Recent advances in epidemic modeling: non-Markov stochastic models and their scaling limits

In this survey paper, we review recent advances in individual based non–Markovian epidemic models. They include epidemic models with a constant infectivity rate, varying infectivity rate or infection-age dependent infectivity, infection-age dependent recovery rate (or equivalently, general law of infectious period), as well as varying susceptibility/immunity. We focus on the scaling limits with a large population, functional law of large numbers (FLLN) and functional central limit theorems (FCLT), while the large and moderate deviations for some Markovian epidemic models are also reviewed. In the FLLN, the limits are either a set of Volterra integral equations, or a system of coupled ODE/PDEs with integral boundary conditions. In the FCLT, the limits are stochastic Volterra integral equations driven by Gaussian processes. We relate our deterministic limits to results of Kermack and McKendrick published in their 1927, 1932 and 1933 seminal papers, where varying infectivity and susceptibility/immunity were already considered. We also discuss some extensions such as models with heterogeneous population, spatial models and control problems, and present some open problems.