Post on 02-Jan-2016
description
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
A stochastic percolation model for disease spread in
crops
Alex Cook (BioSS and Heriot-Watt University)Supervised by: Glenn Marion, Gavin Gibson
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
Experiments
• Hosts: radish• Pathogen: R. solani fungus• Disease: damping-off
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
Experiments
• Hosts: radish• Pathogen: R. solani fungus• Disease: damping-off• Modi operandi: spreads from dead plant material or
infected neighbouring plants
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
Experiments
• Hosts: radish• Pathogen: R. solani fungus• Disease: damping-off• Modi operandi: spreads from dead plant material or
infected neighbouring plants
Picture adapted from Bailey et al (2000), New Phytology 146, pg. 535.
infected host plant
fungal mycelium
20mm
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
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+)
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
Day 4
See Otten et al (2003), Ecology 84, pg.3232
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
Day 5
See Otten et al (2003), Ecology 84, pg.3232
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
Day 6
See Otten et al (2003), Ecology 84, pg.3232
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
Day 7
See Otten et al (2003), Ecology 84, pg.3232
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
Day 8
See Otten et al (2003), Ecology 84, pg.3232
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
Day 9
See Otten et al (2003), Ecology 84, pg.3232
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
Day 10
See Otten et al (2003), Ecology 84, pg.3232
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
Day 11
See Otten et al (2003), Ecology 84, pg.3232
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
Day 12
See Otten et al (2003), Ecology 84, pg.3232
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
Day 13
See Otten et al (2003), Ecology 84, pg.3232
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
Day 14
See Otten et al (2003), Ecology 84, pg.3232
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
Day 15
See Otten et al (2003), Ecology 84, pg.3232
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
Day 16
See Otten et al (2003), Ecology 84, pg.3232
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
Day 17
See Otten et al (2003), Ecology 84, pg.3232
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
Day 18
See Otten et al (2003), Ecology 84, pg.3232
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
Day 19
See Otten et al (2003), Ecology 84, pg.3232
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
Day 20
See Otten et al (2003), Ecology 84, pg.3232
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
Day 21
See Otten et al (2003), Ecology 84, pg.3232
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
Model
• Primary infections at rate α(t) - from inoculum• Secondary infections at rate β(t) - from neighbour
β (t)
β(t)
α(t)
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
Model
• Primary rate α(t) = a
• Secondary rate β(t) = b0 exp{ – b1 log2(tdonor/b2)}
t t
β(t)
β(t)
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
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
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
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
susceptible infectiousα
β δ
symptomatic and
infectious
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
Model
• We therefore want to estimate 5 parameters:– a primary rate of infection
– b0, b1, b2 govern secondary rate of infection
– d rate of symptom development
• Call these θ
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
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
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
Markov chain Monte Carlo
Want to estimate θCan 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
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
Markov chain Monte Carlo
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
Results
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
Results
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
Results
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
Results
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
Results
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
Results
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
Over-sampled?
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
Future work: crop mixtures
• Mix of species or varieties• May help reduce disease levels• May help slow down evolution of virulence
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
Extension to mixtures
• Natural extension of model:
• Implies 16 parameters for 2 host types, or 33 for 3!• But: less estimative power
aR
dR
per host type (R)
b0DR
b1DR
b2DR
per donor-recipient pair (DR)
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
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
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
Grazie mille!
alex@bioss.ac.uk
www.bioss.ac.uk/~alex
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