# Invasion, Stability and Evolution to Criticality in Spatially Extended, Artificial Host-Pathogen Ecologies

D. A. Rand, M. Keeling, H. B. Wilson

## Abstract

We consider an individual-based spatial model of a generic host-pathogen system and explore the differences between such models and mean-field systems. We find a range of new dynamical and evolutionary phenomena, in particular: (i) in this system, selective pressure is substantially reduced compared with the corresponding mean-field models, and artificial suppression of the pathogen population leads to faster evolution and reduces evolutionary stability; (ii) unlike the mean-field models, there exists a critical transmissibility $\tau _{\text{c}}$ above which the pathogen dies out; and (iii) the system displays self-evolved criticality. If the transmissibility $\tau$ is allowed to mutate, it evolves to the critical value $\tau _{\text{c}}$. Thus the system evolves to put itself at the boundary at which it can exist. Observations of the individual-based spatial model motivate an explanation for these phenomena in terms of the dynamics of host patches involving their connections and disconnections. We therefore construct a patch model of this and show that this simplified model behaves in a similar way to the individual-based spatial model.