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

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 separating a phase where the outbreak cluster is finite from the other one where it percolates through the system. 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.

Automated summary

The size and shape of the affected region play a crucial role in understanding disease dynamics and organizing future actions. This study explores a modified SIR model where agents diffuse on a lattice, and disease transmission occurs between infected and susceptible agents that are nearest neighbors. The research investigates the geometric properties of the outbreak and unvisited clusters, finding a hybrid transition separating a finite outbreak cluster from one that percolates through the system. Additionally, the outbreak cluster exhibits similar behavior to the critical cluster of ordinary percolation, while clusters with unvisited sites have a size distribution with a Fisher exponent tau < 2.