Functional Central Limit Theorems for epidemic models with varying infectivity and waning immunity

We study an individual-based stochastic epidemic model in which infected individuals gradually become susceptible again following each infection (generalized SIS model). The epidemic dynamics is described by the average infectivity and susceptibility processes in the population together with the numbers of infected and susceptible/uninfected individuals. In R. Forien et al., Stochastic epidemic models with varying infectivity and susceptibility. arXiv preprint arXiv:2210.04667 (2022), a functional law of large numbers (FLLN) is proved as the population size goes to infinity, and asymptotic endemic behaviors are also studied. In this paper, we prove a functional central limit theorem (FCLT) for the stochastic fluctuations of the epidemic dynamics around the FLLN limit. The FCLT limit for the aggregate infectivity and susceptibility processes is given by a system of stochastic non-linear integral equation driven by a two-dimensional Gaussian process.