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We present a unifying, tractable approach for studying the spread of virusescausing complex diseases requiring to be modeled using a large number of types(e.g., infective stage, clinical state, risk factor class). We show thatrecording each infected individual's infection age, i.e., the time elapsedsince infection, has three benefits.  First, regardless of the number of types, the age distribution of thepopulation can be described by means of a first-order, one-dimensional partialdifferential equation (PDE) known as the McKendrick-von Foerster equation. Thefrequency of type $i$ is simply obtained by integrating the probability ofbeing in state $i$ at a given age against the age distribution. Thisrepresentation induces a simple methodology based on the additional assumptionof Poisson sampling to infer and forecast the epidemic. We illustrate thistechnique using French data from the COVID-19 epidemic.  Second, our approach generalizes and simplifies standard compartmental modelsusing high-dimensional systems of ordinary differential equations (ODEs) toaccount for disease complexity. We show that such models can always berewritten in our framework, thus, providing a low-dimensional yet equivalentrepresentation of these complex models.  Third, beyond the simplicity of the approach, we show that our populationmodel naturally appears as a universal scaling limit of a large class of fullystochastic individual-based epidemic models, where the initial condition of thePDE emerges as the limiting age structure of an exponentially growingpopulation starting from a single individual.

From individual-based epidemic models to McKendrick-von Foerster PDEs: A guide to modeling and inferring COVID-19 dynamics