# SMILE

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

### 2020

We present a unifying, tractable approach for studying the spread of viruses causing complex diseases, requiring to be modeled with a large number of types (infective stage, clinical state, risk factor class...). We show that recording for each infected individual her infection age, i.e., the time elapsed since she was infected,
1. The age distribution $n(t,a)$ of the population at time $t$ is simply described by means of a first-order, one-dimensional partial differential equation (PDE) known as the McKendrick--von 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 three-fold. First, regardless of the number of types, macroscopic observables (e.g., incidence or prevalence of each type) only rely on a one-dimensional PDE decorated'' with types. This representation induces a simple methodology based on the McKendrick-von Foerster PDE with Poisson sampling to infer and forecast the epidemic. This technique is illustrated with French data of the COVID-19 epidemic.
Second, our approach generalizes and simplifies standard compartmental models using high-dimensional systems of 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 and its computational advantages, 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.

# Bibtex

@article {Foutel2020,
author={Foutel-Rodier, Félix and Blanquart, François and Courau, Philibert and Czuppon, Peter and Duchamps, Jean-Jil and Gamblin, Jasmine and Kerdoncuff, Elise and Kulathinal, Rob and R\'egnier, L\'eo and Vuduc, Laura and Lambert, Amaury and Schertzer, Emmanuel},
title = {From individual-based epidemic models to {M}c{K}endrick-von {F}oerster {PDE}s:
A guide to modeling and inferring COVID-19 dynamics},
year = {2020},
abstract = {
We present a unifying, tractable approach for studying the spread of viruses causing complex diseases, requiring to be modeled with a large number of types (infective stage, clinical state, risk factor class...). We show that recording for each infected individual her infection age, i.e., the time elapsed since she was infected,

1. The age distribution $n(t,a)$ of the population at time $t$ is simply described by means of a first-order, one-dimensional partial differential equation (PDE) known as the McKendrick--von 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 three-fold. First, regardless of the number of types, macroscopic observables (e.g., incidence or prevalence of each type) only rely on a one-dimensional PDE decorated'' with types. This representation induces a simple methodology based on the McKendrick-von Foerster PDE with Poisson sampling to infer and forecast the epidemic. This technique is illustrated with French data of the COVID-19 epidemic.

Second, our approach generalizes and simplifies standard compartmental models using high-dimensional systems of 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 and its computational advantages, 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. }
}