From individualbased epidemic models to McKendrickvon Foerster PDEs: A guide to modeling and inferring COVID19 dynamics
Abstract
We present a unifying, tractable approach for studying the spread of viruses causing complex diseases that require 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, 1. The age distribution $n(t, a)$ of the population at time $t$ can be described by means of a firstorder, onedimensional partial differential equation (PDE) known as the McKendrickvon Foerster equation. 2. The frequency of type $i$ at time $t$ is simply obtained by integrating the probability $p(a, i)$ of being in state $i$ at age a against the age distribution $n(t, a)$. The advantage of this approach is threefold. First, regardless of the number of types, macroscopic observables (e.g., incidence or prevalence of each type) only rely on a onedimensional PDE "decorated" with types. This representation induces a simple methodology based on the McKendrickvon Foerster PDE with Poisson sampling to infer and forecast the epidemic. We illustrate this technique using a French data from the COVID19 epidemic. Second, our approach generalizes and simplifies standard compartmental models using highdimensional 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 lowdimensional 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 individualbased epidemic models, here the initial condition of the PDE emerges as the limiting age structure of an exponentially growing population starting from a single individual.
 Publication:

arXiv eprints
 Pub Date:
 July 2020
 arXiv:
 arXiv:2007.09622
 Bibcode:
 2020arXiv200709622F
 Keywords:

 Quantitative Biology  Populations and Evolution;
 Mathematics  Probability;
 Primary: 92D30;
 Secondary: 60J80;
 60J85;
 35Q92
 EPrint:
 38 pages, 8 figures