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Epidemics Among a Population of Households

  • Frank G. Ball
  • Owen D. Lyne
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 126)

Abstract

This paper considers SIR and SIS epidemics among a population partitioned into households. This heterogeneity has important implications for the threshold behaviour of epidemics and optimal vaccination strategies. It is shown that taking into account household structures when modelling public health problems is valuable. An overview of households models is given, including a determination of threshold parameters, the probability of a global epidemic and some new results on vaccination strategies for SIS households epidemics. Simulation and numerical studies are presented which exemplify the results discussed.

Key words

SIR and SIS epidemics household structure threshold behaviour optimal vaccination strategy metapopulation models 

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References

  1. [1]
    CL. Addy, I.M. Longini, and M. Haber, A generalized stochastic model for the analysis of infectious disease final size data, Biometrics 47 (1991), pp. 961–974.zbMATHCrossRefGoogle Scholar
  2. [2]
    H. Andersson, Epidemics in a population with social structures, Math. Biosci. 140 (1997), pp. 79–84.zbMATHCrossRefGoogle Scholar
  3. [3]
    J.P. Aparicio, A.F. Capurro, and C. Castillo-Chavez, Transmission and dynamics of tuberculosis on generalized households, J. Theor. Biol. 206 (2000), pp. 327–341.CrossRefGoogle Scholar
  4. [4]
    N.T.J. Bailey, The Mathematical Theory of Infectious Diseases and its Applications, 2nd edn. Griffin, London, 1975.zbMATHGoogle Scholar
  5. [5]
    F.G. Ball, The threshold behaviour of epidemic models, J. Appl. Prob. 20 (1983) pp. 227–241.zbMATHCrossRefGoogle Scholar
  6. [6]
    F.G. Ball, A unified approach to the distribution of total size and total area under the trajectory of infectives in epidemic models, Adv. Appl. Prob. 18 (1986), pp. 289–310.zbMATHCrossRefGoogle Scholar
  7. [7]
    F.G. Ball, Threshold behaviour in stochastic epidemics among households, in: Athens Conference on Applied Probability and Time Series, Volume I: Applied Probability (eds. C.C. Heyde, Y.V. Prohorov, R. Pyke, and S.T. Rachev), Lecture Notes in Statistics 114 (1996), pp. 253–266.CrossRefGoogle Scholar
  8. [8]
    F.G. Ball, Stochastic and deterministic models for SIS epidemics among a population partitioned into households, Math. Biosci. 156 (1999), pp. 41–67.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    F.G. Ball and P. Donnelly, Strong approximations for epidemic models, Stoch. Proc. Appl. 55 (1995), pp. 1–21.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    F.G. Ball and O.D. Lyne, Stochastic multitype SIR epidemics among a population partitioned into households, Adv. Appl. Prob. 33 (2001) pp. 99–123.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    F.G. Ball, D. Mollison, and G. Scalia-Tomba, Epidemics with two levels of mixing, Ann. Appl. Prob. 7 (1997), pp. 46–89.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    R. BartoszyŃski, On a certain model of an epidemic, Appl. Math. 13 (1972), pp. 139–151.zbMATHGoogle Scholar
  13. [13]
    N.G. Becker, Analysis of Infectious Disease Data, Chapman and Hall, London (1989).Google Scholar
  14. [14]
    N.G. Becker, A. Bahrampour, and K. Dietz, Threshold parameters for epidemics in different community settings, Math. Biosci. 129 (1995), pp. 189–208.zbMATHCrossRefGoogle Scholar
  15. [15]
    N.G. Becker and K. Dietz, The effect of the household distribution on transmission and control of highly infectious diseases, Math. Biosci. 127 (1995), pp. 207–219.zbMATHCrossRefGoogle Scholar
  16. [16]
    N.G. Becker and K. Dietz, Reproduction numbers and critical immunity levels in epidemics in a community of households, in: Athens Conference on Applied Probability and Time Series, Volume I: Applied Probability (eds. C.C. Heyde, Y.V. Prohorov, R. Puke, and S.T. Rachev), Lecture Notes in Statistics 114 (1996), pp. 267–276.Google Scholar
  17. [17]
    N.G. Becker and R. Hall, Immunization levels for preventing epidemics in a community of households made up of individuals of different types, Math. Biosci. 132 (1996), pp. 205–216.zbMATHCrossRefGoogle Scholar
  18. [18]
    N.G. Becker and D.N. Starczak, Optimal vaccination strategies for a community of households, Math. Biosci. 139 (1997), pp. 117–132.zbMATHCrossRefGoogle Scholar
