Mathematical models for epidemic spreading on complex networks

TL;DR

This paper introduces a discrete model for epidemic spreading on finite complex networks, analyzing the epidemic threshold, quasi-stationary distribution, and mixing time, highlighting differences from continuous models.

Contribution

It presents a novel discrete approach to epidemic modeling on complex networks, including a theorem on mixing time scaling with network size and proximity to the epidemic threshold.

Findings

Mixing time scales logarithmically with network size.

Mixing time is inversely proportional to the distance from the epidemic threshold.

The model accounts for at most one contamination per time step.

Abstract

We propose a model for epidemic spreading on a finite complex network with a restriction to at most one contamination per time step. Because of a highly discrete character of the process, the analysis cannot use the continous approximation, widely exploited for most of the models. Using discrete approach we investigate the epidemic threshold and the quasi-stationary distribution. The main result is a theorem about mixing time for the process, which scales like logarithm of the network size and which is proportional to the inverse of the distance from the epidemic threshold. In order to present the model in the full context, we review modern approach to epidemic spreading modeling based on complex networks and present necessary information about random networks, discrete-time Markov chains and their quasi-stationary distributions.