Evolutionary Dynamics in an Individual Spatial and a Mean Field Differential Equation Host-Pathogen Model

Abstract

We examine a host-pathogen model in which three types of species exist: empty sites, healthy hosts, and infected hosts. In this model six different transitions can occur: empty sites can be colonized by healthy hosts, healthy hosts can be infected, and infected hosts can either recover or die. We implement this general model in both a spatial context with discrete time and in a homogeneously mixing model in continuous time. We then explore evolution for pairs of parameters, calculating viable regions in the ODE model and and evolutionary vector fields in both models. Our results show that results from the spatial model do not always converge to our ODE model results, that stochasticity in the spatial evolutionary vector field can be used as a measure of the magnitude of evolutionary pressure and as an indicator of non-viable parameters, and that the evolutionary pressures on different parameters are not necessarily independent. For example, a lower transmissibility greatly lowers the magnitude of evolutionary pressure for all parameters associated with transitions from infected hosts.