[email protected] A stochastic percolation model for disease spread in crops Alex Cook (BioSS and...
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[email protected] .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
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Transcript of [email protected] A stochastic percolation model for disease spread in crops Alex Cook (BioSS and...
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
www.bioss.ac.uk/~alex
Experiments
• Hosts: radish• Pathogen: R. solani fungus• Disease: damping-off
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
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
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+)
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)
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)
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
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
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
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
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
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
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)
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
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