  19. [19]
    N.G. Becker and S. Utev, The effect of community structure on the immunity coverage required to prevent epidemics, Math. Biosci. 147 (1998), pp. 23–39.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    T. Britton, Limit theorems and tests to detect within family clustering in epidemic models, Commun. Statist. 26 (1997a), pp. 953–976.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    T. Britton, Tests to detect clustering of infected individuals within families, Biometrics 53 (1997b), pp. 98–109.zbMATHCrossRefGoogle Scholar
  22. [22]
    T. Britton, A test to detect within-family infectivity when the whole epidemic process is observed, Scand. J. Statist. 24 (1997c), pp. 315–330.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    T. Britton, A test of homogeneity versus a specified heterogeneity in an epidemic model, Math. Biosci. 141 (1997d), pp. 79–100.zbMATHCrossRefGoogle Scholar
  24. [24]
    T. Britton, On critical vaccination coverage in multitype epidemics, J. Appl. Prob. 35 (1998), pp. 1003–1006.MathSciNetzbMATHCrossRefGoogle Scholar
  25. [25]
    D.J. Daley and J. Gani, A deterministic general epidemic model in a stratified population, in: Probability, Statistics and Optimisation — A Tribute to Peter Whittle, (ed. F.P. Kelly), Wiley, Chichester (1994), pp. 117–132.Google Scholar
  26. [26]
    O. Diekmann, M.C.M. De Jong and J.A.J. Metz, A deterministic epidemic model taking account of repeated contacts between the same individuals, J. Appl. Prob. 35 (1998), pp. 448–462.zbMATHCrossRefGoogle Scholar
  27. [27]
    J.A.P. Heesterbeek and K. Dietz, The concept of Ro in epidemic theory, Statistica Neerlandica 50 (1996), pp. 89–110.MathSciNetzbMATHCrossRefGoogle Scholar
  28. [28]
    R.J. Kryscio and C. LefÈvre, On the extinction of the SIS stochastic logistic epidemic, J. Appl. Prob. 26 (1989), pp. 685–694.zbMATHCrossRefGoogle Scholar
  29. [29]
    T.G. Kurtz, Solutions of ordinary differential equations as limits of pure jump Markov processes, J. Appl. Prob. 7 (1970), pp. 49–58.MathSciNetzbMATHCrossRefGoogle Scholar
  30. [30]
    T.G. Kurtz, Limit theorems for sequences of jump Markov processes approximating ordinary differential processes, J. Appl. Prob. 8 (1971), pp. 344–356.MathSciNetzbMATHCrossRefGoogle Scholar
  31. [31]
    A. Lajmanovich and J.A. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population, Math. Biosci. 28 (1976), pp. 221–236.MathSciNetzbMATHCrossRefGoogle Scholar
  32. [32]
    D. Ludwig, Final size distributions for epidemics, Math. Biosci. 23 (1975), pp. 33–46.MathSciNetzbMATHCrossRefGoogle Scholar
  33. [33]
    C.J. Mode, Multitype Branching Processes, Elsevier, New York (1971).zbMATHGoogle Scholar
  34. [34]
    D. Mollison, Spatial contact models for ecological and epidemic spread, J. R. Statist. Soc. B 39 (1977), pp. 283–326.MathSciNetzbMATHGoogle Scholar
  35. [35]
    D. Mollison, V. Isham, and B. Grenfell Epidemics: models and data, J. R. Statist. Soc. A 157, (1994), pp. 115–149.CrossRefGoogle Scholar
  36. [36]
    D. Mollison and S.A. Levin, Spatial dynamics of parasitism, in: Ecology of Infectious Diseases in Natural Populations (eds. Grenfell, B.T. and Dobson, A.), Cambridge Univ. Press (1995), pp. 384–398.CrossRefGoogle Scholar
  37. [37]
    S. Rushton and A.J. Mautner, The deterministic model of a simple epidemic for more than one community, Biometrika 42 (1955), pp. 126–132.MathSciNetzbMATHGoogle Scholar
  38. [38]
    C.E.G. Smith, Factors in the transmission of virus infections from animals to man, Sei. Basis Med. Annu. Rev. (1964), pp. 125–150.Google Scholar
  39. [39]
    R.K. Watson, On an epidemic in a stratified population, J. Appl. Prob. 9 (1972), pp. 659–666.zbMATHCrossRefGoogle Scholar
  40. [40]
    P. Whittle, The outcome of a stochastic epidemica note on Bailey’s paper, Biometrika 42 (1955), pp. 116–122.MathSciNetzbMATHGoogle Scholar
  41. [41]
    T. Williams, An algebraic proof of the threshold theorem for the general stochastic epidemic (abstract), Adv. Appl. Prob. 3 (1971), p. 223.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2002

Authors and Affiliations

  • Frank G. Ball
    • 1
  • Owen D. Lyne
    • 1
  1. 1.School of Mathematical SciencesUniversity of Nottingham, University ParkNottinghamUK

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