A spatial host-parasite model with host immunity: Survival and linear spread of parasites on $\mathbb{Z}$

TL;DR

This paper models parasite invasion in a spatial host population with immunity, showing survival depends on offspring and immunity, and that parasites spread linearly if they survive.

Contribution

It introduces a generalized frog model incorporating host immunity and proves linear spread of parasites conditioned on survival.

Findings

Parasite survival depends on mean offspring and immunity height.

Parasites invade at linear speed given survival.

Survival probability is characterized by offspring and immunity means.

Abstract

We introduce a generalized version of the frog model to describe the invasion of a parasite population in a spatially structured immobile host population with host immunity on the integer line. Parasites move according to simple symmetric random walks and try to infect any host they meet. Hosts, however, own an immunity against the parasites that protects them from infection for a random number of attacks. Once a host gets infected, it and the infecting parasite die, and a random number of offspring parasites is generated. We show that the positivity of the survival probability of parasites only depends on the mean offspring and mean height of immunity. Furthermore, we prove through the construction of a renewal structure that given survival of the parasite population parasites invade the host population at linear speed under relatively mild assumptions on the host immunity distribution.