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Threshold behaviour and final outcome of an epidemic on a random network with household structure

Part of: Markov processes Special processes Graph theory

Published online by Cambridge University Press:  01 July 2016

Frank Ball ,
David Sirl  and
Pieter Trapman
Frank Ball*
Affiliation:
The University of Nottingham
David Sirl*
Affiliation:
The University of Nottingham
Pieter Trapman*
Affiliation:
University Medical Center Utrecht and Vrije Universiteit Amsterdam
*
∗ Postal address: School of Mathematical Sciences, The University of Nottingham, University Park, Nottingham, NG7 2RD, UK.
∗ Postal address: School of Mathematical Sciences, The University of Nottingham, University Park, Nottingham, NG7 2RD, UK.
∗∗∗∗ Postal address: Stochastics Section, Faculty of Sciences, Vrije Universiteit Amsterdam, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands. Email address: [email protected]
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Abstract

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In this paper we consider a stochastic SIR (susceptible→infective→removed) epidemic model in which individuals may make infectious contacts in two ways, both within ‘households’ (which for ease of exposition are assumed to have equal size) and along the edges of a random graph describing additional social contacts. Heuristically motivated branching process approximations are described, which lead to a threshold parameter for the model and methods for calculating the probability of a major outbreak, given few initial infectives, and the expected proportion of the population who are ultimately infected by such a major outbreak. These approximate results are shown to be exact as the number of households tends to infinity by proving associated limit theorems. Moreover, simulation studies indicate that these asymptotic results provide good approximations for modestly sized finite populations. The extension to unequal-sized households is discussed briefly.

Keywords

Branching processcouplingepidemic processfinal outcomehouseholdslocal and global contactsrandom graphsusceptibility setthreshold theorem

MSC classification

Primary: 92D30: Epidemiology 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 05C80: Random graphs 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 41 , Issue 3 , September 2009 , pp. 765 - 796
Copyright
Copyright © Applied Probability Trust 2009 

References

Andersson, H. (1997). Epidemics in a population with social structures. Math. Biosci. 140, 79–84.Google Scholar
Andersson, H. (1998). Limit theorems for a random graph epidemic model. Ann. Appl. Prob. 8, 1331–1349.Google Scholar
Andersson, H. (1999). Epidemic models and social networks. Math. Scientist 24, 128–147.Google Scholar
Ball, F. (1986). A unified approach to the distribution of total size and total area under the trajectory of infectives in epidemic models. Adv. Appl. Prob. 18, 289–310.Google Scholar
Ball, F. and Lyne, O. D. (2001). Stochastic multitype SIR epidemics among a population partitioned into households. Adv. Appl. Prob. 33, 99–123.Google Scholar
Ball, F. and Neal, P. (2002). A general model for stochastic SIR epidemics with two levels of mixing. Math. Biosci. 180, 73–102.CrossRefGoogle ScholarPubMed
Ball, F. and Neal, P. (2003). The great circle epidemic model. Stoch. Process. Appl. 107, 233–268.CrossRefGoogle Scholar
Ball, F. and Neal, P. (2008). Network epidemic models with two levels of mixing. Math. Biosci. 212, 69–87.CrossRefGoogle ScholarPubMed
Ball, F. and O'Neill, P. (1999). The distribution of general final state random variables for stochastic epidemic models. J. Appl. Prob. 36, 473–491.Google Scholar
Ball, F., Mollison, D. and Scalia-Tomba, G. (1997). Epidemics with two levels of mixing. Ann. Appl. Prob. 7, 46–89.CrossRefGoogle Scholar
Becker, N. G. and Dietz, K. (1995). The effect of household distribution on transmission and control of highly infectious diseases. Math. Biosci. 127, 207–219.Google Scholar
Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.Google Scholar
Britton, T., Janson, S. and Martin-Löf, A. (2007). Graphs with specified degree distributions, simple epidemics, and local vaccination strategies. Adv. Appl. Prob. 39, 922–948.CrossRefGoogle Scholar
Britton, T., Deijfen, M., Lagerås, A. N. and Lindholm, M. (2008). Epidemics on random graphs with tunable clustering. J. Appl. Prob. 45, 743–756.Google Scholar
Durrett, R. (2006). Random Graph Dynamics. Cambridge University Press.CrossRefGoogle Scholar
Haccou, P., Jagers, P. and Vatutin, V. (2005). Branching Processes: Variation, Growth, and Extinction of Populations. Cambridge University Press.CrossRefGoogle Scholar
Janson, S. (2009). The probability that a random multigraph is simple. Combinatorics Prob. Comput. 18, 205–225.CrossRefGoogle Scholar
Kenah, E. and Robins, J. M. (2007). Second look at the spread of epidemics on networks. Phys. Rev. E 76, 036113, 12 pp.Google Scholar
Kiss, I. Z., Green, D. M. and Kao, R. R. (2006). The effect of contact heterogeneity and multiple routes of transmission on final epidemic size. Math. Biosci. 203, 124–136.CrossRefGoogle ScholarPubMed
Kuulasmaa, K. (1982). The spatial general epidemic and locally dependent random graphs. J. Appl. Prob. 19, 745–758.Google Scholar
Newman, M. E. J. (2002). Spread of epidemic disease on networks. Phys. Rev. E 66, 016128, 11 pp.Google Scholar
Newman, M. E. J. (2003). The structure and function of complex networks. SIAM Rev. 45, 167–256.Google Scholar
Pellis, L., Ferguson, N. M. and Fraser, C. (2008). The relationship between real-time and discrete-generation models of epidemic spread. Math. Biosci. 216, 63–70.Google Scholar
Trapman, P. (2007). On analytical approaches to epidemics on networks. Theoret. Pop. Biol. 71, 160–173.Google Scholar
Van der Hofstad, R., Hooghiemstra, G. and Znamenski, D. (2007). Distances in random graphs with finite mean and infinite variance degrees. Electron. J. Prob. 12, 703–766.Google Scholar
Watts, D. J. and Strogatz, S. H. (1998). Collective dynamics of ‘small-world’ networks. Nature 393, 440–442.CrossRefGoogle ScholarPubMed
Whittle, P. (1955). The outcome of a stochastic epidemic—a note on Bailey's paper. Biometrika 42, 116–122.Google Scholar