A branching random-walk model of disease outbreaks and the percolation backbone

TL;DR

This paper models disease outbreaks using a branching random walk on a lattice, revealing a mixed-order transition and geometric properties similar to percolation theory, with implications for understanding epidemic spread.

Contribution

It introduces a lattice-based SIR extension with geometric analysis, uncovering a hybrid phase transition and percolation-like properties of outbreak clusters.

Findings

Identifies a mixed-order (hybrid) transition in outbreak size.

Outbreak clusters share exponents with percolation backbone.

Unvisited site clusters follow a Fisher exponent less than 2.

Abstract

The size and shape of the region affected by an outbreak is relevant to understand the dynamics of a disease and help to organize future actions to mitigate similar events. A simple extension of the SIR model is considered, where agents diffuse on a regular lattice and the disease may be transmitted when an infected and a susceptible agents are nearest neighbors. We study the geometric properties of both the connected cluster of sites visited by infected agents (outbreak cluster) and the set of clusters with sites that have not been visited. By changing the density of agents, our results show that there is a mixed-order (hybrid) transition where the region affected by the disease is finite in one phase but percolates through the system beyond the threshold. Moreover, the outbreak cluster seems to have the same exponents of the backbone of the critical cluster of the ordinary percolation while the clusters with unvisited sites have a size distribution with a Fisher exponent $\tau<2$.