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,