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Assessing the Impact of Intervention Delays on Stochastic Epidemics


A stochastic model of disease transmission among a population partitioned into groups is defined. The model is of SEIR (Susceptible-Exposed-Infective-Removed) type and features intervention in response to the progress of the disease, and moreover includes a random delay before the intervention occurs. A threshold parameter for the model, which can be used to assess the efficacy of the intervention, is defined. The threshold parameter can be calculated either in closed form or via recursion, for a number of different choices of exposed, infectious and delay period distributions, both for the epidemic model itself and also a large-group approximation. In particular both constant and Erlang-distributed delay periods are considered. Sufficient conditions under which a constant delay gives the least effective intervention are presented. For a given mean delay, it is shown that the two-point delay distribution provides the optimal intervention.


  • Andersson H, Britton T (2000) Stochastic epidemic models and their statistical analysis. Springer Verlag, New York

    Book  MATH  Google Scholar 

  • Asmussen S (2003) Applied probability and queues, 2nd edn. Springer, New York

    MATH  Google Scholar 

  • Ball FG, Mollison D, Scalia-Tomba, G (1997) Epidemics with two levels of mixing. Ann Appl Probab 7:46–89

    MathSciNet  Article  MATH  Google Scholar 

  • Ball FG, Milne RK, Yeo GF (1994) Continuous-time Markov chains in a random environment, with applications to ion channel modelling. Adv Appl Probab 26:919–946

    MathSciNet  Article  MATH  Google Scholar 

  • Ball FG, O’Neill PD, Pike J (2007) Stochastic epidemic models in structured populations featuring dynamic vaccination and isolation. J Appl Probab 44:571–585

    MathSciNet  Article  MATH  Google Scholar 

  • Bhatt UN (1972) Elements of applied stochastic processes. John Wiley and Sons, New York

    Google Scholar 

  • Daley DJ (1990) The size of epidemics with variable infectious periods. Technical report SMS–012–90, Statistics Research Section, School of Mathematical Sciences, Australian National University

  • Eubank S, Guclu H, Kumar VSA, Marathe MV, Srinivasan A, Toroczkai Z, Wang N (2004) Modelling disease outbreaks in realistic social networks. Nature 429:180–184

    Article  Google Scholar 

  • Feller W (1971) An introduction to probability theory and its applications, vol 1, 3rd edn. John Wiley and Sons, New York

    Google Scholar 

  • Halloran ME, Longini IM, Nizam A, Yang Y (2002) Containing bioterrorist smallpox. Science 298:1428–1432

    Article  Google Scholar 

  • Isham V (1993) Stochastic models for epidemics with special reference to AIDS. Ann Appl Probab 3:1–27

    MathSciNet  Article  MATH  Google Scholar 

  • Isham V, Medley G (eds) (1996) Models for infectious human diseases: their structure and relation to data. Cambridge University Press

  • Jagers P (1975) Branching processes with biological applications. John Wiley and Sons, London

    MATH  Google Scholar 

  • Kaplan EH, Craft DL, Wein LM (2002) Emergency response to a smallpox attack: the case for mass vaccination. Proc Natl Acad Sci 99:10935–10940

    Article  Google Scholar 

  • Keeling M, Woolhouse MEJ, Shaw DJ, Matthews L, Chase-Topping M, Haydon DT, Cornell SJ, Kappey J, Wilesmith J, Grenfell BT (2001) Dynamics of the 2001 UK foot and mouth epidemic: stochastic dispersal in a heterogeneous landscape. Science 294:813–817

    Article  Google Scholar 

  • Kendall D (1948) On the generalized “birth-and-death” process. Ann Math Stat 19:1–15

    MathSciNet  Article  Google Scholar 

  • Lefèvre CP, Picard P (1993) An unusual stochastic order relation with some applications in sampling and epidemic theory. Adv Appl Probab 25:63–81

    Article  MATH  Google Scholar 

  • Longini IM, Halloran ME, Nizam A, Yang Y (2004) Containing pandemic influenza with antiviral agents. Am J Epidemiol 159:623–633

    Article  Google Scholar 

  • Spencer SEF (2007) Stochastic epidemic models for emerging diseases. PhD Thesis, University of Nottingham School of Mathematical Sciences

  • Tildesley MJ, Savill NJ, Shaw DJ, Deardon R, Brooks SP, Woolhouse, MEJ, Grenfell BT, Keeling MJ (2006) Optimal reactive vaccination strategies for a foot-and-mouth outbreak in the UK. Nature 440:83–86

    Article  Google Scholar 

  • Watson RK (1980) A useful random time-scale transformation for the standard epidemic model. J Appl Probab 17:324–332

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Simon Edward Frank Spencer.

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Spencer, S.E.F., O’Neill, P.D. Assessing the Impact of Intervention Delays on Stochastic Epidemics. Methodol Comput Appl Probab 15, 803–820 (2013).

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  • Stochastic epidemic
  • Emerging disease
  • Vaccination
  • Intervention delay
  • Branching process
  • Reproduction number

AMS 2010 Subject Classifications

  • 92D30
  • 60K99