Pair-level approximations to the spatio-temporal dynamics of epidemics on asymmetric contact...
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Pair-level approximations to the spatio-temporal dynamics of epidemics on asymmetric contact networks Roger G Bowers Kieran Sharkey The University of Liverpool
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Transcript of Pair-level approximations to the spatio-temporal dynamics of epidemics on asymmetric contact...
Pair-level approximations to the spatio-temporal dynamics of
epidemics on asymmetric contact networks
Roger G BowersKieran Sharkey
The University of Liverpool
Outline
• Networks• Pair approximation on symmetric networks• Pair approximation on asymmetric
networks• Application• Comparison with simulation• Conclusion
GG
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GGG
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)()1( GGGIG
Networks and Incidence Matrices
Symmetric
Asymmetric
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transposed → open
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IgR
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Dynamics of Singletons (Symmetric
Networks)
Closure – mean field approximation
N nodes nN = ║G ║links
τ transmission; g recovery
S susceptible; I infected; R recovered
[…] the number of …
][][ ISSI
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]
Dynamics of Pairs (Symmetric Networks)
the ratio of the number of triples with no open links to the total number of triples
Closure – Pair Approximation (Symmetric Networks)
n
n
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Dynamics of Singletons (Asymmetric
Networks)
Closure – mean field approximation
Dynamics of Pairs (Asymmetric Networks)
Closure – Pair Approximation (Asymmetric Networks)
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ratios of the number of triples closed by given links to the total number of triples of given type
Application - Disease transmission between fish farms
Nodes
• Fish Farms• Fisheries• Wild populations
Routes of transmission
• Live fish movement• Water flow• Wild fish migration• Fish farm personnel &
equipment
?
Epidemiology of three fish diseasesIHN (Infectious Haematopoietic Necrosis)VHS (Viral Haemorrhagic Septicaemia)GS (Gyrodactylus Salaris)
Nodes
Fish farms
FisheriesWild fish(EA sampling sites)
Slides in this section provided by Mark Thrush at CEFAS
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)
3576 65 65 0
)65653576/(65
)865651714/(8
)865651714/(65
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Route 1: Live Fish Movement
Infectious Time Series
Infectious Time Series
Results obtained by applying symmetric results directly … naïve use of G
Infectious Time Series
Susceptible Time Series
Conclusion
• Pleasing extension of the theory of pair approximation to asymmetric networks.
• Illustration of its efficacy in dealing with applied situations
Closure – Pair Approximation (Asymmetric Networks)
Closure – Pair Approximation (Asymmetric Networks)
Closure – Pair Approximation (Asymmetric Networks)
