Spatial Invasion of Cooperative Parasites

TL;DR

This paper analyzes the invasion probabilities and times of cooperative parasites in spatially structured host populations modeled by random geometric graphs, providing bounds and asymptotic characterizations supported by simulations.

Contribution

It introduces bounds and asymptotic results for invasion probabilities and times of cooperative parasites in spatial networks, extending previous models to spatially structured populations.

Findings

Invasion probabilities relate to survival probabilities of cooperative branching processes.

Asymptotic invasion times are characterized mathematically.

Simulation results support theoretical bounds.

Abstract

In this paper we study invasion probabilities and invasion times of cooperative parasites spreading in spatially structured host populations. The spatial structure of the host population is given by a random geometric graph on $[0,1]^n$, $n\in \mathbb{N}$, with a Poisson($N$)-distributed number of vertices and in which vertices are connected over an edge when they have a distance of at most $r_N\in \Theta\left(N^{\frac{\beta-1}{n}}\right)$ for some $0<\beta<1$ and $N\rightarrow \infty$. At a host infection many parasites are generated and parasites move along edges to neighbouring hosts. We assume that parasites have to cooperate to infect hosts, in the sense that at least two parasites need to attack a host simultaneously. We find lower and upper bounds on the invasion probability of the parasites in terms of survival probabilities of branching processes with cooperation. Furthermore, we characterize the asymptotic invasion time. An important ingredient of the proofs is a comparison with infection dynamics of cooperative parasites in host populations structured according to a complete graph, i.e. in well-mixed host populations. For these infection processes we can show that invasion probabilities are asymptotically equal to survival probabilities of branching processes with cooperation. Furthermore, we build in the proofs on techniques developed in [BP22], where an analogous invasion process has been studied for host populations structured according to a configuration model. We substantiate our results with simulations.