Scale-Free Network Scale-Free Network ModelsModels

in Epidemiology in Epidemiology

Preliminary FindingsPreliminary Findings

Jill BigleyJill Bigley DunhamDunhamF. Brett BerlinF. Brett Berlin

George Mason UniversityGeorge Mason University19 August 200419 August 2004

08/19/2004 Scale-Free Network Models in Epidemiology

Problem/MotivationProblem/Motivation

• Epidemiology traditionally approached as a Epidemiology traditionally approached as a medical/public health medical/public health understanding understanding issueissue– Medical biology => Pathogen behavior– Outbreak history => Outbreak potential– Infectivity characteristics => Threat prioritization

• Outbreak & Control Models = Contact ModelsOutbreak & Control Models = Contact Models– Statistical Models (Historical Patterning)– Contact Tracing and Triage (Reactive)– Network Models (Predictive)

Scale-Free Network Models in Epidemiology

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The Challenge is ChangingThe Challenge is Changing

• Epidemiology is now a Epidemiology is now a securitysecurity issue issue– Complexity of society redefines contact– Potential & reality of pathogens as

weapons

Epidemiology is Now About

DecisionsDecisions

08/19/2004 Scale-Free Network Models in Epidemiology

Modeling OptionsModeling Options

• Current statistical models don’t workCurrent statistical models don’t work– Oversimplified– No superspreader events (SARS)

• Simple network models have limited Simple network models have limited utilityutility

• Recent discoveries suggest Recent discoveries suggest application of scale-free networksapplication of scale-free networks– Broad applicability (cells => society)– Interesting links to Chaos Theory

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Statistical ApproachesStatistical Approaches Susceptible-Infected-Susceptible Susceptible-Infected-Susceptible

Model (SIS)Model (SIS)

R

S SI

E

Susceptible-Infected-Removed Model (SIR)Susceptible-Infected-Removed Model (SIR)

Susceptible-Exposed-Infected- Susceptible-Exposed-Infected- Removed (SEIR)Removed (SEIR)

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Differential EquationsDifferential Equations• SIR ModelSIR Model

• SEIR ModelSEIR Model

s(t), e(t), i(t), r(t) : Fractions of the population in each of the states.S + I + R = 1S + E + I + R = 1

1 / Mean latent period for the disease. Contact rate.1 / Mean infection rate.

08/19/2004 Scale-Free Network Models in Epidemiology

Statistical Systems Presume Randomness

Research QuestionResearch Question:

Is the epidemiological network

Random? …or ??

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Network ModelsNetwork Models

• Differential Equations model assumes the Differential Equations model assumes the population is “fully mixed” (random).population is “fully mixed” (random).

• In real world, In real world, each individual has contact each individual has contact with only a small fraction of the entire with only a small fraction of the entire population.population.

• The number of contacts and the frequency The number of contacts and the frequency of interaction vary from individual to of interaction vary from individual to individual. individual.

• These patterns can be best modeled as a These patterns can be best modeled as a NETWORK.NETWORK.

08/19/2004 Scale-Free Network Models in Epidemiology

Scale-Free NetworkScale-Free Network• A small proportion of the nodes in a scale-A small proportion of the nodes in a scale-

free network have high degree of free network have high degree of connection. connection.

• Power law distribution P(k) Power law distribution P(k) O(k O(k--). ).

A given node has k connections to other A given node has k connections to other nodes with probability as the power law nodes with probability as the power law distribution with distribution with = [2, 3]. = [2, 3].

• Examples of known scale-free networks:Examples of known scale-free networks:– Communication Network - Internet– Ecosystems and Cellular Systems– Social network responsible for spread of disease

08/19/2004 Scale-Free Network Models in Epidemiology

Reprinted from Linked: The New Science of Networks by Albert-Laszlo Barabasi

08/19/2004 Scale-Free Network Models in Epidemiology

Generation of Scale-Free Generation of Scale-Free NetworkNetwork

• The vertices are distributed at random in a The vertices are distributed at random in a plane. plane.

• An edge is added between each pair of An edge is added between each pair of vertices with probability vertices with probability pp. .

• Waxman Model: Waxman Model: P(u,v) = * exp( -d / (*L) ), 0 , 1.

