Local and global survival for infections with recovery

TL;DR

This paper proves conditions under which an infection modeled by particles on a lattice survives indefinitely, including cases with recovery, by solving open problems related to infection persistence and density parameters.

Contribution

It establishes the survival of infection with recovery in particle systems, addressing two open problems from prior research and analyzing the impact of initial density and recovery rates.

Findings

Infection survives infinitely often at the origin for small recovery rates.

Existence of density thresholds ensuring survival regardless of recovery rate.

Proof of open problems from Kesten and Sidoravicius.

Abstract

We establish two open problems from Kesten and Sidoravicius [8]. Particles are initially placed on $\Z^{d}$ with a given density and evolve as independent continuous-time random walks. Particles initially placed at the origin are declared as infected. Infection transmits instantaneously to healthy particles on the same site and infected particles become healthy with a positive rate. We prove that, for small enough recovery rates, the infection process survives and visits the origin infinitely many times on the event of survival. Second, we establish the existence of density parameters for which the infection survives for all choices of the recovery rate.