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Mathematical modelling of epidemics among fish farms in the UK ISVEE X1 (2006) Cairns, Australia Kieran Sharkey The University of Liverpool 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 Outline Pair-level equations and Foot&Mouth disease Application to fish farms Overview of modified model Results from new model applied to fish farm networks The Foot & Mouth Model Total animal movement ban Remaining transmission is symmetric Contact Network B C A D A B C D A 0 0 0 1 B 0 0 1 1 C 0 1 0 0 D 1 1 0 0 S Infection I Removal R S S I SI Pair S . I [ S ] [ SI ] S I [ S ] [ SI ] [ I ] [ SI ] g[ I ] [ R ] g[ I ] Insoluble I S [ S ] [ SI ] [ I ] [ SI ] g[ I ] [ R ] g[ I ] [ SI ] n[ S ][ I ] N [ S ][ I ] N Mean Field S I [ S ] [ SI ] [ I ] [ SI ] g[ I ] [ R ] g[ I ] [SI ] S S I [ SS ] 2 [ SSI ] Pair-wise Equations 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] Triples Approximation B [ AB][ BC ] [ ABC ] [ B] A C B A B C A B + C A C Transmission routes between fish farms Nodes Fish farms Nodes Fish farms Fisheries Nodes Fish farms Fisheries Wild fish (EA sampling sites) Thames Avon Test Itchen Stour Route 1: Live Fish Movement Thames Avon Test Itchen Stour Route 2: Water flow (down stream) Route 2: Water flow (down stream) Transmission routes for disease Transmission Mechanisms Foot&Mouth Fish disease Transportation Waterways Non-symmetric Non-symmetric Transmission Transmission Local Symmetric Transmission Local Symmetric Transmission General pair-wise model Asymmetric Contact Network B C A D A B C D A 0 0 0 0 B 0 0 1 0 C 0 1 0 0 D 1 1 0 0 S I S←I S I S→I S I S↔I I S S [S S ] -τ[I→S→S] S S I [S S ] -τ[I→S→S] -τ[S→S←I] B A B C A B + C A C Some results from the model Nodes Fish farms Transport network (Live fish movement Database) 3576 65 65 0 1714 65 65 8 829 65 0 0 32 8 0 0 16 0 0 0 0 Infectious Time Series Infectious Time Series Infectious Time Series Susceptible Time Series Summary Symmetric pair-wise equations generalise to include asymmetric transmission Asymmetric equations perform better on asymmetric networks. 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/s00285-0060377-3, URL http://dx.doi.org/10.1007/s00285-006-0377-3