Survival and parasite spread in a spatial host-parasite model with host immunity

Abstract

We introduce a stochastic model for the invasion of a parasite population in a spatially structured host population, which includes an individual-based adaptive immune response. We will call this the "Spatial Infection Model with Host Immunity" or SIMI for short. In the SIMI, parasites move as independent simple random walks on a graph until they meet a vertex that is inhabited by a host. With a given probability, the host repels the infection, kills the parasite, and adapts its probability to repel the next infection. After a successful infection attempt, both the host and the attacking parasite die, and only the parasite leaves a random number of offspring. We study the SIMI on the integer line and show that parasites have a positive survival probability if and only if the mean offspring are greater than the mean number of needed infection attempts. Furthermore, we study the speed at which the parasites invade the host population. If the probability of a host after repelling an infection, to also repel the next one does not grow fast enough, then parasites propagate across the host population at a linear speed. However, if that probability grows quickly enough, the propagation speed is polynomial with exponent less than 1. Finally, we investigate the SIMI on higher-dimensional graphs with hosts that are either totally immune and never get infected or get infected in the first attempt. We show that the survival probability undergoes a phase transition in the frequency of totally immune hosts.