1. Animal transport: spread of disease and …. Uno Wennergren Tom Lindström Annie Jonsson Nina Håkansson Jenny Lennartsson Spatio-Temporal Biology Division of Theoretical Biology Linköping University Sweden

2. Animal transport 2006 - Aims of different projects The context – Research groups, their expertise – Data base on animal movements Specific research questions Estimating probability of animal movements – Tom Lindström

3. Projects - aims - groups Spread of disease: Foot and mouth disease. – Prepare to optimize intervention Animal welfare – Reduce stress and distance transported

4. Spread of disease: Foot and mouth disease. Funded by Swedish Civil Contingencies Agency (Swedish DHS): 2 grants, PI’s: UW and SSL at SVA – Prepare to optimize intervention Spatio-Temporal Biology (4 persons) – Biology/Ecology – Mathematics – Scientific Computing National Veterinary Institute (SVA) (3 persons) – Disease control and epidemiology – Veterinary medicine

5. Animal welfare – Reduce stress and distance per animal Funded by Swedish Board of Agriculture (Swedish USDA) PI: UW Spatio-Temporal Biology (3 persons) – Biology/Ecology – Mathematics – Scientific Computing Dept. of Animal Environment and Health, Swedish University of Agricultural Sciences (2 persons) – Animal welfare – Veterinary medicine Skogforsk, LiU, NHH (3 persons) – Optimization –Logistics – route planning –

6. Database All animal (cattle and pigs) movements between farms and farms to slaughterhouses. Not per vehicle – Cattle on individual level: birth, sale purchase, export, import, temporarily away (pasture), return from pasture, slaughter/house, death – Pig, on group level: as above Report within seven days Farms and slaughterhouses in Sweden. Dots: blue –farms, red – large slaughterhouses. Green - smaller slaughterhouses. From Håkansson et al 2007. Sweden

7. Database -specifics 12 months - cattle approximately 1 000 000 reports of sales and purhase Important: errors in reports 10% – Possible to edit the database and reduce to 1% error by logical corrections (database cleaning) Spatial and temporal investigation of reported movements, births and deaths of cattle and pigs in Sweden. Submitted. Nöremark, Håkansson, Lindström, Wennergren, and Sternberg Lewerin.

8. Specific research questions 1.Other contacts between farms - questionnaire to farmers (SVA) 2.From measured contacts to probability of contact 3.Spread: Modeling disease specifics 4.Route planning of animal transport – effect on contacts and movement distance. 5.Production units: composition and configuration 6.Networks 1.Analysing transport network 2.Testing efficiency of network measures as predictors 1.Generating netorks 2.Testing linkdensity on network formation 3.Testing measures as predictors

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10. Specific research questions From measured contacts to probability of contact Estimating probabilities Tom Lindström

11. Animal movements between holdings Which farms are likely to have contacts through animal movements? – Mathematical description. – Estimation from data. Distance – Contacts between nearby farms are more common – Several different processes – Preventive Veterinary Medicine (any day now…)

12. A word on the data Should be good… Pigs reported at transport level by the receiving farmer Cattle reported at individual level by farmers at both origin and end. – Cattle moved on the same day between same farms constitute one transport – Mismatch – “Cleaning” using the identity of cattle Locations of many cattle farms not in the database but areas of valid for subsidies Inactive farms in the data base

13. Quantifying distance dependence Distance dependence needs two measurements. Probability of contacts has – Scale Measured as Variance (or Squared Displacement) – Shape Measured as Kurtosis

14. Variance Distance P

15. Kurtosis Distance P

16. Why these measures? Important to have quantities for comparison – Between epidemics – Between types of contacts – Between years Theoretical connection to biological invasions – Squared displacement relates to diffusion constant. – Discrete representation of space (i.e. farms has X,Y coordinates) => Fat tails more important

17. Kernel function A generalized normal distribution Variance and Kurtosis given by a and b. Extended to two dimensions (X,Y coordinates) – S normalizes the kernel, Volume=1.

18. Kernel function normalization With discrete representation of farm distribution normalization over all possible destination farms d is distance, i is start farm, k is possible destination farms (k≠i) and N is number of farms.

