Transcript Document

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