Multi-patch epidemic models with general exposed and infectious periods

We study   multi-patch epidemic models where individuals may migrate from one patch to another in either of the susceptible, exposed/latent, infectious and recovered states.  We assume that infections occur both locally with a rate that depends on the patch as well as ``from distance" from all the other patches. The exposed and infectious periods have general distributions, and are not affected by the possible migrations of the individuals. The migration processes in either of the three states are assumed to be Markovian, and independent of the exposed and infectious periods. We establish a functional law of large number (FLLN) and a function central limit theorem (FCLT) for the susceptible, exposed/latent, infectious and recovered processes. In the FLLN, the limit is determined by a set of Volterra integral equations. In the special case of deterministic exposed and infectious periods, the limit becomes a system of ODEs with delays. In the FCLT, the limit is given by a set of stochastic Volterra integral equations driven by a sum of independent Brownian motions and   continuous Gaussian processes with an explicit covariance structure.