Scale-Free Network Models in Epidemiology Preliminary Findings

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) 08/19/2004 Scale-Free Network Models in Epi demiology

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 Epi demiology

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 Epi demiology

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

Differential Equations SIR Model SEIR Model 1 / Mean latent period for the disease. Contact rate. 1 / Mean infection rate. 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 Epi demiology

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

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 Epi demiology

Scale-Free Network A small proportion of the nodes in a scalefree 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 Epi demiology

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

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 Epi demiology

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 Epi demiology

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 08/19/2004 Decision functions calculate exposure, infection, removal Numerical experiments with differing decision functions/parameters Scale-Free Network Models in Epi demiology

Proposed Simulator 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 Epi demiology

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 Epi demiology

INTERACTION NETWORK Population connectivity graph Key Challenges – Data Temporality: Input data (even nearreal 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 Epi demiology

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} 08/19/2004 Scale-Free Network Models in Epi demiology

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 Epi demiology

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 Epi demiology

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 Epi demiology

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