Pandemic Spread in Communities via Random Graphs

TL;DR

This paper models pandemic spread across multiple communities using multi-type random graphs, deriving conditions for outbreaks, calculating infected fractions, and analyzing effects of interventions and herd immunity.

Contribution

It introduces a multi-type random contact graph framework to analyze pandemic dynamics, including outbreak conditions and the impact of countermeasures.

Findings

The basic reproduction number equals the Perron-Frobenius eigenvalue of the contact matrix.

The model predicts the size of the infected population based on eigenvector analysis.

Simulations show differences between homogeneous and heterogeneous contact structures.

Abstract

Working in the multi-type Galton-Watson branching-process framework we analyse the spread of a pandemic via a general multi-type random contact graph. Our model consists of several communities, and takes, as input, parameters that outline the contacts between individuals in distinct communities. Given these parameters, we determine whether there will be an outbreak and if yes, we calculate the size of the giant connected component of the graph, thereby, determining the fraction of the population of each type that would be infected before it ends. We show that the pandemic spread has a natural evolution direction given by the Perron-Frobenius eigenvector of a matrix whose entries encode the average number of individuals of one type expected to be infected by an individual of another type. The corresponding eigenvalue is the basic reproduction number of the pandemic. We perform numerical simulations that compare homogeneous and heterogeneous spread graphs and quantify the difference between them. We elaborate on the difference between herd immunity and the end of the pandemic and the effect of countermeasures on the fraction of infected population.