Let v be a
three!
twovapplications of graph theory to bovine
epidemiology
Jess Enright, University of Stirling
Kitty Meeks, University of Glasgow
The Cattle Tracing System
The Cattle Tracing System
Life Trajectories
Animal A
Animal B
Farm 2
1st Jan
Farm 1
Farm 4
SH 1
Farm 3
Animal A
Born,Farm 2,Farm 1,Farm 3, SH 1,Dead
Animal B
Born,Farm 1,Farm 3,Farm 4,SH 1, Dead
Bigram Counts
Bigram
Born ,Farm 1
Born,Farm 2
Farm 1, Farm 3
Farm 2,Farm 1
Farm 3,SH 1
Farm 4,SH 1
SH 1,Dead
Count
1
1
2
1
1
1
2
Trade partnerships: “CTS Links”
Contagion on graphs
Contagion on graphs
How can we limit expected epidemic size?
One method: limit maximum component size
by deleting edges
Trade partnerships: “CTS Links”
Edge deletion work
Most edge deletion problems are NP-Hard
Yannakakis: deletion to planar, outer-planar, line, transitive digraphs
are hard
Watanabe, Ae, Nakamura: deletion to anything 3-connected is hard
Goldberg et al., deletion to unit interval graphs is hard
El-Mallah and Colburn: deletion to cographs is hard
Margot: deletion to threshold graphs is hard
Natanzon, Shamir, Sharan: deletion to bipartite graph, disjoint unions
of cliques, perfect, chain, chordal, split, AT-free graphs is hard,
but poly time for some with limited degree
Limiting component size
Deleting edges costs: delete minimum
number of edges to achieve a fixed
maximum component size
NP-Complete
Minimum edge deletion to limit max component size
Try: limited treewidth graphs
Important thing
about treewidth:
If we can get a tree
decomposition
with limited
treewidth, can
often employ
dynamic
programming
approach to get a
feasible algorithm
for fixed
parameters
Why treewidth?
Treewidth
12
Why limited treewidth?
Will other disease-relevant graphs have limited treewidth?
Some will!
Some will not.
How does the algorithm work?
Standard treewidth trick:
calculate optimum answer for
vertices at each bag.
Contrive so you only need to
know the answer for a node’s
children to calculate optimum for
that node.
This usually takes time
exponential in the size of the bag
at each node.
But this is ok on a decomposition
of limited treewidth!
How does the algorithm work?
Calculate a signature for each bag,
which tells us how many edges would
have to be deleted in a subgraph
induced by this bag and below
Key: for each bag, need only
consider signatures of children
2w
O((wh) n)
the treewidth
the max component
size
What’s next?
Directed graphs
Planar graphs
Other ways of limiting epidemic size
Overall goal: developing graph-theoretic
approaches to network epidemiology
A change of pace:
How big will an epidemic be?
Hard on general graphs from vertex feedback set.
Our contacts have time, and our contagion moves forward in time.
Farm 1
Farm 2
Farm 2
Farm 3
1st Jan
1st March
Farm 1
Farm 4
Farm 3
Farm 4
1
1
st
March
Farm 5
st
Farm 5
Feb
BFS/DP to the rescue
Farm 1
1
0.5
0.25
0.125
Farm 2
Farm 3
Farm 4
Farm 5
0.125
BFS/DP to the rescue
Farm 1
1
1
1
1
Farm 2
Farm 3 0.5
Farm 4
Farm 5
0.5
0.5
Now this node
needs to fail to
avoid infection
from both
parents
Estimate of mean outbreak size
How many simulations must we do to find the expectation?
Number of simulations
Wall-clock time
Let v be a heifer:
A much bigger challenge than farm-wise graph epidemiology
Cow contact graph
Say there’s only one farm in the UK
?
Cow contact graph
Say there’s only one farm in the UK
Cow contact graph
Ok, fine. 2 farms
Cow contact graph
m farms, each cow is a feasible set of intervals
Cow contact graph
each cow is a feasible set of
intervals
two cows are adjacent if they have
interval intersection
Q: Minimum number of sentinel animals?
Sentinel animals: a set of dominating cows.
Dominating set is P on interval graphs
Cow contact graph
?
Tolerance version: each cow is a
feasible set of intervals
two cows are adjacent if they have
enough interval intersection
Q: What’s the worst tolerance?
A bad tolerance: gives us a large component
without detection.
Possible detection criteria: number of farms involved,
number of simultaneously infected animals, number
of dead infected animals, explicit testing set, etc.
Q: What’s the worst tolerance?
A: depends on the time of year, and the
detection method
Experiments!
Thanks!
Kitty Meeks
[email protected]
[email protected]
@researcherjess
Land Parcel Data