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We study multi-patch epidemic models where individuals may migrate from onepatch to another in either of the susceptible, exposed/latent, infectious andrecovered states. We assume that infections occur both locally with a rate thatdepends on the patch as well as "from distance" from all the other patches. Theexposed and infectious periods have general distributions, and are not affectedby the possible migrations of the individuals. The migration processes ineither of the three states are assumed to be Markovian, and independent of theexposed 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 isdetermined by a set of Volterra integral equations. In the special case ofdeterministic exposed and infectious periods, the limit becomes a system ofODEs with delays. In the FCLT, the limit is given by a set of stochasticVolterra integral equations driven by a sum of independent Brownian motions andcontinuous Gaussian processes with an explicit covariance structure.

Multi-patch epidemic models with general exposed and infectious periods