Percolation and the pandemic

TL;DR

This paper explores how percolation theory models epidemic outbreaks, analyzing the critical thresholds related to infectivity and contact range across different network structures.

Contribution

It links percolation thresholds to epidemiological parameters like R0 and examines effects of contact range on epidemic percolation models.

Findings

Smaller contact ranges raise the critical R0 needed for epidemic spread.

Percolation thresholds vary with network topology and connectivity.

Extended-range contacts lower the epidemic threshold in lattice models.

Abstract

This paper is dedicated to the memory of Dietrich Stauffer, who was a pioneer in percolation theory and applications of it to problems of society, such as epidemiology. An epidemic is a percolation process gone out of control, that is, going beyond the critical transition threshold $p_c$. Here we discuss how the threshold is related to the basic infectivity of neighbors $R_0$, for trees (Bethe lattice), trees with triangular cliques, and in non-planar lattice percolation with extended-range connectivity. It is shown how having a smaller range of contacts increases the critical value of $R_0$ above the value $R_{0,c}=1$ appropriate for a tree, an infinite-range system or a large completely connected graph.