Journal of Applied Probability

Acquaintance vaccination in an epidemic on a random graph with specified degree distribution

Frank Ball and David Sirl

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Abstract

We consider a stochastic SIR (susceptible → infective → removed) epidemic on a random graph with specified degree distribution, constructed using the configuration model, and investigate the `acquaintance vaccination' method for targeting individuals of high degree for vaccination. Branching process approximations are developed which yield a post-vaccination threshold parameter, and the asymptotic (large population) probability and final size of a major outbreak. We find that introducing an imperfect vaccine response into the present model for acquaintance vaccination leads to sibling dependence in the approximating branching processes, which may then require infinite type spaces for their analysis and are generally not amenable to numerical calculation. Thus, we propose and analyse an alternative model for acquaintance vaccination, which avoids these difficulties. The theory is illustrated by a brief numerical study, which suggests that the two models for acquaintance vaccination yield quantitatively very similar disease properties.

Article information

Source
J. Appl. Probab., Volume 50, Number 4 (2013), 1147-1168.

Dates
First available in Project Euclid: 10 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.jap/1389370105

Digital Object Identifier
doi:10.1239/jap/1389370105

Mathematical Reviews number (MathSciNet)
MR3161379

Zentralblatt MATH identifier
1301.92071

Subjects
Primary: 92D30: Epidemiology
Secondary: 05C80: Random graphs [See also 60B20] 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Branching process epidemic process final size network random graph threshold behaviour vaccination

Citation

Ball, Frank; Sirl, David. Acquaintance vaccination in an epidemic on a random graph with specified degree distribution. J. Appl. Probab. 50 (2013), no. 4, 1147--1168. doi:10.1239/jap/1389370105. https://projecteuclid.org/euclid.jap/1389370105


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References

  • \item[] Andersson, H. (1997). Epidemics in a population with social structures. Math. Biosci. 140, 79–84.
  • \item[] Andersson, H. (1998). Limit theorems for a random graph epidemic model. Ann. Appl. Prob. 8, 1331–1349.
  • \item[] Andersson, H. (1999). Epidemic models and social networks. Math. Scientist 24, 128–147.
  • \item[] Ball, F. and Neal, P. (2008). Network epidemic models with two levels of mixing. Math. Biosci. 212, 69–87.
  • \item[] Ball, F. and Sirl, D. (2012). An SIR epidemic model on a population with random network and household structure, and several types of individuals. Adv. Appl. Prob. 44, 63–86.
  • \item[] Ball, F., Sirl, D. and Trapman, P. (2009). Threshold behaviour and final outcome of an epidemic on a random network with household structure. Adv. Appl. Prob. 41, 765–796.
  • \item[] Ball, F., Sirl, D. and Trapman, P. (2010). Analysis of a stochastic SIR epidemic on a random network incorporating household structure. Math. Biosci. 224, 53–73.
  • \item[] Ball, F., Sirl, D. and Trapman, P. (2013). Epidemics on random intersection graphs. To appear in Ann. Appl. Prob.
  • \item[] Becker, N. G. and Starczak, D. N. (1998). The effect of random vaccine response on the vaccination coverage required to prevent epidemics. Math. Biosci. 154, 117–135.
  • \item[] 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.
  • \item[] Britton, T., Deijfen, M., Lagerås, A. N. and Lindholm, M. (2008). Epidemics on random graphs with tunable clustering. J. Appl. Prob. 45, 743–756.
  • \item[] Cohen, R., Havlin, S. and ben Avraham, D. (2003). Efficient immunization strategies for computer networks and populations. Phys. Rev. Lett. 91, 247901.
  • \item[] Diekmann, O., de Jong, M. C. M. and Metz, J. A. J. (1998). A deterministic epidemic model taking account of repeated contacts between the same individuals. J. Appl. Prob. 35, 448–462.
  • \item[] Durrett, R. (2007). Random Graph Dynamics. Cambridge University Press.
  • \item[] Halloran, M. E., Haber, M. and Longini, I. M., Jr. (1992). Interpretation and estimation of vaccine efficacy under heterogeneity. Amer. J. Epidemiol. 136, 328–343.
  • \item[] Kenah, E. and Robins, J. M. (2007). Second look at the spread of epidemics on networks. Phys. Rev. E 76, 036113.
  • \item[] 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.
  • \item[] Mode, C. J. (1971). Multitype Branching Processes. Theory and Applications (Modern Analytic Comput. Meth. Sci. Math. 34). Elsevier, New York.
  • \item[] Newman, M. E. J. (2002). Spread of epidemic disease on networks. Phys. Rev. E 66, 016128.
  • \item[] Newman, M. E. J., Strogatz, S. H. and Watts, D. J. (2001). Random graphs with arbitrary degree distributions and their applications. Phys. Rev. E 64, 026118.
  • \item[] Olofsson, P. (1996). Branching processes with local dependencies. Ann. Appl. Prob. 6, 238–268.
  • \item[] Seneta, E. (1973). Non-Negative Matrices. Halsted Press, New York. \endharvreferences

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