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

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

• Epidemiology is now a security issue
– Complexity of society redefines contact
– Potential & reality of pathogens as weapons
08/19/2004
Scale-Free Network Models in
Epidemiology

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

 Susceptible-Infected-Susceptible Model (SIS)
 Susceptible-Infected-Removed Model (SIR)
 Susceptible-Exposed-Infected- Removed
(SEIR)
S I S
E R
08/19/2004 Scale-Free Network Models in
Epidemiology

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

08/19/2004
Statistical Systems Presume Randomness
Research Question:
Is the epidemiological network
Random? …or ??
Scale-Free Network Models in
Epidemiology

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

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

08/19/2004
Reprinted from Linked: The New Science of Networks by Albert-Laszlo Barabasi
Scale-Free Network Models in
Epidemiology

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

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

1. Create scale-free network
• Georgia Tech - Internetwork Topology Model and ns2 with
Waxman model
• Deterministic scale-free network generation -- Barabasi, et.al.
2. Apply simulation parameters
• Numerical experiments, etc.
3. Step simulation through time
• Decision functions calculate exposure, infection, removal
• Numerical experiments with differing decision functions/parameters
08/19/2004 Scale-Free Network Models in
Epidemiology

• Multi-stage Computation
• Separate Interaction and Decision
Networks
• Multi-dimensional Network Layering
• Extensible Data Sources
• Decomposable/Recomposable Nodes
• Introduce concept of SuperStopper
08/19/2004 Scale-Free Network Models in
Epidemiology

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

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

• Set of j Nodes N={ n
I
, n
II
, … , n j
}
• Set of k Unordered Pairs (Links) L = {( n,n )
I
,
( n,n )
II
, ... , ( n,n ) k
}
• Set of m Communities C={ c
I
, c
II
, …, c m
}
• Set of p Attributes A={ a
I
, a
II
, …, a p
}
• Set of q Functions F={ f
I
, f
II
, …, f q
}
08/19/2004 Scale-Free Network Models in
Epidemiology

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

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

• Refine theoretical framework
• Computational capability/architecture
• Simulator development
• Extensible data source compilation
• Host systems acquisition
• Partnering for research and implementation
08/19/2004 Scale-Free Network Models in
Epidemiology

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