Competition-exclusion and coexistence in a two-strain SIS epidemic model in patchy environments

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.