arXiv subjects

Sascha Franck

Publications and source records attributed to Sascha Franck.

2 recordsLinked to original sources

arXiv2025

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

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.

math.PR↗

arXiv2025

A phase transition for a spatial host-parasite model with extreme host immunities on $\mathbb{Z}^d$ and $\mathbb{T}_d$

We investigate a model of a parasite population invading spatially distributed immobile hosts on a graph, which is a modification of the frog model. Each host has an unbreakable immunity against infection with a certain probability $1-p$ and parasites move as simple symmetric random walks attempting to infect any host they encounter and subsequently reproduce themselves. We show that, on $\mathbb{Z}^d$ with $d\ge 2$ and the $d$-regular tree $\mathbb{T}_d$ with $d\ge 3$, the survival probability of parasites exhibits a phase transition at a critical value of $p_c\in(0,1)$. Also, we show that adding vertices and edges to the underlying graph can, in general, both increase or decrease the value of $p_c$. Finally, we show that on quasi-vertex-transitive graphs, with probability $1$, a fixed vertex is only visited finitely often by a parasite under mild assumptions on the offspring distribution of parasites.

math.PR↗