of time. We model the agent and subjects as random walkers on a complete graph of N sites, jumping with equal rates from site to site. When one of the walkers is at the same site as the agent for a length of time τ, we assume that the infection probability is given by an exponential law with parameter γ, i.e. q(τ) = 1 - e-γ τ . We introduce the boundary condition that all walkers return to their initial site (‘home’) at the end of a fixed period T. We also assume that the incubation period is longer than T, so that there is no immediate propagation of the infection. In this model, we find that for short periods T, i.e. such that γ T ≪ 1 and T ≪ 1, the infection probability is remarkably small and behaves like T 3. On the other hand, for large T, the probability tends to 1 (as might be expected) exponentially. However, the dominant exponential rate is given approximately by 2γ/[(2+γ)N] and is therefore small for large N.

Keywords

Random walkcomplete graphmodel for infectionMarkov chain

MSC classification

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 62P10: Applications to biology and medical sciences
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 4 , December 2004 , pp. 1008 - 1021
Copyright
Copyright © Applied Probability Trust 2004 

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