Functional central limit theorems for epidemic models with varying infectivity

Pang, Guodong; Pardoux, Etienne

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 at the elapsed time (that is, infection-age dependent). 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 FCLT for the generalized SEIR model (including SIR as a special case), the limits for 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 exposed/latent, infected 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.