1. Pair-level approximations to the spatio-temporal dynamics of epidemics on asymmetric contact networks Roger G Bowers Kieran Sharkey The University of Liverpool

2. Outline Networks Pair approximation on symmetric networks Pair approximation on asymmetric networks Application Comparison with simulation Conclusion

3. 12 34 12 34 Networks and Incidence Matrices Symmetric Asymmetric 12 34 12 3 ●4 transposed → open

4. Dynamics of Singletons ( Symmetric Networks ) Closure – mean field approximation N nodes nN = ║G ║ links τ transmission; g recovery S susceptible; I infected; R recovered […] the number of …

5. 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] Dynamics of Pairs ( Symmetric Networks )

6. the ratio of the number of triples with no open links to the total number of triples Closure – Pair Approximation ( Symmetric Networks )

7. Dynamics of Singletons ( Asymmetric Networks ) Closure – mean field approximation

8. Dynamics of Pairs ( Asymmetric Networks )

9. Closure – Pair Approximation ( Asymmetric Networks ) ratios of the number of triples closed by given links to the total number of triples of given type

10. Application - Disease transmission between fish farms Nodes Fish Farms Fisheries Wild populations Routes of transmission Live fish movement Water flow Wild fish migration Fish farm personnel & equipment ? Epidemiology of three fish diseases IHN (Infectious Haematopoietic Necrosis) VHS (Viral Haemorrhagic Septicaemia) GS (Gyrodactylus Salaris)

11. Nodes Fish farms Fisheries Wild fish (EA sampling sites) Slides in this section provided by Mark Thrush at CEFAS

12. Avon Test Thames Itchen Stour

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

14. Route 2: Water flow (down stream)

16. 3576 65 0 171465 8 Route 1: Live Fish Movement

17. Infectious Time Series

18. Results obtained by applying symmetric results directly … naïve use of G

19. Infectious Time Series

20. Susceptible Time Series

21. Conclusion Pleasing extension of the theory of pair approximation to asymmetric networks. Illustration of its efficacy in dealing with applied situations

23. Closure – Pair Approximation ( Asymmetric Networks )