Limits of a multi-patch SIS epidemic model

We start from a stochastic SIS model for the spread of epidemics among a population partitioned into M sites, each containing N individuals; epidemic spread occurs through within-site ('local') contacts and global contacts. We analyse the limit behaviour of the system as M and N increase to infinity. Two limit procedures are considered, according to the order in which M and N go to infinity; independently of the order, the limiting distribution of infected individuals across sites is a probability measure, whose evolution in time is governed by the weak form of a PDE. Existence and uniqueness of the solutions to this problem is shown. Finally, it is shown that the infected distribution converges, as time goes to infinity, to a Dirac measure at the value x(*), the equilibrium of a single-patch SIS model with contact rate equal to the sum of local and global contacts.