Modeling Disease Transmission
Across Social Networks
DIMACS seminar
February 7, 2005
Stephen Eubank
Virginia Bioinformatics Institute
Virginia Tech
eubank@vt.edu
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Variations on a Theme
I. Estimating a Social Network
II. Varieties of Social Networks
III. Characterizing Networks for Epidemiology
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Translation
• Compute structural properties of very large graphs
– Which ones?
• Are local properties enough?
• Structural properties should be robust
– How? need efficient algorithms
• Generate constrained random graphs
– for experiment
• Chung-Lu, Reed-Molloy, MCMC
– for analysis
• preserve independence as much as possible
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If not uniform mixing, what?
Network model
ODE model
Homogenous
Isotropic
?
...
N 2 alternative
~2
networks
Do Local Constraints Fix Global Properties?
• N vertices  ~ 2N2 graphs
(non-identical vertices  few symmetries)
• E edges  ~ N2E graphs
• Degree distribution  ?? graphs
• Clustering coefficient  ?? graphs
• What additional constraints  ?? graphs equivalent w.r.t.
epidemics?
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Estimating a social network
• Synthetic population
• Survey (diary) based activity templates
• Iterative solution to a large game
– Assigning locations for activities (depends on travel times)
– Planning routes
– Estimating travel times (depends on activity locations)
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Example Synthetic Household
QuickTime™ and a Graphi cs decompressor are needed to see thi s picture.
Qu i c k T i m e ™ a n d a Gra p h i c s d e c o m p r e s s o r a re n e e d e d to s e e th i s p i c tu re .
Qu i c k T i m e ™ a n d a Gra p h i c s d e c o m p r e s s o r a re n e e d e d to s e e th i s p i c tu re .
QuickTime™ and a Graphi cs decompressor are needed to see thi s picture.
QuickTime™ and a Graphi cs decompressor are needed to see thi s picture.
Age
26
26
7
Income
$27k
$16k
$0
Status
worker
worker
student
Automobile
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Example Route Plans
SHOP
WORK
first person in household
HOME
second person in household
LUNCH
WORK
SHOP
DOCTOR
HOME
Estimating Travel Times by Microsimulation
intersection with multiple
turn buffers (not internally
divided into grid cells)
single-cell vehicle
multiple-cell vehicle
7.5 meter  1 lane cellular
automaton grid cells
Typical Family’s Day
Carpool
Work
Lunch
Work
Carpool
Shopping
Home
Car
Home
Car
Daycare
Bus
time
School
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Bus
Others Use the Same Locations
time
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Time Slice of a Social Network
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Activities Adapt to Situation
Home
Home
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# deaths per initial infected by day 100
Example: Smallpox Response Efficacy
Part II: Varieties of Social Networks
• Definition of vertex
– People
– Concepts (location, role in society, group)
• Definition of edge
– Effective contact
– Proximity
• Weights
– Edges: Interaction strength / probability of transmission
– Vertices: “importance”
• Time dependence
• Directionality
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A Social Network: multipartite labeled graph
People (8.8 million)
Vertex attributes:
• age
• household size
• gender
• income
•…
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A Social Network: bipartite labeled graph
Locations (1 million)
Vertex attributes:
• (x,y,z)
• land use
•…
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A Social Network: bipartite labeled graph
Edge attributes:
• activity type: shop, work, school
• (start time 1, end time 1)
• probability of transmitting
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A Social Network: projection onto people
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A Social Network: projection onto people
[t1,t2]
[t2,t3]
[t3,t4]
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[t4,t5]
A Social Network: projection over time
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Dendrogram: actual path disease takes
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A Social Network: bipartite labeled graph
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A Social Network: projection onto locations
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A Social Network: projection onto locations
t2
t3
t4
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A Social Network: projection over time
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Disease Dynamics & Scenario
Determine Relevant Projections
• People projection: edge if people co-located
– communicable disease + vaccination/isolation
• Location projection: directed edge if travel between locations
– contamination, quarantine
• Time dependence: almost periodic
– Important time scales set by disease dynamics:
• Infectious period
• Duration of contact for transmission
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Example: Person-person graph
Person-person graph
(~ dendrogram with ptransmission = 1)
Dendrogram with ptransmission << 1
Geographic spread
Characterizing EpiSims Networks
•
Degree distributions
•
Pointwise clustering: ratio of # triangles to # possible
•
Assortative mixing by degree, age, …
•
Shortest path length distribution
•
Expansion
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Degree Distribution, location-location
Degree Distribution, people-people
Sensitivity to parameters
Sensitivity to parameters
Assortative Mixing in EpiSims Graphs
• Static people - people projection is assortative
– by degree (~0.25)
– but not as strongly by age, income, household size, …
This is
• Like other social networks
• Unlike
– technological networks,
– Erdos-Renyi random graphs
– Barabasi-Albert networks
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Removing high degree people useless
Removing high degree locations better
Clustering coefficient vs degree
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Characterizing Networks for Epidemiology
•
•
Question: how to change a network to reduce [casualties]?
Constraints:
–
–
–
–
•
Don’t know ahead of time where outbreak begins
Minimize impact on other social functions of network
Don’t know true network, only estimated one
Incorporate dependence on pathogen properties
Optimization:
– Propose edge/vertex removal based on measurable (local)
properties
– Quickly estimate effect of new structure
•
How does propagation depend on structure?
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Suggested Metric
Nk(i) = Number of distinct people connected to person i
by a (shortest) path of length k
 “k-betweenness”, “pointwise k-expansion”
 Important k values are related to ratio of incubation to response
times
 Shortest path vs any path: depends on probability of transmission
– Given N1(i), ..., Nk(i), can construct analog for non-shortest path of
length k
x Assumes static graph, but expect graph to change
 Simple cases incorporate intuitively important properties
– For k=1, N1(i) = d(i)
– For k=2, includes degree distribution, clustering, assortativity by
degree
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Comparison to “usual suspects”
x Harder to measure in real networks
x Difficult to work with analytically
 Perturbative expansions (say, around tree-like structure) are lacking
a small parameter to expand in
 Describes how clustering should be combined with degree
 Degree alone determines neither vulnerability nor criticality
 Betweenness is global, sensitive to small changes
 Usual statistics don’t incorporate time scales naturally
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Degree alone determines neither
vulnerability nor criticality
Same degree distribution
Different assortative mixing by degree
Introduce index case uniformly at random,
what color (degree) is vulnerable?
Top graph: degree 1, 80% of the time
Bottom graph: degree 4, 80% of the time
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Critical vertex
Use depends on how disease is introduced
• Introduction uniformly distributed,
consider distribution over all people: mean, variance, …
• Introduction concentrated on specific part of graph,
consider distribution over k-neighborhood
• Introduction by malicious agent,
consider worst case or tail
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Conclusion
Progress on many fronts, but plenty more to be done:
• Estimating large social networks
• Building efficient, scalable simulations
• Understanding structure of social networks
• Determining how structure affects disease spread
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