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We study a class of individual-based, fixed-population size epidemic modelsunder general assumptions, e.g., heterogeneous contact rates encapsulatingchanges in behavior and/or enforcement of control measures. We show that thelarge-population dynamics are deterministic and relate to theKermack-McKendrick PDE. Our assumptions are minimalistic in the sense that theonly important requirement is that the basic reproduction number of theepidemic $R_0$ be finite, and allow us to tackle both Markovian andnon-Markovian dynamics. The novelty of our approach is to study the "infectiongraph" of the population. We show local convergence of this random graph to aPoisson (Galton-Watson) marked tree, recovering Markovian backward-in-timedynamics in the limit as we trace back the transmission chain leading to afocal infection. This effectively models the process of contact tracing in alarge population. It is expressed in terms of the Doob $h$-transform of acertain renewal process encoding the time of infection along the chain. Ourresults provide a mathematical formulation relating a fundamentalepidemiological quantity, the generation time distribution, to the successivetime of infections along this transmission chain.

General epidemiological models: Law of large numbers and contact tracing