Mathematical modelling of epidemics among fish farms in

the UKISVEE 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 diseasesIHN (Infectious Haematopoietic Necrosis)VHS (Viral Haemorrhagic Septicaemia)GS (Gyrodactylus Salaris)

Liverpool University Applied Maths Dept Liverpool University Veterinary Epidemiology GroupLancaster University Statistics DeptStirling University Institute for AquacultureCEFAS – 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 0 0 0 10 0 1 10 1 0 01 1 0 0

A

B

C

D

Contact Network

SI

R

Infection

Removal

S

S I SI Pair

S I

][][.

SIS

S I

Insoluble

][][

][][][

][][

IgR

IgSII

SIS

S I

N

IS

N

ISnSI

]][[]][[][

Mean Field

][][

][][][

][][

IgR

IgSII

SIS

S I

][][

][][][

][][

IgR

IgSII

SIS

][SI

S S

I

][2][ SSISS

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

][

]][[][

B

BCABABC

A

B

C

A

B

CA

B

CA

B

C+

Transmission routes between fish farms

Nodes

Fish farms

Nodes

Fish farms

Fisheries

Nodes

Fish farms

FisheriesWild fish(EA sampling sites)

AvonTest

Thames

Itchen

Stour

AvonTest

Thames

Itchen

Stour

Route 1: Live Fish Movement

Route 2: Water flow (down stream)

Route 2: Water flow (down stream)

Transmission routes fordisease

Transportation

Non-symmetric

Transmission

Waterways

Non-symmetric

Transmission

Fish disease

LocalSymmetricTransmission

Foot&Mouth

Transmission Mechanisms

LocalSymmetricTransmission

General pair-wise

model

A

D

B

C

A B C D 0 0 0 10 0 1 10 1 0 00 0 0 0

A

B

C

D

Asymmetric Contact Network

S I

S I

S I

S→I

S←I

S↔I

I S S

-τ[I→S→S]

][ SS

S S I

-τ[S→S←I]-τ[I→S→S]

][ SS

A

B

CA

B

CA

B

C+

Some results from the model

Nodes

Fish farms

Transport network(Live fish movementDatabase)

3576

1714

829

32

16

0

65 65 0

65 65 8

65 0 0

8 0 0

0 0 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/s00285-006-0377-3,

URL http://dx.doi.org/10.1007/s00285-006-0377-3