Global dynamics–convergence to equilibria–of epidemic patch models with immigration

We consider SIR and SIS epidemic models with bilinear incidence and migration between two patches, where infected individuals cannot migrate from one patch to another due to medical screening. We find the thresholds classifying the global dynamics of the models in terms of the model parameters, and we obtain the global asymptotical stability of the disease free and the disease endemic (in one patch or in both patches) equilibria. This global asymptotic stability of endemic equilibria is established by using a novel Lyapunov function.