  We introduce an epidemic model with varying infectivity and general exposed
and infectious periods, where the infectivity of each individual is a random
function of the elapsed time since infection, those function being i.i.d. for
the various individuals in the population. This approach models infection-age
dependent infectivity, and extends the classical SIR and SEIR models. We focus
on the infectivity process (total force of infection at each time), and prove a
functional law of large number (FLLN). In the deterministic limit of this LLN,
the infectivity process and the susceptible process are determined by a
two-dimensional deterministic integral equation. From its solutions, we then
derive the exposed, infectious and recovered processes, again using integral
equations. For the early phase, we study the stochastic model directly by using
an approximate (non--Markovian) branching process, and show that the epidemic
grows at an exponential rate on the event of non-extinction, which matches the
rate of growth derived from the deterministic linearized equations. We also use
these equations to derive the basic reproduction number $R_0$ during the early
stage of an epidemic, in terms of the average individual infectivity function
and the exponential rate of growth of the epidemic.
