Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-10T15:48:50.918Z Has data issue: false hasContentIssue false
  • English
  • Français

Random walks on a complete graph: a model for infection

Part of: Applications Stochastic processes

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
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aldous, D. (1991). Applications of random walks on finite graphs. In Selected Proc. Sheffield Symp. Appl. Prob. (Sheffield, 1989) (IMS Lecture Notes Monogr. Ser. 18), Institute of Mathematical Statistics, Hayward, CA, pp. 12–26.Google Scholar
Aldous, D. (1991). Meeting times for independent Markov chains. Stoch. Process. Appl. 38, 185–193.Google Scholar
Bailey, N. T. J. (1975). The Mathematical Theory of Infectious Diseases and Its Applications, 2nd edn. Griffin, London.Google Scholar
Becker, N. G. (1989). Analysis of Infectious Disease Data. Chapman and Hall, London.Google Scholar
Brydges, D. C. and Martin, Ph. A. (1999). Coulomb systems at low density: a review. J. Statist. Phys. 96, 1163–1330.Google Scholar
Coppersmith, D., Tetali, P., and Winkler, P. (1993). Collisions among random walks on a graph. SIAM J. Discrete Math. 6, 363–374.Google Scholar
Daley, D. J., and Gani, J. (1999). Epidemic Modelling: An Introduction (Camb. Studies Math. Biol. 15). Cambridge University Press.Google Scholar
De Gennes, P.-G. (1979). Scaling Concepts in Polymer Physics. Cornell University Press.Google Scholar
Doi, M. (1996). Introduction to Polymer Physics. Clarendon Press, Oxford.Google Scholar
Ginibre, J. (1971). Some applications of functional integration in statistical mechanics. In Statistical Mechanics and Quantum Field Theory (Summer School of Theoretical Physics, Les Houches, 1970), eds De Witt, C. M. and Stora, R., Gordon and Breach, New York, pp. 327–427.Google Scholar
Tóth, B. (1990). Phase transition in an interacting Bose system. An application of the theory of Ventsel′ and Freŭdlin. J. Statist. Phys. 61, 749–764.CrossRefGoogle Scholar
Winkler, P. (2000). Dependent percolation and colliding random walks. Random Structures Algorithms 16, 58–84.Google Scholar
Ziman, J. M. (1979). Models of Disorder. Cambridge University Press.Google Scholar