Importance sampling for partially observed temporal epidemic models

  • Andrew J. Black


We present an importance sampling algorithm that can produce realisations of Markovian epidemic models that exactly match observations, taken to be the number of a single event type over a period of time. The importance sampling can be used to construct an efficient particle filter that targets the states of a system and hence estimate the likelihood to perform Bayesian inference. When used in a particle marginal Metropolis Hastings scheme, the importance sampling provides a large speed-up in terms of the effective sample size per unit of computational time, compared to simple bootstrap sampling. The algorithm is general, with minimal restrictions, and we show how it can be applied to any continuous-time Markov chain where we wish to exactly match the number of a single event type over a period of time.


Importance sampling Markov chain Epidemic modelling Particle filter 



This research was supported by an ARC DECRA fellowship (DE160100690). AJB also acknowledges support from both the ARC Centre of Excellence for Mathematical and Statistical Frontiers (CoE ACEMS), and the Australian Government NHMRC Centre for Research Excellence in Policy Relevant Infectious diseases Simulation and Mathematical Modelling (CRE PRISM\(^2\)). Supercomputing resources were provided by the Phoenix HPC service at the University of Adelaide. AJB would also like to thank Joshua Ross and James Walker for comments on an earlier draft of the manuscript.

Supplementary material

11222_2018_9827_MOESM1_ESM.pdf (72 kb)
Supplementary material 1 (pdf 72 KB)


