  We present a unifying, tractable approach for studying the spread of viruses
causing complex diseases requiring to be modeled using a large number of types
(e.g., infective stage, clinical state, risk factor class). We show that
recording each infected individual's infection age, i.e., the time elapsed
since infection, has three benefits.
  First, regardless of the number of types, the age distribution of the
population can be described by means of a first-order, one-dimensional partial
differential equation (PDE) known as the McKendrick-von Foerster equation. The
frequency of type $i$ is simply obtained by integrating the probability of
being in state $i$ at a given age against the age distribution. This
representation induces a simple methodology based on the additional assumption
of Poisson sampling to infer and forecast the epidemic. We illustrate this
technique using French data from the COVID-19 epidemic.
  Second, our approach generalizes and simplifies standard compartmental models
using high-dimensional systems of ordinary differential equations (ODEs) to
account for disease complexity. We show that such models can always be
rewritten in our framework, thus, providing a low-dimensional yet equivalent
representation of these complex models.
  Third, beyond the simplicity of the approach, we show that our population
model naturally appears as a universal scaling limit of a large class of fully
stochastic individual-based epidemic models, where the initial condition of the
PDE emerges as the limiting age structure of an exponentially growing
population starting from a single individual.
