Statistics and Computing

, Volume 16, Issue 3, pp 309–317 | Cite as

Computation of final outcome probabilities for the generalised stochastic epidemic

  • Nikolaos DemirisEmail author
  • Philip D. O’Neill


This paper is concerned with methods for the numerical calculation of the final outcome distribution for a well-known stochastic epidemic model in a closed population. The model is of the SIR (Susceptible→Infected→ Removed) type, and the infectious period can have any specified distribution. The final outcome distribution is specified by the solution of a triangular system of linear equations, but the form of the distribution leads to inherent numerical problems in the solution. Here we employ multiple precision arithmetic to surmount these problems. As applications of our methodology, we assess the accuracy of two approximations that are frequently used in practice, namely an approximation for the probability of an epidemic occurring, and a Gaussian approximation to the final number infected in the event of an outbreak. We also present an example of Bayesian inference for the epidemic threshold parameter.


Stochastic epidemic models Markov chain Monte Carlo Methods Limit theorems Multiple precision arithmetic Final size 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Addy C. L., Longini I. M. and Haber M. 1991. A generalized stochastic model for the analysis of infectious disease final size data. Biometrics 47: 961–974.Google Scholar
  2. Andersson H. and Britton T. 2000. Stochastic Epidemic Models and Their Statistical Analysis, Springer Lecture Notes in Statistics, New York.Google Scholar
  3. Bailey N. T. J. 1975. The Mathematical Theory of Infectious Diseases and its Applications, 2nd edn. London: Griffin.Google Scholar
  4. Ball F. G. 1986. A unified approach to the distribution of total size and total area under the trajectory of infectives in epidemic models. Adv. Appl. Probab. 18: 289–310.zbMATHCrossRefGoogle Scholar
  5. Ball F. G. and Donnelly P. 1995. Strong approximations for epidemic models. Stoch. Proc. Appl. 55: 1–21.MathSciNetCrossRefGoogle Scholar
  6. Ball F. G., Mollison D. and Scalia-Tomba G. 1997. Epidemics with two levels of mixing. Ann. Appl. Probab. 7: 46–89.MathSciNetCrossRefGoogle Scholar
  7. Bartlett M. S. 1949. Some evolutionary stochastic processes. J. Roy. Statist. Soc. B, 11: 211–229.zbMATHMathSciNetGoogle Scholar
  8. Becker N. G. 1989. Analysis of Infectious Disease Data, Chapman and Hall, London.Google Scholar
  9. Becker N. G. and Britton T. 2001. Design issues for studies of infectious diseases. J. Stat. Plan. Inf. 96: 41–66.CrossRefGoogle Scholar
  10. Becker N. G. and Dietz K. 1995. The effect of the household distribution on transmission and control of highly infectious diseases. Math. Biosci. 127: 207–219.CrossRefGoogle Scholar
  11. Brent R. P. 1978. A Fortran multiple precision arithmetic package. ACM Trans. Math. Soft. 4: 57–70.CrossRefGoogle Scholar
  12. Demiris N. and O’Neill P. D. 2005a. Bayesian inference for epidemics with two levels of mixing. Scand. J. Stat 32: 265–280.MathSciNetCrossRefGoogle Scholar
  13. Demiris N. and O’Neill P. D. 2005b. Bayesian inference for stochastic multitype epidemics in structured populations via random graphs. J. Roy. Statist. Soc. B 67: 731–745.MathSciNetCrossRefGoogle Scholar
  14. Dietz K. 1993. The estimation of the basic reproduction number for infectious diseases. Statistical methods in medical research 2: 23–41.Google Scholar
  15. Gilks W. R., Richardson S. and Spiegelhalter D. J. 1996. Markov Chain Monte Carlo in Practice. London: Chapman and Hall.Google Scholar
  16. Lefévre C. and Utev S. 1999. Branching approximation for the collective epidemic model. Methodology and Computing in Applied Probability 1: 211–228.MathSciNetCrossRefGoogle Scholar
  17. Rida W. N. 1991. Asymptotic properties of some estimators for the infection rate in the general stochastic epidemic. J. R. Statist. Soc. B 53: 269–283.zbMATHMathSciNetGoogle Scholar
  18. Smith D. M. 1991. A Fortran Package For Floating-Point Multiple-Precision Arithmetic. ACM Trans. Math. Soft. 17: 273–283.zbMATHCrossRefGoogle Scholar
  19. Whittle P. 1955. The outcome of a stochastic epidemic—a note on Bailey’s paper. Biometrika 42: 116–122.zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  1. 1.Medical Research Council Biostatistics UnitCambridgeUK
  2. 2.University of NottinghamUK

Personalised recommendations