  We develop a new perturbation method for studying quasi-neutral competition
in a broad class of stochastic competition models, and apply it to the analysis
of fixation of competing strains in two epidemic models. The first model is a
two-strain generalization of the stochastic Susceptible-Infected-Susceptible
(SIS) model. Here we extend previous results due to Parsons and Quince (2007),
Parsons et al (2008) and Lin, Kim and Doering (2012). The second model, a
two-strain generalization of the stochastic Susceptible-Infected-Recovered
(SIR) model with population turnover, has not been studied previously. In each
of the two models, when the basic reproduction numbers of the two strains are
identical, a system with an infinite population size approaches a point on the
deterministic coexistence line (CL): a straight line of fixed points in the
phase space of sub-population sizes. Shot noise drives one of the strain
populations to fixation, and the other to extinction, on a time scale
proportional to the total population size. Our perturbation method explicitly
tracks the dynamics of the probability distribution of the sub-populations in
the vicinity of the CL. We argue that, whereas the slow strain has a
competitive advantage for mathematically "typical" initial conditions, it is
the fast strain that is more likely to win in the important situation when a
few infectives of both strains are introduced into a susceptible population.
