Brownian snails with removal die out in one dimension

Brownian snails with removal is a spatial epidemic model defined as follows. Initially, a homogeneous Poisson process of susceptible particles on ${\mathbb{R}}^d$ with intensity $\lambda >0$ is deposited and a single infected one is added at the origin. Each particle performs an independent standard Brownian motion. Each susceptible particle is infected immediately when it is within distance 1 from an infected particle. Each infected particle is removed at rate $\alpha >0$, and removed particles remain such forever. Answering a question of Grimmett and Li, we prove that in one dimension, for all values of $\lambda$ and $\alpha$, the infection almost surely dies out.