19. Kernel function normalization This separates spatial pattern of farms from distance dependence in contacts. Important if farm distribution is non random. Farm density in Sweden (farms/km 2 )

20. And USA From Shields and Mathews, 2003

21. Is the kernel function good enough? A single distribution may not be sufficient to fit data on multiple scales (both short and long distance contacts). An alternative model – A mixture model – Part distance dependent and part uniform (Mass Action Mixing) Models applied to pig and cattle transports (all transports during one year)..

22. An alternative model f 1 is distance dependent part: f 2 is MAM part: w is proportion of distance dependence.

23. Fitting to data Bayesian approach Increasingly common at least in ecological literature. Ellison 2008

24. Markov Chain Monte Carlo Parameters obtained through Markov Chain Monte Carlo (MCMC). Well suitable for epidemiological problems. A simple model can be expanded to include complexity. Drawback is computation time, and effective parallelization is difficult.

25. Markov Chain Monte Carlo Repeated (correlated) random draws from the posterior distribution of parameters. Gibbs Sampling – Direct draws from known distributions conditional on other parameter values Metropolis-Hastings – Values are proposed and subsequently accepted or rejected dependent on likelihood ratios

26. Markov Chain Monte Carlo Also allows for model selection by comparing the full posterior distribution of model probabilities. In our study, the mixture model was a much better model. Pigs Cattle

27. Comparing models and observed data Bars: observed transport distances. Dotted line: predictions by Model 1. Solid line: predictions by Model 2 CattlePigs

28. Network measures Will differences have consequences for estimation of disease spread dynamics? Networks generated with the different models Network measures Nodes (farms) and links (transports) A B C D

29. Network measures Density – proportion of farms connected Model 1 Model 2

30. Network measures Clustering Coefficient – proportion of “triplets” – If A is connected to B and C, are B and C connected? A B C D Model 1 Model 2

31. Network measures Fragmentation index – measures the amount of fragments not connected to the rest. A B C D E F D Model 1 Model 2

32. Network measures Betweeness centralization index – Are some nodes more central than others? A B CD E F D A B CD E F D Model 1 Model 2

33. Animal transports Higher Cluster Coefficient and lower Density for Model 2 – Depends on difference in short distance contacts – Depletion of susceptibles Group Betweeness higher for Model 1 in Cattle. – Due to long distance transport being more rare Conclusion: Model 2 is a better model (higher likelihood) and the difference may have impact on disease spread prediction.

34. More than distance? Why not compare to observed networks? Is there something but distance that matter? Some work in progress…

35. More than distance? Pig industry very structured, production types – Multiplying herd – Sow pool central unit – Sow pool satellite herd – Fattening herd – Farrow to finish herd – Piglet producing herd – Nucleus herd http://www.swedishmeats.com

36. From To

37. Production types in cattle? Dairy and beef producers Male calves on dairy farms are often sold to beef producers (at lest in Sweden) Other differences in production types? – Roping? – Organic farming? – Climate/geografic factors

38. More than distance? Reality is messy… – Data base not perfect – Missing production types – Several production types per farms Weights in the model – A farm is a fraction of each possible type. One parameter estimation per combination (sender/receiver) of production types.

39. More than distance? Size dependence – Size (Capacity) – Two different sizes Adult sows Piglets Different production types have different response – Different for sending or receiving – Total 64x4 parameters just for size… – Modeled as power function (Size θ )

40. More than distance? Distance dependence – Different for different production types – Variance, Kurtosis and mixing parameter for each combination

41. More than distance? Many parameters… 9*64=576 Some combinations of production types have few transports => uncertain estimations. Variance and kurtosis not clearly different from ∞. Using a prior may help – But it’s nicer to be objective…

42. Hierarchical Bayesian We can let the data decide the prior – Hyper parameters Hierarchical Bayesian model “Borrowing strength”

43. Animal transports – part 2 θ1θ1 θ2θ2 θ3θ3 θnθn Data P(θ )

44. Hierarchical Bayesian When would this make sense? – If parameters values are expected to be different but not totally different – E.g. distance… Parameter estimations based on much data… – Little influence of hierarchical prior Parameter estimations with little data… – Highly influenced by the hierarchical prior. Increases the variance of the prior distribution.

45. Thank you Questions?