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In this survey paper, we review the recent advances in individual basednon--Markovian epidemic models. They include epidemic models with a constantinfectivity rate, varying infectivity rate or infection-age dependentinfectivity, infection-age recovery rate (or equivalently, general law ofinfectious period), as well as varying susceptibility/immunity. We focus on thescaling limits with a large population, functional law of large numbers (FLLN)and functional central limit theorems (FCLT), while the large and moderatedeviations for some Markovian epidemic models are also reviewed. In the FLLN,the limits are a set of Volterra integral equations, and in the FCLT, thelimits are stochastic Volterra integral equations driven by Gaussian processes.We relate our deterministic limits to the results in the seminal papers byKermack and McKendrick published in 1927, 1932 and 1933, where the varyinginfectivity and susceptibility/immunity were already considered. We alsodiscuss some extensions, including models with heterogeneous population,spatial models and control problems, as well as open problems.

Recent Advances in Epidemic Modeling: Non-Markov Stochastic Models and their Scaling Limits