Transcript Scale-Free Network Models in Epidemiology

Scale-Free Network Models
in Epidemiology
Preliminary Findings
Jill Bigley Dunham
F. Brett Berlin
George Mason University
19 August 2004
Problem/Motivation
• Epidemiology traditionally approached as a
medical/public health understanding issue
– Medical biology => Pathogen behavior
– Outbreak history => Outbreak potential
– Infectivity characteristics => Threat prioritization
• Outbreak & Control Models = Contact Models
– Statistical Models (Historical Patterning)
– Contact Tracing and Triage (Reactive)
– Network Models (Predictive)
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Scale-Free Network Models in
Epidemiology
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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Scale-Free Network Models in
Epidemiology
Modeling Options
• Current statistical models don’t work
– Oversimplified
– No superspreader events (SARS)
• Simple network models have limited utility
• Recent discoveries suggest application of
scale-free networks
– Broad applicability (cells => society)
– Interesting links to Chaos Theory
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Scale-Free Network Models in
Epidemiology
Statistical Approaches
 Susceptible-Infected-Susceptible Model (SIS)
 Susceptible-Infected-Removed Model (SIR)
 Susceptible-Exposed-Infected- Removed
(SEIR)
S
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Scale-Free Network Models in
Epidemiology
Differential Equations
• SIR Model
1 /   Mean latent period
for the disease.

 Contact rate.
1 /   Mean infection
rate.
• SEIR Model
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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Scale-Free Network Models in
Epidemiology
Statistical Systems Presume Randomness
Research Question:
Is the epidemiological network
Random? …or ??
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Scale-Free Network Models in
Epidemiology
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 Models in
Epidemiology
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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Scale-Free Network Models in
Epidemiology
Reprinted from Linked: The New Science of Networks by Albert-Laszlo Barabasi
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Scale-Free Network Models in
Epidemiology
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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Scale-Free Network Models in
Epidemiology
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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Scale-Free Network Models in
Epidemiology
Procedure
1. Create scale-free network
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Georgia Tech - Internetwork Topology Model and ns2 with
Waxman model
Deterministic scale-free network generation -- Barabasi, et.al.
2. Apply simulation parameters
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Numerical experiments, etc.
3. Step simulation through time
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Decision functions calculate exposure, infection, removal
Numerical experiments with differing decision
functions/parameters
Scale-Free Network Models in
Epidemiology
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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Scale-Free Network Models in
Epidemiology
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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Scale-Free Network Models in
Epidemiology
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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Scale-Free Network Models in
Epidemiology
PRIMITIVES
• Set of j Nodes N={nI, nII, … , nj}
• Set of k Unordered Pairs (Links) L = {(n,n)I,
(n,n)II, ... , (n,n)k}
• Set of m Communities C={cI, cII, …, cm}
• Set of p Attributes A={aI, aII, …, ap}
• Set of q Functions F={fI, fII, …, fq}
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Scale-Free Network Models in
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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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Scale-Free Network Models in
Epidemiology
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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Scale-Free Network Models in
Epidemiology
FUTURE WORK
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Refine theoretical framework
Computational capability/architecture
Simulator development
Extensible data source compilation
Host systems acquisition
Partnering for research and
implementation
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Scale-Free Network Models in
Epidemiology
Concluding Perspectives
• Computational Opportunities
• Theory and Policy
• Chaos and Complexity
• Imperative for Alchemy
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Scale-Free Network Models in
Epidemiology