Imagine a pathogen that is spreading radially as a circular wave through a population of susceptible hosts. In the interior of this circular region, the infection dies out due to a subcritical density of susceptibles. If a mutant pathogen, having some advantage over wild-type pathogens, arises in this region it is likely to die out without leaving a noticeable trace. Mutants that arise closer to the infection wavefront have access to more susceptible hosts and thus are more likely to become established and perhaps (locally) out-compete the original pathogen. Among the factors (position, transmission rate, pathogen-induced death rate) that influence the fate of a mutant, which are most important? What does this tell us about the types of mutants that are likely to invade and become established? How do such tendencies serve to steer the evolution of pathogens in a spatial setting? Do different types of models of the same phenomena lead to similar conclusions? We address these issues from the point of view of an individual-based stochastic spatial model of host-pathogen interactions. We consider the probability of a successful invasion by a single mutant as a function of the transmissibility and virulence strengths and the mutant position in the wavefront. Next, for a version of the model in which mutations arise spontaneously, we obtain analytical and simulation results on the mean time to a successful invasion. We also use our model predictions to gain insight into experimental data on bacteriophage plaques. Finally, we compare our results to those based on ordinary and partial differential equations to better understand how different models might influence our predictions on the fate of a mutant pathogen.