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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.
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