We formulate epidemiological models for the transmission of a pathogen that can mutate in the host to create a second infectious mutant strain. The models account for mutation rates that depend on how long the host has been infected. We derive explicit formulas for the reproductive number of the epidemic based on the local stability of the infection-free equilibrium. We analyze the existence and stability of the boundary equilibrium, whose infection components are zero and positive, respectively, and the endemic equilibrium, whose components are all positive. We establish the conditions for global stability of the infection-free and boundary equilibria and local stability of the endemic equilibrium for the case where there is no age structure for the pathogen in the infected population. We show that under certain circumstances, there is a Hopf bifurcation where the endemic equilibrium loses its stability, and periodic solutions appear. We provide examples and numerical simulations to illustrate the Hopf bifurcation.