Recent biomathematical literature has suggested that, under the assumption of a trade-off between replication speed and fidelity, a pathogen can evolve to more than one optimal mutation rate. O'Fallon (2011) presents a particularly compelling case grounded in simulation. In this thesis, we treat the subject analytically, approaching it through the lens of adaptive dynamics. We formulate a within-host model of the pathogen load starting from assumptions at the genomic level, explicitly accounting for the fact that most mutations are deleterious and stunt growth. We single out the pathogen's mutation probability as the evolving trait that distinguishes strains from one another. Our between-host dynamics take the form of an SI model, first without superinfection and later with two types of non-smooth superinfection function. The pathogen's virulence and transmission rate are functions of the within-host equilibrium pathogen densities. In the case of our mechanistically defined superinfection function, we uncover evolutionary branching in conjunction with two transmission functions, one a caricatural (expansion) example, the other a more biologically realistic (logistic) one. Because of the non-smoothness of the mechanistic superinfection function, our branching points are actually one-sided ESSs à la Boldin and Diekmann (2014). When branching occurs, two strains with different mutation probabilities both ultimately persist on the evolutionary timescale.
