  This work examines the dynamics of solutions of a two-strain SIS epidemic
model in patchy environments. The basic reproduction number $\mathcal{R}_0$ is
introduced, and sufficient conditions are provided to guarantee the global
stability of the disease-free equilibrium (DFE). In particular, the DFE is
globally stable when either: (i) $\mathcal{R}_0\le \frac{1}{k}$, where $k\ge 2$
is the total number of patches, or (ii) $\mathcal{R}_0<1$ and the dispersal
rate of the susceptible population is large. Moreover, the questions of
competition-exclusion and coexistence of the strains are investigated when the
single-strain reproduction numbers are greater than one. In this direction,
under some appropriate hypotheses, it is shown that the strain whose basic
reproduction number and local reproduction function are the largest always
drives the other strain to extinction in the long run. Furthermore, the
asymptotic dynamics of the solutions are presented when either both strain's
local reproduction functions are spatially homogeneous or the population
dispersal rate is uniform. In the latter case, the invasion numbers are
introduced and the existence of coexistence endemic equilibrium (EE) is proved
when these invasion numbers are greater than one. Numerical simulations are
provided to complement the theoretical results.
