1. Mathematical modelling of epidemics among fish farms in the UK ISVEE X1 (2006) Cairns, Australia Kieran Sharkey The University of Liverpool

2. Funded by Defra (Department for Environment, Food and Rural Affairs) Investigate epidemiology of three fish diseases IHN (Infectious Haematopoietic Necrosis) VHS (Viral Haemorrhagic Septicaemia) GS (Gyrodactylus Salaris) Liverpool University Applied Maths Dept Liverpool University Veterinary Epidemiology Group Lancaster University Statistics Dept Stirling University Institute for Aquaculture CEFAS – Defra funded Laboratory

3. Pair-level equations and Foot&Mouth disease Application to fish farms Overview of modified model Results from new model applied to fish farm networks Outline

4. The Foot & Mouth Model Total animal movement ban Remaining transmission is symmetric

5. A D B C A B C D 0 0 0 1 0 0 1 1 0 1 0 0 1 1 0 0 ABCDABCD Contact Network

6. S I R Infection Removal

7. S

8. S I SI Pair

9. S I

10. S I Insoluble

11. S I Mean Field

12. S I

13. SS I 

14. d[SS]/dt = -2  [SSI] d[SI]/dt =  ([SSI]-[ISI]-[SI])-g[SI] d[SR]/dt = -  [RSI]+g[SI] d[II]/dt = 2  ([ISI]+[SI])-2g[II] d[IR]/dt =  [RSI]+g([II]-[IR]) d[RR]/dt = 2g[IR] Pair-wise Equations

15. Triples Approximation A B C A B C A B C A B C +

16. Transmission routes between fish farms

17. Nodes Fish farms

18. Nodes Fish farms Fisheries

19. Nodes Fish farms Fisheries Wild fish (EA sampling sites)

20. Avon Test Thames Itchen Stour

21. Avon Test Thames Itchen Stour Route 1: Live Fish Movement

22. Route 2: Water flow (down stream)

24. Transmission routes for disease Transportation Non-symmetric Transmission Waterways Non-symmetric Transmission Fish disease Local Symmetric Transmission Foot&Mouth Transmission Mechanisms Local Symmetric Transmission

25. General pair-wise model

26. A D B C A B C D 0 0 0 1 0 0 1 1 0 1 0 0 0 0 ABCDABCD Asymmetric Contact Network

27. SI SI SI S→I S←I S↔I

28. ISS -τ[I→S→S]

29. SSI -τ[S→S←I]-τ[I→S→S]

32. A B C A B C A B C +

34. Some results from the model

35. Nodes Fish farms Transport network (Live fish movement Database)

36. 3576 1714 829 32 16 0 65 0 8 0 0 800 0 00 000

37. Infectious Time Series

40. Susceptible Time Series

41. Summary Symmetric pair-wise equations generalise to include asymmetric transmission Asymmetric equations perform better on asymmetric networks.

42. Pair-level approximations to the spatio-temporal dynamics of epidemics on asymmetric contact networks Journal of Mathematical Biology, Volume 53, Issue 1, Jul 2006, Pages 61 - 85, DOI 10.1007/s00285-006- 0377-3, URL http://dx.doi.org/10.1007/s00285-006-0377-3