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

TL;DR

This paper introduces a unified PDE-based framework for modeling complex epidemic dynamics, simplifying high-dimensional models, and enabling inference and forecasting of COVID-19 spread from individual infection data.

Contribution

It demonstrates how recording infection age leads to a PDE approach that simplifies complex models and connects to stochastic individual-based models, with practical application to COVID-19 data.

Findings

The PDE framework accurately models COVID-19 epidemic data in France.

High-dimensional ODE models can be reformulated into low-dimensional PDEs.

The approach provides a universal scaling limit for stochastic epidemic models.

Abstract

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.