Functional limit theorems for non-Markovian epidemic models

We study non-Markovian stochastic epidemic models (SIS, SIR, SIRS, and SEIR), in which the infectious (and latent/exposing, immune) periods have a general distribution. provide representation of evolution dynamics using time epochs infection latency/exposure, immunity). Taking limit as size population tends to infinity, we prove both functional law large number (FLLN) central theorem (FCLT) for processes interest these models. In FLLN, limits are unique solution system deterministic Volterra integral equations, while FCLT, multidimensional Gaussian solutions linear equations. proof an important Poisson random measures diffusion-scaled converging components driving process.