  We consider a simple stochastic model for the spread of a disease caused by
two virus strains in a closed homogeneously mixing population of size N. The
spread of each strain in the absence of the other one is described by the
stochastic logistic SIS epidemic process, and we assume that there is perfect
cross-immunity between the two strains, that is, individuals infected by one
are temporarily immune to re-infections and infections by the other. For the
case where one strain has a strictly larger basic reproductive ratio than the
other, and the stronger strain on its own is supercritical (that is, its basic
reproductive ratio is larger than 1), we derive precise asymptotic results for
the distribution of the time when the weaker strain disappears from the
population, that is, its extinction time. We further extend our results to
certain parameter values where the difference between the two reproductive
ratios may tend to 0 as $N \to \infty$.
  In proving our results, we illustrate a new approach to a fluid limit
approximation for a sequence of Markov chains in the vicinity of a stable fixed
point of the limit.
