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We study a stochastic epidemic model with multiple patches (locations), whereindividuals in each patch are categorized into three compartments, Susceptible,Infected and Recovered/Removed, and may migrate from one patch to another inany of the compartments. Each individual is associated with a randominfectivity function which dictates the force of infectivity while theinteractive infection process depends on the age of infection (elapsed timesince infection). We prove a functional law of large number for the epidemicevolution dynamics including the aggregate infectivity process, the numbers ofsusceptible and recovered individuals as well as the number of infectedindividuals at each time that have been infected for a certain amount of time.From the limits, we derive a PDE model for the density of the number ofinfected individuals with respect to the infection age, which is a systems oflinear PDE equations with a boundary condition that is determined by a set ofintegral equations.

PDE model for multi-patch epidemic models with migration and infection-age dependent infectivity