Scale-Free Network Models in Epidemiology Preliminary Findings

1. Scale-Free Network Models in EpidemiologyPreliminary Findings Jill BigleyDunham F. Brett Berlin George Mason University 19 August 2004

2. 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) Scale-Free Network Models in Epidemiology

3. 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 Scale-Free Network Models in Epidemiology

4. 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 Scale-Free Network Models in Epidemiology

5. S I S E R Statistical Approaches • Susceptible-Infected-Susceptible Model (SIS) • Susceptible-Infected-Removed Model (SIR) • Susceptible-Exposed-Infected- Removed (SEIR) Scale-Free Network Models in Epidemiology

6. Differential Equations 1 /   Mean latent period for the disease.   Contact rate. 1 /  Mean infection rate. • SIR Model • 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 Scale-Free Network Models in Epidemiology

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

8. 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. Scale-Free Network Models in Epidemiology

9. 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 Scale-Free Network Models in Epidemiology

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

11. 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. Scale-Free Network Models in Epidemiology

12. 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 Scale-Free Network Models in Epidemiology

13. 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 Scale-Free Network Models in Epidemiology

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

15. 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) Scale-Free Network Models in Epidemiology

16. 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 Scale-Free Network Models in Epidemiology

17. 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} Scale-Free Network Models in Epidemiology

18. 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. Scale-Free Network Models in Epidemiology

19. 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 Scale-Free Network Models in Epidemiology

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

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