The Ross–Macdonald model in a patchy environment

We generalize to n   patches the Ross–Macdonald model which describes the dynamics of malaria. We incorporate in our model the fact that some patches can be vector free. We assume that the hosts can migrate between patches, but not the vectors. The susceptible and infectious individuals have the same dispersal rate. We compute the basic reproduction ratio R0R0. We prove that if R0⩽1R0⩽1, then the disease-free equilibrium is globally asymptotically stable. When R0>1R0>1, we prove that there exists a unique endemic equilibrium, which is globally asymptotically stable on the biological domain minus the disease-free equilibrium.