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Contact network uncertainty in individual level models of infectious disease transmission

Waleed Almutiry and Rob Deardon


Infectious disease transmission between individuals in a heterogeneous population is often best modelled through a contact network. This contact network can be spatial in nature, with connections between individuals closer in space being more likely. However, contact network data are often unobserved. Here, we consider the fit of an individual level model containing a spatially-based contact network that is either entirely, or partially, unobserved within a Bayesian framework, using data augmented Markov chain Monte Carlo (MCMC). We also incorporate the uncertainty about event history in the disease data. We also examine the performance of the data augmented MCMC analysis in the presence or absence of contact network observational models based upon either knowledge about the degree distribution or the total number of connections in the network. We find that the latter tend to provide better estimates of the model parameters and the underlying contact network.

Corresponding author: Waleed Almutiry, Mathematics, Arts and Science College in Ar Rass, Qassim University, Buraidah, Saudi Arabia, E-mail:

Funding source: Ontario Ministry of Agriculture, Food and Rural Affairs

Funding source: Canada Foundation for Innovation

Funding source: Qassim University

Funding source: Natural Sciences and Engineering Research Council of Canada


This work was funded by the Ontario Ministry of Agriculture, Food and Rural Affairs (OMAFRA), the Natural Sciences and Engineering Research Council of Canada (NSERC), Qassim University through the Saudi Arabian Cultural Bureau in Canada, and was carried out on equipment funded by the Canada Foundation for Innovation – Leading Edge Fund project “Centre for Public Health and Zoonoses” at the University of Guelph.

  1. Research funding: None declared.

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Competing interests: Authors state no conflict of interest.


Barthélemy, M., A. Barrat, R. Pastor-Satorras, and A. Vespignani. 2005. “Dynamical Patterns of Epidemic Outbreaks in Complex Heterogeneous Networks.” Journal of Theoretical Biology 235 (2): 275–88, in Google Scholar

Beaumont, M. A., W. Zhang, and D. J. Balding. 2002. “Approximate Bayesian Computation in Population Genetics.” Genetics 162 (4): 2025–35.Search in Google Scholar

Bifolchi, N., R. Deardon, and Z. Feng. 2013. “Spatial Approximations of Network-based Individual Level Infectious Disease Models.” Spatial and Spatio-temporal Epidemiology 6: 59–70, in Google Scholar

Britton, T., and P. D. O’Neill. 2002. “Bayesian Inference for Stochastic Epidemics in Populations with Random Social Structure.” Scandinavian Journal of Statistics 29 (3): 375–90, in Google Scholar

Danon, L., A. P. Ford, T. House, C. P. Jewell, M. J. Keeling, G. O. Roberts, J. V. Ross, and M. C. Vernon. 2011. “Networks and the Epidemiology of Infectious Disease.” Interdisciplinary Perspectives on Infectious Diseases 2011: 3–30, in Google Scholar

Deardon, R., S. P. Brooks, B. T. Grenfell, M. J. Keeling, M. J. Tildesley, N. J. Savill, D. J. Shaw, and M. E. Woolhouse. 2010. “Inference for Individual-level Models of Infectious Diseases in Large Populations.” Statistica Sinica 20 (1): 239.Search in Google Scholar

Deeth, L. E. and R. Deardon. 2013. “Latent Conditional Individual-level Models for Infectious Disease Modeling.” International Journal of Biostatistics 9 (1): 75–93, in Google Scholar

Gamerman, D., and H. F. Lopes. 2006. Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference. New York: CRC Press.Search in Google Scholar

Gelman, A., J. B. Carlin, H. S. Stern, and D. B. Rubin. 2004. Bayesian Data Analysis. New York: Chapman & Hall/CRC.Search in Google Scholar

Gog, J. R., S. Ballesteros, C. Viboud, L. Simonsen, O. N. Bjornstad, J. Shaman, D. L. Chao, F. Khan, and B. T. Grenfell. 2014. “Spatial Transmission of 2009 Pandemic Influenza in the US.” PLoS Computational Biology 10 (6): e1003635, in Google Scholar

Groendyke, C., D. Welch, and D. R. Hunter. 2012. “A Network-based Analysis of the 1861 Hagelloch Measles Data.” Biometrics 68 (3): 755–65, in Google Scholar

Jewell, C. P., T. Kypraios, P. Neal, and G. O. Roberts. 2009. “Bayesian Analysis for Emerging Infectious Diseases.” Bayesian Analysis 4 (3): 465–96, in Google Scholar

Keeling, M. J., and K. T. Eames. 2005. “Networks and Epidemic Models.” Journal of the Royal Society Interface 2 (4): 295–307, in Google Scholar

Keeling, M. J., M. E. Woolhouse, D. J. Shaw, L. Matthews, M. Chase-Topping, D. T. Haydon, S. J. Cornell, J. Kappey, J. Wilesmith, and B. T. Grenfell. 2001. “Dynamics of the 2001 UK Foot and Mouth Epidemic: Stochastic Dispersal in a Heterogeneous Landscape.” Science 294 (5543): 813–7, in Google Scholar

Malik, R., R. Deardon, and G. P. Kwong. 2016. “Parameterizing Spatial Models of Infectious Disease Transmission that Incorporate Infection Time Uncertainty Using Sampling-based Likelihood Approximations.” PLoS One 11 (1): e0146253, in Google Scholar

McKinley, T. J., A. R. Cook, and R. Deardon. 2009. “Inference in Epidemic Models without Likelihoods.” The International Journal of Biostatistics 5 (1), in Google Scholar

Meyers, L. A., M. Newman, and B. Pourbohloul. 2006. “Predicting Epidemics on Directed Contact Networks.” Journal of Theoretical Biology 240 (3): 400–18, in Google Scholar

Neal, P., and G. Roberts. 2005. “A Case Study in Non-centering for Data Augmentation: Stochastic Epidemics.” Statistics and Computing 15 (4): 315–27, in Google Scholar

Pokharel, G., and R. Deardon. 2014. “Supervised Learning and Prediction of Spatial Epidemics.” Spatial and Spatio-temporal Epidemiology 11: 59–77, in Google Scholar

Pokharel, G., and R. Deardon. 2016. “Gaussian Process Emulators for Spatial Individual-level Models of Infectious Disease.” Canadian Journal of Statistics 44 (4): 480–501, in Google Scholar

Ster, I. C., and N. M. Ferguson. 2007. “Transmission Parameters of the 2001 Foot and Mouth Epidemic in Great Britain.” PLoS One 2 (6): e502.Search in Google Scholar

Toni, T., D. Welch, N. Strelkowa, A. Ipsen, and M. P. Stumpf. 2009. “Approximate Bayesian Computation Scheme for Parameter Inference and Model Selection in Dynamical Systems.” Journal of the Royal Society Interface 6 (31): 187–202, in Google Scholar

Volz, E. 2008. “SIR Dynamics in Random Networks with Heterogeneous Connectivity.” Journal of Mathematical Biology 56 (3): 293–310, in Google Scholar

Supplementary Material

Full description and results of the extra analysis of the two larger epidemics, and tables of the posterior means and 95% credible intervals of the model parameters for the known, partial unknown, and complete unknown contact network analyses of both intense and sparse network data sets of population sizes = 25 and 50 individuals.

The online version of this article offers supplementary material (

Received: 2019-06-02
Accepted: 2020-11-20
Published Online: 2021-01-08

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