  1. Aho, A.V., Ullman, J.D.: Foundations of Computer Science. W. H. Freeman and Company, New York (1995)zbMATHGoogle Scholar
  2. Anderson, D.F.: A modified next reaction method for simulating chemical systems with time dependent propensities and delays. J. Chem. Phys. 127, 214107 (2007)CrossRefGoogle Scholar
  3. Andrieu, C., Doucet, A., Holenstein, R.: Particle Markov chain Monte Carlo methods (with discussion). J. R. Stat. Soc. B 72, 269–342 (2010)MathSciNetCrossRefGoogle Scholar
  4. Black, A.J., Geard, N., McCaw, J.M., McVernon, J., Ross, J.V.: Characterising pandemic severity and transmissibility from data collected during first few hundred studies. Epidemics 19, 61–73 (2017)CrossRefGoogle Scholar
  5. Black, A.J., McKane, A.J.: Stochastic formulation of ecological models and their applications. Trends Ecol. Evol. 27, 337–345 (2012)CrossRefGoogle Scholar
  6. Black, A.J., Ross, J.V.: Estimating a Markovian epidemic model using household serial interval data from the early phase of an epidemic. PLoS ONE 8, e73420 (2013)CrossRefGoogle Scholar
  7. Black, A.J., Ross, J.V.: Computation of epidemic final size distributions. J. Theor. Biol. 367, 159–165 (2015)CrossRefGoogle Scholar
  8. David, H.A., Nagaraja, H.N.: Order Statistics, 3rd edn. Wiley, New York (2005)Google Scholar
  9. Del Moral, P., Jasra, P., Lee, A., Yau, C., Zhang, X.: The alive particle filter and its use in particle Markov chain Monte Carlo. Stoch. Anal. Appl. 33, 943–974 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Doucet, A., de Freitas, N., Gordon, N.J. (eds.): Sequential Monte Carlo Methods in Practice. Springer, New York (2001)Google Scholar
  11. Doucet, A., Johansen, A.M.: A tutorial on particle filtering and smoothing: fifteen years later. Handb. Nonlinear Filter. 12(656–704), 3 (2009)zbMATHGoogle Scholar
  12. Doucet, A., Pitt, M.K., Deligiannidis, G., Kohn, R.: Efficient implementation of Markov chain Monte Carlo when using an unbiased likelihood estimator. Biometrika 102, 295–313 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  13. Drovandi, C.C.: Pseudo-marginal algorithms with multiple CPUs (2014).
  14. Drovandi, C.C., McCutchan, R.A.: Alive SMC\(^2\): Bayesian model selection for low-count time series models with intractable likelihoods. Biometrics 72, 344–353 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. Gibson, G.J., Renshaw, E.: Estimating parameters in stochastic compartmental models using Markov chain methods. Math. Med. Biol. 15, 19–40 (1998). CrossRefzbMATHGoogle Scholar
  16. Gibson, M.A., Bruck, J.: Efficient exact stochastic simulation of chemical systems with many species and many channels. J. Phys. Chem. A 104, 1876–1889 (2000)CrossRefGoogle Scholar
  17. Gillespie, D.T.: A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comput. Phys. 22, 403–434 (1976)MathSciNetCrossRefGoogle Scholar
  18. Golightly, A., Kypraios, T.: Efficient SMC\(^2\) schemes for stochastic kinetic models. Stat. Comput. (2017).
  19. Golightly, A., Wilkinson, D.J.: Bayesian parameter inference for stochastic biochemical network models using particle Markov chain Monte Carlo. Interface Focus 1, 807–820 (2011). CrossRefGoogle Scholar
  20. Gordon, N.J., Salmond, D.J., Smith, A.F.M.: Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEE Proc. F 140, 107–113 (1993)Google Scholar
  21. Jenkinson, G., Goutsias, J.: Numerical integration of the master equation in some models of stochastic epidemiology. PLoS ONE 7, e36160 (2012)CrossRefGoogle Scholar
  22. Jewell, C.P., Kypraios, T., Neal, P., Roberts, G.O.: Bayesian analysis for emerging infectious diseases. Bayesian Anal. 4, 465–496 (2009). MathSciNetCrossRefzbMATHGoogle Scholar
  23. Keeling, M.J., Rohani, P.: Modeling Infectious Diseases in Humans and Animals. Princeton University Press, Princeton, NJ (2007)zbMATHGoogle Scholar
  24. Knuth, D.: The Art of Computer Programming, vol. 1. Addison-Wesley, Reading, MA (1997)zbMATHGoogle Scholar
  25. Kroese, D.P., Taimre, T., Botev, Z.I.: Handbook of Monte Carlo Methods. Wiley, New York (2011)CrossRefzbMATHGoogle Scholar
  26. Lau, M.S.Y., Cowling, B.J., Cook, A.R., Riley, S.: Inferring influenza dynamics and control in households. Proc. Natl. Acad. Sci. 112, 9094–9099 (2015)CrossRefGoogle Scholar
  27. McKinley, T.J., Ross, J.V., Deardon, R., Cook, A.R.: Simulation-based Bayesian inference for epidemic models. Comput. Stat. Data Anal. 71, 434–447 (2014). MathSciNetCrossRefGoogle Scholar
  28. O’Neill, P.D., Roberts, G.O.: Bayesian inference for partially observed stochastic epidemics. J. R. Stat. Soc. A 162, 121–130 (1999). CrossRefGoogle Scholar
  29. Pitt, M.K., Silva, R., Giordani, P., Kohn, R.: On some properties of Markov chain Monte Carlo simulation methods based on the particle filter. J. Econom. 171, 134–151 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  30. Pooley, C.M., Bishop, S.C., Marion, G.: Using model-based proposals for fast parameter inference on discrete state space, continuous-time Markov processes. J. R. Soc. Interface 12, 20150225 (2015)CrossRefGoogle Scholar
  31. Regan, D.G., Wood, J.G., Benevent, C., et al.: Estimating the critical immunity threshold for preventing hepatitis a outbreaks in men who have sex with men. Epidemiol. Infect. 144, 1528–1537 (2016). CrossRefGoogle Scholar
  32. Roh, M.K., Gillespie, D.T., Petzold, L.R.: State-dependent biasing method for importance sampling in the weighted stochastic simulation algorithm. J. Chem. Phys. 133, 174106 (2010)CrossRefGoogle Scholar
  33. Sherlock, C., Thiery, A.H., Roberts, G.O., Rosenthal, J.S.: On the efficiency of pseudo-marginal random walk metropolis algorithms. Ann. Stat. 43, 238–275 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  34. Stockdale, J.E., Kypraios, T., O’Neill, P.D.: Modelling and bayesian analysis of the Abakaliki smallpox data. Epidemics 19, 13–23 (2017). CrossRefGoogle Scholar
  35. van Kampen, N.G.: Stochastic Processes in Physics and Chemistry. Elsevier, Amsterdam (1992)zbMATHGoogle Scholar
  36. Walker, J.N., Ross, J.V., Black, A.J.: Inference of epidemiological parameters from household stratified data. PLoS ONE 12, e0185910 (2017)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of AdelaideAdelaideAustralia
  2. 2.ACEMS, School of Mathematical SciencesUniversity of AdelaideAdelaideAustralia

Personalised recommendations