Spread of disease in a patchy environment

For a two patches SIR model, it is shown that its dynamic behavior is determined by several quantities. We have shown that if R0 < 1, then the disease-free equilibrium is globally asymptotically stable, otherwise it is unstable. Some sufficient conditions for the local stability of boundary equilibria are obtained. Numerical simulations indicate that travel between patches can reduces oscillations in both patches; may enhances oscillations in both patches; or travel switches oscillations from one patch to another.