  We study a class of individual-based, fixed-population size epidemic models
under general assumptions, e.g., heterogeneous contact rates encapsulating
changes in behavior and/or enforcement of control measures. We show that the
large-population dynamics are deterministic and relate to the
Kermack-McKendrick PDE. Our assumptions are minimalistic in the sense that the
only important requirement is that the basic reproduction number of the
epidemic $R_0$ be finite, and allow us to tackle both Markovian and
non-Markovian dynamics. The novelty of our approach is to study the "infection
graph" of the population. We show local convergence of this random graph to a
Poisson (Galton-Watson) marked tree, recovering Markovian backward-in-time
dynamics in the limit as we trace back the transmission chain leading to a
focal infection. This effectively models the process of contact tracing in a
large population. It is expressed in terms of the Doob $h$-transform of a
certain renewal process encoding the time of infection along the chain. Our
results provide a mathematical formulation relating a fundamental
epidemiological quantity, the generation time distribution, to the successive
time of infections along this transmission chain.
