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

Functional law of large numbers and PDEs for epidemic models with infection-age dependent infectivity