ScaleFree Network Models in Epidemiology Preliminary Findings Jill

Scale-Free Network Models in Epidemiology Preliminary Findings Jill Bigley Dunham F. Brett Berlin George Mason University 08/19/2004 19 August Scale-Free Network 2004 Models in Epidemiology

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

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

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

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

Procedure 1. Create scale-free network • • Georgia Tech - Internetwork Topology Model and ns 2 with Waxman model Deterministic scale-free network generation -- Barabasi, et. al. 2. Apply simulation parameters • Numerical experiments, etc. 3. Step simulation through time • • 08/19/2004 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 Super. Stopper 08/19/2004 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) 08/19/2004 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 08/19/2004 Scale-Free Network Models in Epidemiology

PRIMITIVES • Set of j Nodes N={n. I, n. II, … , nj} • 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, …, cm} • Set of p Attributes A={a. I, a. II, …, ap} • Set of q Functions F={f. I, f. II, …, fq} 08/19/2004 Scale-Free Network Models in Epidemiology

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

FUTURE WORK • • • 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

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