Bounding basic characteristics of spatial epidemics with a new percolation model

TL;DR

This paper introduces a new percolation model for epidemic spread on directed graphs, providing bounds on outbreak probabilities and sizes, and generalizing many existing models in epidemiology.

Contribution

The paper presents a novel percolation framework that accounts for correlated infectivity and susceptibility, extending previous models and offering bounds on epidemic characteristics.

Findings

Bounds on percolation probability established

Expected cluster size bounded by independent percolation models

Model generalizes many existing epidemic models

Abstract

We introduce a new percolation model to describe and analyze the spread of an epidemic on a general directed and locally finite graph. We assign a two-dimensional random weight vector to each vertex of the graph in such a way that the weights of different vertices are i.i.d., but the two entries of the vector assigned to a vertex need not be independent. The probability for an edge to be open depends on the weights of its end vertices, but conditionally on the weights, the states of the edges are independent of each other. In an epidemiological setting, the vertices of a graph represent the individuals in a (social) network and the edges represent the connections in the network. The weights assigned to an individual denote its (random) infectivity and susceptibility, respectively. We show that one can bound the percolation probability and the expected size of the cluster of vertices that can be reached by an open path starting at a given vertex from above and below by the corresponding quantities for respectively independent bond and site percolation with certain densities; this generalizes a result of Kuulasmaa. Many models in the literature are special cases of our general model.