Local and global survival for infections with recovery

Abstract

We treat infection models with recovery and establish two open problems from Kesten and Sidoravicius [8]. Particles are initially placed on the d-dimensional integer lattice with a given density and evolve as independent continuous-time simple random walks. Particles initially placed at the origin are declared as infected, and healthy particles immediately become infected when sharing a site with an already infected particle. Besides, infected particles become healthy with a positive rate. We prove that, provided the recovery rate is small enough, the infection process not only survives, but also 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 possible choices of the recovery rate.