Résumé : We investigate the spread of a population of pathogens infecting a spatially distributed host population with immunity. We model this situation by placing susceptible immobile hosts on the vertices of $\mathbb{Z}$. Pathogens diffuse in space according to symmetric simple random walks on $\mathbb{Z}$ and attempt an infection when they meet a host. As hosts often have an immune response against infections, we assume that each host needs to be attacked a random number of times, according to some distribution $I$, before it will be infected. Otherwise parasite reproduction is prevented and the parasite gets killed. In case of an successful infection the parasite kills the host and sets free a random number of offspring, according to some distribution $A$. We characterize the survival probability of the pathogen population depending on the initial distribution of pathogens and show under some relatively mild conditions on $I$ and $A$, that conditioned on survival of the parasite population, the infection spreads a.s. asymptotically at least and at most linearly fast. This talk is based on joint work in progress with Sascha Franck.