  In this paper, we prove functional central limit theorems (FCLTs) for a
stochastic epidemic model with varying infectivity and general infectious
periods recently introduced in Forien, Pang and Pardoux (2020).The infectivity
process (total force of infection at each time) is composed of the independent
infectivity random functions of each infectious individual, which starts at the
time of infection. These infectivity random functions induce the infectious
periods (as well as exposed, recovered or immune periods in full generality),
whose probability distributions can be very general. The epidemic model
includes the generalized non-Markovian SIR, SEIR, SIS, SIRS models with
infection-age dependent infectivity. In the FCLTs for the generalized SIR and
SEIR models, the limits of the diffusion-scaled fluctuations of the infectivity
and susceptible processes are a unique solution to a two-dimensional
Gaussian-driven stochastic Volterra integral equations, and then given these
solutions, the limits for the infected (exposed/infectious) and recovered
processes are Gaussian processes expressed in terms of the solutions to those
stochastic Volterra integral equations. We also present the FCLTs for the
generalized SIS and SIRS models.
