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

We study a stochastic epidemic model with multiple patches (locar tions), where individuals in each patch are categorized into three compartments, Susceptible, Infected and Recovered/Removed, and may migrate from one patch to another in any of the compartments. Each individual is associated with a random infectivity function which dictates the force of infection depending upon the age of infection (elapsed time since infection). We prove a functional law of large number for the epidemic evolution dynamics including the aggregate infec¬tivity process, the numbers of susceptible and recovered individuals as well as the number of infected individuals 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 of infected individuals with respect to the infection age, which is a system of linear PDE equations with a boundary condition that is determined by a set of integral equations.