1. Scale-Free Network Models in Epidemiology Preliminary Findings Jill Bigley Dunham F. Brett Berlin George Mason University 19 August 2004

2. 08/19/2004Scale-Free Network Models in Epidemiology Problem/Motivation Epidemiology traditionally approached as a medical/public health understanding issueEpidemiology 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 ModelsOutbreak & Control Models = Contact Models –Statistical Models (Historical Patterning) –Contact Tracing and Triage (Reactive) –Network Models (Predictive)

3. 08/19/2004Scale-Free Network Models in Epidemiology The Challenge is Changing Epidemiology is now a security issueEpidemiology is now a security issue –Complexity of society redefines contact –Potential & reality of pathogens as weapons Epidemiology is Now About Decisions

4. 08/19/2004Scale-Free Network Models in Epidemiology Modeling Options Current statistical models don’t workCurrent statistical models don’t work –Oversimplified –No superspreader events (SARS) Simple network models have limited utilitySimple network models have limited utility Recent discoveries suggest application of scale-free networksRecent discoveries suggest application of scale-free networks –Broad applicability (cells => society) –Interesting links to Chaos Theory

5. 08/19/2004Scale-Free Network Models in Epidemiology Statistical Approaches  Susceptible-Infected-Susceptible Model (SIS) R SSI E  Susceptible-Infected-Removed Model (SIR)  Susceptible-Exposed-Infected- Removed (SEIR)

6. 08/19/2004Scale-Free Network Models in Epidemiology Differential 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 = 1 S + E + I + R = 1 1 /   Mean latent period for the disease.   Contact rate. 1 /   Mean infection rate.

7. 08/19/2004Scale-Free Network Models in Epidemiology Statistical Systems Presume Randomness Research Question Research Question: Is the epidemiological network Random? …or ??

8. 08/19/2004Scale-Free Network Models in Epidemiology Network Models Differential Equations model assumes the population is “fully mixed” (random).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.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. The number of contacts and the frequency of interaction vary from individual to individual. These patterns can be best modeled as a NETWORK. These patterns can be best modeled as a NETWORK.

9. 08/19/2004Scale-Free Network Models in Epidemiology Scale-Free Network A small proportion of the nodes in a scale-free network have high degree of connection. A small proportion of the nodes in a scale-free network have high degree of connection. Power law distribution P(k)  O(k -  ). 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]. 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: Examples of known scale-free networks: – Communication Network - Internet – Ecosystems and Cellular Systems – Social network responsible for spread of disease

10. 08/19/2004Scale-Free Network Models in Epidemiology Reprinted from Linked: The New Science of Networks by Albert-Laszlo Barabasi

11. 08/19/2004Scale-Free Network Models in Epidemiology Generation of Scale-Free Network The vertices are distributed at random in a plane. The vertices are distributed at random in a plane. An edge is added between each pair of vertices with probability p. An edge is added between each pair of vertices with probability p. 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.

12. 08/19/2004Scale-Free Network Models in Epidemiology Problems with this Approach Waxman model inappropriate for creating scale-free networksWaxman model inappropriate for creating scale-free networks Most current topology generators are not up to this task!Most current topology generators are not up to this task! One main characteristic of scale-free networks is addition of nodes over timeOne main characteristic of scale-free networks is addition of nodes over time

13. 08/19/2004Scale-Free Network Models in Epidemiology Procedure 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

14. 08/19/2004Scale-Free Network Models in Epidemiology Proposed Simulator Multi-stage ComputationMulti-stage Computation Separate Interaction and Decision NetworksSeparate Interaction and Decision Networks Multi-dimensional Network LayeringMulti-dimensional Network Layering Extensible Data SourcesExtensible Data Sources Decomposable/Recomposable NodesDecomposable/Recomposable Nodes Introduce concept of SuperStopperIntroduce concept of SuperStopper

15. 08/19/2004Scale-Free Network Models in Epidemiology TWO-PHASE COMPUTATION Separate Progression & TransmissionSeparate Progression & Transmission Progression: Track internal factorsProgression: Track internal factors –Node susceptibility (e.g., general health) –Token infectiousness Transmission: Track inter-nodal transitionTransmission: Track inter-nodal transition –External catalytic effects –Token dynamics (e.g., spread, blockage, etc)

16. 08/19/2004Scale-Free Network Models in Epidemiology INTERACTION NETWORK Population connectivity graphPopulation 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

17. 08/19/2004Scale-Free Network Models in Epidemiology PRIMITIVES Set of j Nodes N={n I, n II, …, n j }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 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 m Communities C={c I, c II, …, c m } Set of p Attributes A={a I, a II, …, a p }Set of p Attributes A={a I, a II, …, a p } Set of q Functions F={f I, f II, …, f q }Set of q Functions F={f I, f II, …, f q }

18. 08/19/2004Scale-Free Network Models in Epidemiology DECISION NETWORK Separate overlay network defining control decision parameters which are applied to the Interaction 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.

19. 08/19/2004Scale-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 NOAA weather/environmental data Multi-source intelligence assessments Multi-source intelligence assessments

20. 08/19/2004Scale-Free Network Models in Epidemiology FUTURE WORK Refine theoretical frameworkRefine theoretical framework Computational capability/architectureComputational capability/architecture Simulator developmentSimulator development Extensible data source compilationExtensible data source compilation Host systems acquisitionHost systems acquisition Partnering for research and implementationPartnering for research and implementation

21. 08/19/2004Scale-Free Network Models in Epidemiology Concluding Perspectives Computational Opportunities Theory and Policy Chaos and Complexity Imperative for Alchemy