6 Epilogue

Hopefully these practicals have given you some insight into the types of appraoches that we can use in order to fit infectious disease models to real-data, and some understanding of the complexities involved. There are many other techniques available, each with their own advantages and disadvantages. Some key (but not exhaustive) references (with a focus on epidemic modelling) are:

  • Data-augmented MCMC (e.g. Gibson and Renshaw 1998; O’Neill and Roberts 1999; S. Cauchemez and Ferguson 2008; Jewell et al. 2009)
  • Sequential Monte Carlo (Simon Cauchemez et al. 2008)
  • Maximum likelihood via iterated filtering (Ionides, Bretó, and King 2006)
  • Approximate Bayesian Computation (e.g. Toni et al. 2009; McKinley, Cook, and Deardon 2009; Conlan et al. 2012; Brooks Pollock, Roberts, and Keeling 2014)
  • Pseudo-marginal methods (e.g. O’Neill et al. 2000; Beaumont 2003; Andrieu and Roberts 2009; McKinley et al. 2014)
  • Particle MCMC (Andrieu, Doucet, and Holenstein 2010; Drovandi, Pettitt, and McCutchan 2016)
  • Synthetic likelihood (Wood 2010)
  • History Matching and emulation (Andrianakis et al. 2015; McKinley et al. 2018)
  • Bayesian Optimization for Likelihood-Free Inference of simulator-based statistical models (BOLFI) (Gutmann and Corander 2016)

Some other important points to remember:

  • Here we have use small numbers of particles in our ABC-SMC routines so that we can feasibly run the practicals in time. If you use these techniques in your own research make sure that you use a much larger number of particles to produce more accurate estimates.
  • The same goes for the number of iterations for the PMCMC chains. In practice you will need to run for much longer to properly assess convergence and mixing.

References

Andrianakis, Ioannis, Ian Vernon, Nicky McCreesh, Trevelyan J. McKinley, Jeremy E. Oakley, Rebecca N. Nsubuga, Michael Goldstein, and Richard G. White. 2015. “Bayesian History Matching of Complex Infectious Disease Models Using Emulation: A Tutorial and a Case Study on Hiv in Uganda.” PLoS Computational Biology 11 (1): e1003968.

Andrieu, Christophe, Arnaud Doucet, and Roman Holenstein. 2010. “Particle Markov Chain Monte Carlo Methods.” Journal of the Royal Statistical Society, Series B (Methodological) 72 (3): 269–342.

Andrieu, Christophe, and Gareth O. Roberts. 2009. “The Pseudo-Marginal Approach for Efficient Monte Carlo Simulation.” The Annals of Statistics 37 (2): 697–725.

Beaumont, Mark A. 2003. “Estimation of Population Growth and Decline in Genetically Monitored Populations.” Genetics 164: 1139–60.

Brooks Pollock, Ellen, Gareth O. Roberts, and Matt J. Keeling. 2014. “A Dynamic Model of Bovine Tuberculosis Spread and Control in Great Britain.” Nature 511: 228–31. https://doi.org/10.1038/nature13529.

Cauchemez, S., and Neil M. Ferguson. 2008. “Likelihood-Based Estimation of Continuous-Time Epidemic Models from Time-Series Data: Application to Measles Transmission in London.” Journal of the Royal Society Interface 5 (25): 885–97.

Cauchemez, Simon, Alain-Jacques Valleron, Pierre-Yves Boëlle, Antoine Flahault, and Neil M. Ferguson. 2008. “Estimating the Impact of School Closure on Influenza Transmission from Sentinel Data.” Nature 452: 750–55. https://doi.org/10.1038/nature06732.

Conlan, Andrew J. K., Trevelyan J. McKinley, Katerina Karolemeas, Ellen Brooks Pollock, Anthony V. Goodchild, Andrew P. Mitchell, Colin P. D. Birch, Richard S. Clifton-Hadley, and James L. N. Wood. 2012. “Estimating the Hidden Burden of Bovine Tuberculosis in Great Britain.” PLoS Computational Biology 8 (10): e1002730.

Drovandi, Christopher C., Anthony N. Pettitt, and Roy A. McCutchan. 2016. “Exact and Approximate Bayesian Inference for Low Integer-Values Time Series Models with Intractable Likelihoods.” Bayesian Analysis 11 (2): 325–52.

Gibson, Gavin J., and Eric Renshaw. 1998. “Estimating Parameters in Stochastic Compartmental Models Using Markov Chain Methods.” IMA Journal of Mathematics Applied in Medicine and Biology 15: 19–40.

Gutmann, Michael U., and Jukka Corander. 2016. “Bayesian Optimization for Likelihood-Free Inference of Simulator-Based Statistical Models.” Journal of Machine Learning Research 17: 1–47.

Ionides, E.L., C. Bretó, and A.A. King. 2006. “Inference for Nonlinear Dynamical Systems.” Proceedings of the National Academy of Sciences USA 103: 18438–43.

Jewell, Chris P., Theodore Kypraios, Peter Neal, and Gareth O. Roberts. 2009. “Bayesian Analysis for Emerging Infectious Diseases.” Bayesian Analysis 4 (4): 465–96.

McKinley, Trevelyan J., Alex R. Cook, and Robert Deardon. 2009. “Inference in Epidemic Models Without Likelihoods.” The International Journal of Biostatistics 5 (1). https://doi.org/10.2202/1557-4679.1171.

McKinley, Trevelyan J., Joshua V. Ross, Rob Deardon, and Alex R. Cook. 2014. “Simulation-Based Bayesian Inference for Epidemic Models.” Computational Statistics and Data Analysis 71: 434–47.

McKinley, Trevelyan J., Ian Vernon, Ioannis Andrianakis, Nicky McCreesh, Jeremy E. Oakley, Rebecca N. Nsubuga, Michael Goldstein, and Richard G. White. 2018. “Approximate Bayesian Computation and Simulation-Based Inference for Complex Stochastic Epidemic Models.” Statistical Science 33 (1): 4–18. https://doi.org/10.1214/17-STS618.

O’Neill, P.D., D.J. Balding, N.G. Becker, M. Eerola, and D. Mollison. 2000. “Analyses of Infectious Disease Data from Household Outbreaks by Markov Chain Monte Carlo Methods.” Applied Statistics 49: 517–42.

O’Neill, Philip D., and Gareth O. Roberts. 1999. “Bayesian Inference for Partially Observed Stochastic Epidemics.” Journal of the Royal Statistical Society. Series A (General) 162: 121–29.

Toni, Tina, David Welch, Natalja Strelkowa, Andreas Ipsen, and Michael P.H. Strumpf. 2009. “Approximate Bayesian Computation Scheme for Parameter Inference and Model Selection in Dynamical Systems.” Journal of the Royal Society Interface 6: 187–202.

Wood, Simon N. 2010. “Statistical Inference for Noisy Nonlinear Ecological Dynamic Systems.” Nature 466: 1102–4.