– L is the maximum distance between any two nodes. – Increase in alpha increases the number of edges in the

graph. – Increase in beta increases the number of long edges

relative to short edges. – d is the Euclidean distance from u to v in Waxman-1. – d is a random number between [0, L] in Waxman-2.

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Problems with this Problems with this ApproachApproach

• Waxman model inappropriate for Waxman model inappropriate for

creating scale-free networkscreating scale-free networks

• Most current topology generators are Most current topology generators are

not up to this task!not up to this task!

• One main characteristic of scale-free One main characteristic of scale-free

networks is addition of nodes over networks is addition of nodes over

timetime

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ProcedureProcedure

1.1. Create scale-free networkCreate scale-free network• Georgia Tech - Internetwork Topology Model and ns2

with Waxman model• Deterministic scale-free network generation -- Barabasi,

et.al.

2.2. Apply simulation parametersApply simulation parameters• Numerical experiments, etc.

3.3. Step simulation through timeStep simulation through time• Decision functions calculate exposure, infection, removal• Numerical experiments with differing decision

functions/parameters

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Proposed SimulatorProposed Simulator

• Multi-stage ComputationMulti-stage Computation• Separate Interaction and Decision Separate Interaction and Decision

NetworksNetworks• Multi-dimensional Network LayeringMulti-dimensional Network Layering• Extensible Data SourcesExtensible Data Sources• Decomposable/Recomposable NodesDecomposable/Recomposable Nodes• Introduce concept of SuperStopperIntroduce concept of SuperStopper

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TWO-PHASE TWO-PHASE COMPUTATIONCOMPUTATION

• Separate Progression & TransmissionSeparate Progression & Transmission• Progression: Track internal factorsProgression: Track internal factors

– Node susceptibility (e.g., general health)– Token infectiousness

• Transmission: Track inter-nodal Transmission: Track inter-nodal transitiontransition– External catalytic effects– Token dynamics (e.g., spread, blockage,

etc)

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INTERACTION NETWORKINTERACTION NETWORK

• Population connectivity graph Population connectivity graph • Key ChallengesKey Challenges

– Data Temporality: Input data (even near-real time observation) generally limited to past history & statistical analysis.

– Data Integration: Sources, sensor/observer characteristics, precision & context often poorly defined, unknown or incompatible

– Dimensionality of connectivity

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PRIMITIVESPRIMITIVES

• Set of j Nodes N=Set of j Nodes N={{nnII, , nnIIII, … , , … , nnjj} }

• Set of k Unordered Pairs (Links) L = Set of k Unordered Pairs (Links) L =

{({(n,nn,n))II, (, (n,nn,n))IIII, ... , (, ... , (n,nn,n))kk} }

• Set of m Communities C=Set of m Communities C={{ccII, , ccIIII, …, , …, ccmm} }

• Set of p Attributes A=Set of p Attributes A={{aaII, , aaIIII, …, , …, aapp} }

• Set of q Functions F=Set of q Functions F={{ffII, , ffIIII, …, , …, ffqq} }

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DECISION NETWORKDECISION NETWORK

• Separate overlay network defining Separate overlay network defining control decision parameters which are control decision parameters which are applied to the Interaction Network.applied to the Interaction Network.– Shutting down public transportation– Implementing preferential vaccination

strategies

The Interaction Network models societal and system realities and dynamics. The Decision

Network models policy maker options.

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EXTENSIBLE DATA EXTENSIBLE DATA SOURCESSOURCES

Model and simulation must be Model and simulation must be dynamically extensible -- designed to dynamically extensible -- designed to reconfigure and recompute based on reconfigure and recompute based on insertion of external source databases, insertion of external source databases, and real-time changeand real-time change• NOAA weather/environmental dataNOAA weather/environmental data• Multi-source intelligence Multi-source intelligence assessmentsassessments

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FUTURE WORKFUTURE WORK

• Refine theoretical frameworkRefine theoretical framework• Computational Computational

capability/architecture capability/architecture • Simulator developmentSimulator development• Extensible data source compilationExtensible data source compilation• Host systems acquisitionHost systems acquisition• Partnering for research and Partnering for research and

implementationimplementation

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Concluding PerspectivesConcluding Perspectives

• Computational Opportunities

• Theory and Policy

• Chaos and Complexity

• Imperative for Alchemy