Minimal percolating sets for mutating infectious diseases

TL;DR

This paper introduces a time-dependent percolation model to better understand the spread of mutating infectious diseases, providing algorithms to identify minimal initial infection sets on trees.

Contribution

It generalizes bootstrap percolation by incorporating a dynamic percolation function and develops an algorithm to find minimal percolating sets on finite trees.

Findings

Defined F(t)-bootstrap percolation as a new model

Proved properties of the generalized percolation process

Developed a polynomial-time algorithm for minimal sets on trees

Abstract

This paper is dedicated to the study of the interaction between dynamical systems and percolation models, with views towards the study of viral infections whose virus mutate with time. Recall that r-bootstrap percolation describes a deterministic process where vertices of a graph are infected once r neighbors of it are infected. We generalize this by introducing F(t)-bootstrap percolation, a time-dependent process where the number of neighbouring vertices which need to be infected for a disease to be transmitted is determined by a percolation function F(t) at each time t. After studying some of the basic properties of the model, we consider smallest percolating sets and construct a polynomial-timed algorithm to find one smallest minimal percolating set on finite trees for certain F(t)-bootstrap percolation models.