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A stochastic percolation model for disease spread in crops

A stochastic percolation model for disease spread in crops. Alex Cook (BioSS and Heriot-Watt University) Supervised by: Glenn Marion, Gavin Gibson. Experiments. Hosts: radish Pathogen: R. solani fungus Disease: damping-off. Experiments. Hosts: radish

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A stochastic percolation model for disease spread in crops

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  1. A stochastic percolation model for disease spread in crops Alex Cook (BioSS and Heriot-Watt University) Supervised by: Glenn Marion, Gavin Gibson

  2. Experiments • Hosts: radish • Pathogen: R. solani fungus • Disease: damping-off

  3. Experiments • Hosts: radish • Pathogen: R. solani fungus • Disease: damping-off • Modi operandi: spreads from dead plant material or infected neighbouring plants

  4. Experiments • Hosts: radish • Pathogen: R. solani fungus • Disease: damping-off • Modi operandi: spreads from dead plant material or infected neighbouring plants fungal mycelium infected host plant Picture adapted from Bailey et al (2000), New Phytology 146, pg. 535. 20mm

  5. Experiments • Hosts: radish • Pathogen: R. solani fungus • Disease: damping-off • Modi operandi: spreads from dead plant material or infected neighbouring plants • 2 treatments (high/low inoculum)  13 replicates  414 seedlings planted » 10 000 observations of day of first symptoms (4,…,21,21+)

  6. Day 4 See Otten et al (2003), Ecology 84, pg.3232

  7. Day 5 See Otten et al (2003), Ecology 84, pg.3232

  8. Day 6 See Otten et al (2003), Ecology 84, pg.3232

  9. Day 7 See Otten et al (2003), Ecology 84, pg.3232

  10. Day 8 See Otten et al (2003), Ecology 84, pg.3232

  11. Day 9 See Otten et al (2003), Ecology 84, pg.3232

  12. Day 10 See Otten et al (2003), Ecology 84, pg.3232

  13. Day 11 See Otten et al (2003), Ecology 84, pg.3232

  14. Day 12 See Otten et al (2003), Ecology 84, pg.3232

  15. Day 13 See Otten et al (2003), Ecology 84, pg.3232

  16. Day 14 See Otten et al (2003), Ecology 84, pg.3232

  17. Day 15 See Otten et al (2003), Ecology 84, pg.3232

  18. Day 16 See Otten et al (2003), Ecology 84, pg.3232

  19. Day 17 See Otten et al (2003), Ecology 84, pg.3232

  20. Day 18 See Otten et al (2003), Ecology 84, pg.3232

  21. Day 19 See Otten et al (2003), Ecology 84, pg.3232

  22. Day 20 See Otten et al (2003), Ecology 84, pg.3232

  23. Day 21 See Otten et al (2003), Ecology 84, pg.3232

  24. Model • Primary infections at rate α(t) - from inoculum • Secondary infections at rate β(t) - from neighbour α(t) β (t) β(t)

  25. Model • Primary rate α(t) = a • Secondary rate β(t) = b0 exp{ – b1 log2(tdonor/b2)} β(t) β(t) t t

  26. Model • Primary rate α(t) = a • Secondary rate β(t) = b0 exp{ – b1 log2(tdonor/b2)} • Data not entirely consistent with this model! • Some non-connectivity (<5%) • Subsequent infection of intermediate hosts

  27. Model • Primary rate α(t) = a • Secondary rate β(t) = b0 exp{ – b1 log2(tdonor/b2)} • Distinguish infection and symptoms • Infection as above, but unseen • After infection, development of symptoms at rate δ(t) = d α susceptible infectious symptomatic and infectious β δ

  28. Model • We therefore want to estimate 5 parameters: • a primary rate of infection • b0, b1, b2govern secondary rate of infection • d rate of symptom development • Call these θ

  29. Parameter estimation • Otten et al (2003) use least squares • identify primary, secondary rates? • requires assumptions for β(t) • Gibson et al (submitted) take Bayesian approach & use McMC • their model unable to deal with non-connectivity • Our approach also uses McMC • non-connectivity no problem See Otten et al (2003), Ecology 84, pg.3232

  30. Markov chain Monte Carlo Want to estimate θ • Can derive joint posterior density for θ • Cannot analyse numerically • Draw a sample from posterior, treating θ and t as random • Use sample to make inference on θ McMC: e.g. Gilks et al (1996) Markov chain Monte Carlo in Practice

  31. Markov chain Monte Carlo

  32. Results

  33. Results

  34. Results

  35. Results

  36. Results

  37. Results

  38. Over-sampled?

  39. Future work: crop mixtures • Mix of species or varieties • May help reduce disease levels • May help slow down evolution of virulence

  40. Extension to mixtures • Natural extension of model: • Implies 16 parameters for 2 host types, or 33 for 3! • But: less estimative power

  41. Summary • Improved the model of Gibson et al (submitted) • Fitted model using McMC • expect infection 1.5d before first observe symptoms • Little between treatment variation • Lots of between replicate variation • Investigated more efficient sampling scheme

  42. Grazie mille!

  43. Acknowledgements • Work financed by Biomathematics and Statistics, Scotland. • Experiments carried out by Gilligan et al of the botanical epidemiology and modelling group of the Department of Plant Sciences, University of Cambridge, England. • Copies of these slides are available from www.bioss.ac.uk/~alex/cooktrento.ppt

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