Bayesian Analysis

Using Individual-Level Models for Infectious Disease Spread to Model Spatio-Temporal Combustion Dynamics

Irene Vrbik, Rob Deardon, Zeny Feng, Abbie Gardner, and John Braun

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Individual-level models (ILMs), as defined by Deardon et al. (2010), are a class of models originally designed to model the spread of infectious disease. However, they can also be considered as a tool for modelling the spatio-temporal dynamics of fire. We consider the much simplified problem of modelling the combustion dynamics on a piece of wax paper under relatively controlled conditions. The models are fitted in a Bayesian framework using Markov chain Monte Carlo (MCMC) methods. The focus here is on choosing a model that best fits the combustion pattern.

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Bayesian Anal. Volume 7, Number 3 (2012), 615-638.

First available: 28 August 2012

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Vrbik, Irene; Deardon, Rob; Feng, Zeny; Gardner, Abbie; Braun, John. Using Individual-Level Models for Infectious Disease Spread to Model Spatio-Temporal Combustion Dynamics. Bayesian Analysis 7 (2012), no. 3, 615--638. doi:10.1214/12-BA721.

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  • Beer, T. and Enting, I. G. (1990). “Fire spread and percolation modelling.” Mathematical Computer Modelling, 13: 77 – 96.
  • Berjak, S. G. and Hearne, J. W. (2002). “An improved cellular automaton model for simulating fire in a spatially heterogeneous Savanna system.” Ecological Modelling, 148: 133 – 151.
  • Besag, J. (1972). “Nearest-Neighbour Systems and the Auto-Logistic Model for Binary Data.” Journal of the Royal Statistical Society, Series B, 34: 75 – 83.
  • Byram, G. M. (1959). Forest Fire: Control and Use, chapter Combustion of forest fuels, 65 – 89. New York: McGraw Hill. Ed: K. P. Davis.
  • Calder, C. A. (2008). “A dynamic process convolution approach to modeling ambient particulate matter concentrations.” Environmetrics, 19: 39 – 48.
  • Caragea, P. C. and Kaiser, M. S. (2009). “Autologistic Models With Interpretable Parameters.” Journal of Agricultural, Biological, and Environmental Statistics, 14: 281 – 300.
  • Cressie, N. and Johannesson, G. (2008). “Fixed rank kriging for very large spatial data sets.” Journal of the Royal Statistical Society, Series B, 70: 209 – 226.
  • Deardon, R., Brooks, S., Grenfell, B., Keeling, M., Tildesley, M., Savill, N., Shaw, D., and Woolhouse, M. (2010). “Inference for individual-level models of infectious diseases in large populations.” Statistica Sinica, 20: 239–261.
  • Finny, M. (1998). “FARSITE: Fire area simulator–Model development and evaluation.” United States Department of Agriculture Forest Service Research Paper RMRS-RP-4.
  • Gamerman, D. and Lopes, H. (2006). Markov chain Monte Carlo: stochastic simulation for Bayesian inference. Chapman & Hall/CRC Texts in Statistical Science, 2nd edition.
  • Garcia, T., Braun, J., Bryce, R., and Tymstra, C. (2008). “Smoothing and bootstrapping the PROMETHEUS fire growth model.” Environmetrics, 19: 836–848.
  • Gelman, A., Carlin, J. B., Stern, H. S., and Rubin, D. B. (2004). Bayesian Data Analysis. Chapman & Hall, 2nd edition.
  • Gibson, G. J. and Austin, E. J. (1996). “Fitting and testing spatio-temporal stochastic models with application in plant epidemiology.” Plant Pathology, 45: 172–184.
  • Higdon, D. (1998). “A process-convolution approach to modelling temperatures in the North Atlantic Ocean.” Environmental and Ecological Statistics, 5: 173 – 190.
  • McArthur, A. G. (1966). “Weather and grassland fire behaviour.” Leaflet 100, Australian Forestry and Timber Bureau.
  • Neal, P. J. and Roberts, G. O. (2004). “Statistical inference and model selection for the 1861 Hagelloch measles epidemic.” Biostatistics, 5: 249–261.
  • Noble, I. R., Bary, G. A. V., and Gill, A. M. (1980). “McArthur’s fire-danger meters expressed as equations.” Australian Journal of Ecology, 5: 201 – 203.
  • Ntaimo, L., Zeigler, B. P., Vasconcelos, M. J., and Khargharia, B. (2004). “Forest fire spread and suppression in DEVS.” Simulation, 80: 479 – 500.
  • Perry, G. L. W. (1998). “Current approaches to modelling the spread of wildland fire: a review.” Progress in Physical Geography, 22: 222 – 245.
  • Podur, J., Martell, D. L., and Knight, K. (2002). “Statistical quality control analysis of forest fire activity in Canada.” Canadian Journal of Forest Research, 32: 195–205.
  • R Development Core Team (2009). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing.
  • Rothermel, R. C. (1972). “A mathematical model for predicting fire spread in wildland fuels.” Research Paper INT - 115, USDA Forest Service.
  • Spiegelhalter, D., Best, N., Carlin, B., and van der Linde, A. (2002). “Bayesian measures of model complexity and fit.” Journal of the Royal Statistical Society, Series B, 64: 583–639.
  • Tymstra, C., Bryce, R. W., Wotton, B. M., and Armitage, O. B. (2010). “Development and structure of Prometheus: the Canadian wildland fire growth simulation model.” National Resources Canada, Canadian Forest Services Information Report NOR-X-417.
  • Waltman, P. and Hoppensteadt, F. (1970). “A Problem in the Theory of Epidemics I.” Mathematical Biosciences, 9: 71 – 91.
  • — (1971). “A Problem in the Theory of Epidemics II.” Mathematical Biosciences, 12: 133 – 145.
  • Zheng, Y. and Zhu, J. (2008). “Markov chain Monte Carlo for a Spatial-Temporal Autologistic Regression Model.” Journal of Computational and Graphical Statistics, 17: 123 – 137.