Simulation Science Laboratory Modeling Disease Transmission Across Social Networks DIMACS seminar...

Simulation Science Laboratory

Modeling Disease Transmission Across Social Networks

DIMACS seminar

February 7, 2005

Stephen Eubank

Virginia Bioinformatics Institute

Virginia Tech

[email protected]


Variations on a Theme

I. Estimating a Social Network

II. Varieties of Social Networks

III. Characterizing Networks for Epidemiology


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

If not uniform mixing, what?

Homogenous

Isotropic

??

. . .. . . ~~ 22NN 22 alternative

networks

ODE modelODE model Network modelNetwork model


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?


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)


Example Synthetic Household

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QuickTime™ and a Graphics decompressor are needed to see this picture.

Age 26 26 7

Income $27k $16k $0

Status worker worker student

Automobile

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Example Route Plans

HOME

WORK

LUNCH

WORK

DOCTOR

SHOP

HOME

HOME

WORK

SHOP

second person in household

first person in household

Estimating Travel Times by Microsimulation

7.5 meter 1 lane cellularautomaton grid cells

intersection with multipleturn buffers (not internallydivided into grid cells)

single-cell vehicle

multiple-cell vehicle


Typical Family’s Day

Carpool

HomeHome

Work Lunch WorkCarpool

Bus

Shopping

Car

Daycare

Car

School

time

Bus


Others Use the Same Locations

time


Time Slice of a Social Network


HomeHome

Activities Adapt to SituationActivities Adapt to Situation

Example: Smallpox Response Efficacy#

deat

hs p

er in

itia

l inf

ecte

d by

day

100


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


A Social Network: multipartite labeled graph

People (8.8 million)People (8.8 million)

Vertex attributes:Vertex attributes:• ageage• household sizehousehold size• gendergender• incomeincome• … …


A Social Network: bipartite labeled graph

Vertex attributes:Vertex attributes:• (x,y,z)(x,y,z)• land useland use• … …

Locations (1 million)Locations (1 million)



Edge attributes:Edge attributes:• activity type: shop, work, schoolactivity type: shop, work, school• (start time 1, end time 1)(start time 1, end time 1)• probability of transmittingprobability of transmitting


A Social Network: projection onto people


A Social Network: projection onto people

[t1,t2][t1,t2] [t2,t3][t2,t3] [t3,t4][t3,t4] [t4,t5][t4,t5]


A Social Network: projection over time


Dendrogram: actual path disease takes




A Social Network: projection onto locations


A Social Network: projection onto locations

t3t3 t4t4t2t2


A Social Network: projection over time


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

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

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

Removing high degree people useless

Removing high degree locations better


Clustering coefficient vs degreeClustering coefficient vs degree


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?


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


Comparison to “usual suspects”

x Harder to measure in real networksx 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


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 timeBottom graph: degree 4, 80% of the time

Critical vertexCritical 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


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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Simulation Science Laboratory Modeling Disease Transmission Across Social Networks DIMACS seminar...

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