Pandemic spread in communities via random graphs

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.

Automated summary

Working within the multi-type Galton-Watson branching-process framework, we analyzed the spread of a pandemic through a general multi-type random contact graph. Our model, consisting of multiple communities, determines outbreak likelihood and calculates the size of the giant-connected-component of the graph to predict population infection rates. The pandemic spread is shown to have a natural evolution direction determined by the Perron-Frobenius eigenvector, with the basic reproduction number as the corresponding eigenvalue. Numerical simulations compare homogeneous and heterogeneous spread graphs, emphasizing the impact of countermeasures on infected population fractions.