Epidemic modeling on real networks with spectral graph theory

Abstract

Compartmental epidemiology models have been used to model and study epidemics since the early 20th century. Questions such as "in what conditions will a virus become an epidemic?'' and "how long will it take for the virus to be controlled?" can be better understood with the SI and SIR models. However, in many ways compartment models are overly simplistic and unrepresentative of real networks. We attempt to address some of these simplifications by modeling a virus spreading through a population as a graph, where spectral graph theory ideas can be applied.

In this paper, we first present an introduction to the underlying mathematics and basic properties of both spectral graph theory and epidemic modeling. Then, we present a SIR-like system that can model viral propagation through any graph structure. We experimentally examine the behavior of an epidemic for graphs of various structures and show that the epidemic threshold depends on the inverse of the largest eigenvalue of the graph's adjacency matrix, as was previously proven for the SIS model.