  We study epidemic models where the infectivity of each individual is a random
function of the infection age (the elapsed time of infection). To describe the
epidemic evolution dynamics, we use a stochastic process that tracks the number
of individuals at each time that have been infected for less than or equal to a
certain amount of time, together with the aggregate infectivity process. We
establish the functional law of large numbers (FLLN) for the stochastic
processes that describe the epidemic dynamics. The limits are described by a
set of deterministic integral equations, which has a further characterization
using PDEs under some regularity conditions. The solutions are characterized
with boundary conditions that are given by a system of Volterra equations. We
also characterize the equilibrium points for the PDEs in the SIS model with
infection-age dependent infectivity. To establish the FLLNs, we employ a useful
criterion for weak convergence for the two-parameter processes together with
useful representations for the relevant processes via Poisson random measures.
