Analysis of a two-patch SIS model with saturating contact rate and one- directing population dispersal

In this paper, a two-patch SIS model with saturating contact rate and one-directing population dispersal is proposed. In the model, individuals can only migrate from patch 1 to patch 2. The basic reproduction number $ R_0^1 $ of patch 1 and the basic reproduction number $ R_0^2 $ of patch 2 is identified. The global dynamics are completely determined by the two reproduction numbers. It is shown that if $ R_0^1 < 1 $ and $ R_0^2 < 1 $, the disease-free equilibrium is globally asymptotically stable; if $ R_0^1 < 1 $ and $ R_0^2 > 1 $, there is a boundary equilibrium which is globally asymptotically stable; if $ R_0^1 > 1 $, there is a unique endemic equilibrium which is globally asymptotically stable. Finally, numerical simulations are performed to validate the theoretical results and reveal the influence of saturating contact rate and migration rate on basic reproduction number and the transmission scale.