Equivalence of the Erlang SEIR epidemic model and the renewal equation

Most compartmental epidemic models can be represented using the renewal equation. The value of the renewal equation is not widely appreciated in the epidemiological modelling community, perhaps because its equivalence to standard models has not been presented rigorously in non-trivial cases. Here, we provide analytical expressions for the intrinsic generation interval distribution that must be used in the renewal equation in order to yield epidemic dynamics that are identical to those of the susceptible-exposed-infectious-recovered (SEIR) compartmental model with Erlang-distributed latent and infectious periods. This class of models includes the standard (exponentially-distributed) SIR and SEIR models as special cases.