Spatio-temporal dynamics, fish farms and pair-approximations

Maths2005

The University of LiverpoolKieran Sharkey, Roger Bowers, Kenton Morgan

DEFRA fundedInvestigate epidemiology of three fish diseasesIHN (Infectious Haematopoietic Necrosis)VHS (Viral Haemorrhagic Septicaemia)GS (Gyrodactylus Salaris)

Collaboration between:Liverpool University Veterinary Epidemiology GroupLiverpool University Applied Maths DeptLancaster University Statistics DeptStirling University Institute for AquacultureCEFAS – Defra funded Laboratory

The symmetric pair-wise model and Foot&Mouth disease

Application to fish farms

Overview of non-symmetric model

Results from non-symmetric model applied to fish farm data

Outline

The Symmetric Pair-wise model

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

2001 Foot&Mouth Outbreak

Total ban on livestock movement

Route of transmission assumes to be local & symmetric

S I

][][

][][][

][][

IgR

IgSII

SIS

N

ISnSI

]][[][

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

A

B

C

A

B

CA

B

CA

B

C+

][

]][[][

B

BCABABC

Disease transmission between fish farms

Slides in this section provided by Mark Thrush at CEFAS

Disease transmission matrix

Nodes

• Fish Farms• Fisheries• Wild populations

Routes of transmission

• Live fish movement• Water flow• Wild fish migration• Fish farm personnel &

equipment

?

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)

General pair-wise

model

Contact network: eg

0 1 11 0 00 1 0

G =

Gs0 1 01 0 00 0 0= Ga

0 0 10 0 00 1 0

=

S I

S I

S I

S→I

S←I

S↔I

A

B

CA

B

CA

B

C+

Some results from the model

Nodes

Fish farms

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

The symmetric pair-wise equations can be generalised to include asymmetric transmission.

Summary

The non-symmetric model can give significantly different predictions to the symmetric model.

Summary

The non-symmetric model is closer to stochastic simulation than the symmetric model on one non-symmetric network.