Scale-Free Network Models in Epidemiology Preliminary Findings

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Scale-Free Network Models in
EpidemiologyPreliminary Findings

• Jill Bigley Dunham
• F. Brett Berlin
• George Mason University
• 19 August 2004

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Problem/Motivation

• Epidemiology traditionally approached as a
  medical/public health understanding issue
• Medical biology gt Pathogen behavior
• Outbreak history gt Outbreak potential
• Infectivity characteristics gt Threat
  prioritization
• Outbreak Control Models Contact Models
• Statistical Models (Historical Patterning)
• Contact Tracing and Triage (Reactive)
• Network Models (Predictive)

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The Challenge is Changing

• Epidemiology is now a security issue
• Complexity of society redefines contact
• Potential reality of pathogens as weapons

Epidemiology is Now About Decisions
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Modeling Options

• Current statistical models dont work
• Oversimplified
• No superspreader events (SARS)
• Simple network models have limited utility
• Recent discoveries suggest application of
  scale-free networks
• Broad applicability (cells gt society)
• Interesting links to Chaos Theory

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Statistical Approaches

• Susceptible-Infected-Susceptible Model (SIS)

• Susceptible-Infected-Removed Model (SIR)

• Susceptible-Exposed-Infected- Removed (SEIR)

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Differential Equations

• SIR Model
• SEIR Model

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

s(t), e(t), i(t), r(t) Fractions of the
population in each of the states. S I R
1 S E I R 1
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Statistical Systems Presume Randomness
Research Question Is the epidemiological
network Random? or ??
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Network Models

• Differential Equations model assumes the
  population is fully mixed (random).
• In real world, each individual has contact with
  only a small fraction of the entire population.
• The number of contacts and the frequency of
  interaction vary from individual to individual.
• These patterns can be best modeled as a NETWORK.

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Scale-Free Network

• A small proportion of the nodes in a scale-free
  network have high degree of connection.
• Power law distribution P(k) ? O(k-?).
• A given node has k connections to other nodes
  with probability as the power law distribution
  with ? 2, 3.
• Examples of known scale-free networks
• Communication Network - Internet
• Ecosystems and Cellular Systems
• Social network responsible for spread of disease

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Reprinted from Linked The New Science of
Networks by Albert-Laszlo Barabasi
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Generation of Scale-Free Network

• The vertices are distributed at random in a
  plane.
• An edge is added between each pair of vertices
  with probability p.
• 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 Approach

• Waxman model inappropriate for creating
  scale-free networks
• Most current topology generators are not up to
  this task!
• One main characteristic of scale-free networks is
  addition of nodes over time

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Procedure

• Create scale-free network
• Georgia Tech - Internetwork Topology Model and
  ns2 with Waxman model
• Deterministic scale-free network generation --
  Barabasi, et.al.
• Apply simulation parameters
• Numerical experiments, etc.
• Step simulation through time
• Decision functions calculate exposure, infection,
  removal
• Numerical experiments with differing decision
  functions/parameters

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Proposed Simulator

• Multi-stage Computation
• Separate Interaction and Decision Networks
• Multi-dimensional Network Layering
• Extensible Data Sources
• Decomposable/Recomposable Nodes
• Introduce concept of SuperStopper

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TWO-PHASE COMPUTATION

• Separate Progression Transmission
• Progression Track internal factors
• Node susceptibility (e.g., general health)
• Token infectiousness
• Transmission Track inter-nodal transition
• External catalytic effects
• Token dynamics (e.g., spread, blockage, etc)

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INTERACTION NETWORK

• Population connectivity graph
• Key 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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PRIMITIVES

• Set of j Nodes NnI, nII, , nj
• Set of k Unordered Pairs (Links) L (n,n)I,
  (n,n)II, ... , (n,n)k
• Set of m Communities CcI, cII, , cm
• Set of p Attributes AaI, aII, , ap
• Set of q Functions FfI, fII, , fq

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DECISION NETWORK

• Separate overlay network defining control
  decision parameters which are 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 SOURCES

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

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FUTURE WORK

• Refine theoretical framework
• Computational capability/architecture
• Simulator development
• Extensible data source compilation
• Host systems acquisition
• Partnering for research and implementation

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Concluding Perspectives

• Computational Opportunities
• Theory and Policy
• Chaos and Complexity
• Imperative for Alchemy