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

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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

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

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

A D B C A B C D ABCDABCD Contact Network

S I R Infection Removal

S

S I SI Pair

S I

S I Insoluble

S I Mean Field

S I

SS I 

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

Triples Approximation A B C A B C A B C A B C +

Transmission routes between fish farms

Nodes Fish farms

Nodes Fish farms Fisheries

Nodes Fish farms Fisheries Wild fish (EA sampling sites)

Avon Test Thames Itchen Stour

Avon Test Thames Itchen Stour Route 1: Live Fish Movement

Route 2: Water flow (down stream)

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

General pair-wise model

A D B C A B C D ABCDABCD Asymmetric Contact Network

SI SI SI S→I S←I S↔I

ISS -τ[I→S→S]

SSI -τ[S→S←I]-τ[I→S→S]

A B C A B C A B C +

Some results from the model

Nodes Fish farms Transport network (Live fish movement Database)

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 , DOI /s , URL