Bayesian Analysis

Bayesian estimation of the basic reproduction number in stochastic epidemic models

Damian Clancy and Philip D. O'Neill

Full-text: Open access


In recent years there has been considerable activity in the development and application of Bayesian inferential methods for infectious disease data using stochastic epidemic models. Most of this activity has employed computationally intensive approaches such as Markov chain Monte Carlo methods. In contrast, here we address fundamental questions for Bayesian inference in the setting of the standard SIR (Susceptible-Infective-Removed) epidemic model via simple methods. Our main focus is on the basic reproduction number, a quantity of central importance in mathematical epidemic theory, whose value essentially dictates whether or not a large epidemic outbreak can occur. We specifically consider two SIR models routinely employed in the literature, namely the model with exponentially distributed infectious periods, and the model with fixed length infectious periods. It is assumed that an epidemic outbreak is observed through time. Given complete observation of the epidemic, we derive explicit expressions for the posterior densities of the model parameters and the basic reproduction number. For partial observation of the epidemic, when the entire infection process is unobserved, we derive conservative bounds for quantities such as the mean of the basic reproduction number and the probability that a major epidemic outbreak will occur. If the time at which the epidemic started is observed, then linear programming methods can be used to derive suitable bounds for the mean of the basic reproduction number and similar quantities. Numerical examples are used to illustrate the practical consequences of our findings. In addition, we also examine the implications of commonly-used prior distributions on the basic model parameters as regards inference for the basic reproduction number.

Article information

Bayesian Anal., Volume 3, Number 4 (2008), 737-757.

First available in Project Euclid: 22 June 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Basic reproduction number Bayesian inference; Epidemics Linear programming Stochastic epidemic models


Clancy, Damian; O'Neill, Philip D. Bayesian estimation of the basic reproduction number in stochastic epidemic models. Bayesian Anal. 3 (2008), no. 4, 737--757. doi:10.1214/08-BA328.

Export citation


  • Andersson, H. and Britton, T. (2000). Stochastic Epidemic Models and their Statistical Analysis. Lecture Notes in Statistics 151. New York: Springer-Verlag.
  • Auranen, K., Arjas, E., Leino, T. and Takala, A. K. (2000). “Transmission of Pneumococcal carriage in families: a latent Markov process model for binary longitudinal data.” Journal of the American Statistical Association, 95: 1044–1053.
  • Bailey, N.T.J. (1975). The Mathematical Theory of Infectious Diseases and its Applications (2nd ed). London: Griffin.
  • Becker, N. G. (1989). Analysis of Infectious Disease Data. London: Chapman and Hall.
  • Bhoj, D. S. and Schiefermayr, K. (2001). “Approximations to the distribution of weighted combination of independent probabilities.” Journal of Statistical Computation and Simulation, 68: 153–159.
  • Boys, R. J. and Giles, P. R. (2007). “Bayesian inference for stochastic epidemic models with time-inhomogeneous removal rates.” Mathematical Biology, 55: 223–247.
  • Britton, T. and O'Neill, P. D. (2002). “Bayesian inference for stochastic epidemics in populations with random social structure.” Scandinavian Journal of Statistics, 29: 375–390.
  • Cauchemez, S., Carrat, F., Viboud, C., Valleron, A. J. and Böelle, P. Y. (2004). “A Bayesian MCMC approach to study transmission of influenza: application to household longitudinal data.” Statistics in Medicine, 23: 3469–3487.
  • Charnes, A. and Cooper, W.W. (1962). “Programming with linear fractional functionals.” Naval Research Logistics Quarterly, 9: 181–186.
  • Chowel, G., Hengartner, N. W., Castillo-Chavez, C., Fenimore, P. W. and Hyman, J. M. (2004). “The basic reproduction number of Ebola and the effects of public health measures: the cases of Congo and Uganda.” Journal of Theoretical Biology, 229: 119–126.
  • Demiris, N. and O'Neill, P. D. (2005). “Bayesian inference for stochastic multitype epidemics in structured populations via random graphs.” Journal of the Royal Statistical Society, Series B, 67: 731–746.
  • Dietz, K. (1993). “The estimation of the basic reproduction number for infectious diseases.” Statistical methods in medical research, 2: 23–41.
  • Gibson, G. J. and Renshaw, E. (1998). “Estimating parameters in stochastic compartmental models using Markov Chain methods.” IMA Journal of Mathematics Applied in Medicine and Biology, 15: 19–40.
  • Höhle, M., Jørgensen, E. and O'Neill, P. D. (2005). “Inference in disease transmission experiments by using stochastic epidemic models.” Applied Statistics, 54: 349–366.
  • Ibaraki, T. (1981). “Solving mathematical programming problems with fractional objective functions.” In Generalized Concavity in Optimization and Economics, eds. S. Schaible and W.T. Ziemba, 441–472. New York: Academic Press.
  • Kijima, M. and Seneta, E. (1991). “Some results for quasistationary distributions of birth-death processes.” Journal of Applied Probability, 28: 503–511.
  • Lekone, P. E. and Finkenstädt, B. F. (2006). “Statistical inference in a stochastic epidemic SEIR model with control intervention: Ebola as a case study.” Biometrics 62: 1170–1177.
  • Li, N., Qian, G., and Huggins, R. (2002). “Analysis of between-household heterogeneity in disease transmission from data on outbreak sizes.” Australia and New Zealand Journal of Statistics 44: 401–411.
  • Martos, B. (1965). “The direct power of adjacent vertex programming methods.” Management Science, 12: 241–252.
  • McBryde, E.S., Gibson, G., Pettitt, A.N., Zhang, Y., Zhao, B. and McElwain, D.L.S. (2006). “Bayesian modelling of the severe acute respiratory syndrome epidemic.” Bulletin of Mathematical Biology, 68: 889–917.
  • Neal, P. and Roberts, G. O. (2005). “A case study in non-centering for data augmentation: stochastic epidemics.” Statistics and Computing, 15: 315–327.
  • O'Neill, P. D., Balding, D. J., Becker, N. G., Eerola, M. and Mollison, D. (2000). “Analyses of infectious disease data from household outbreaks by Markov Chain Monte Carlo methods.” Applied Statistics, 49: 517–542.
  • O'Neill, P. D. and Marks, P. J. (2005). “Bayesian model choice and infection route modelling in an outbreak of Norovirus.” Statistics in Medicine, 24: 2011–2024.
  • O'Neill, P. D. and Roberts, G. O. (1999). “Bayesian inference for partially observed stochastic epidemics.” Journal of the Royal Statistical Society, Series A, 162: 121–129.
  • Streftaris, G. and Gibson, G.J. (2004). “Bayesian inference for stochastic epidemics in closed populations.” Statistical Modelling, 4: 63–75.