SMILE

Stochastic Models for the Inference of Life Evolution

Presentation

SMILE is an interdisciplinary research group gathering probabilists, statisticians, bio-informaticians and biologists.
SMILE is affiliated to the Stochastics and Biology group of LPSM (Lab of Probability, Statistics and Modeling) at Sorbonne Université (ex Université Pierre et Marie Curie Paris 06).
SMILE is hosted within the CIRB (Center for Interdisciplinary Research in Biology) at Collège de France.
SMILE is supported by Collège de France and CNRS.
Visit also our homepage at CIRB.

Recent contributions of the SMILE group related to SARS-Cov2 and COVID-19.

Directions

SMILE is hosted at Collège de France in the Latin Quarter of Paris. To reach us, go to 11 place Marcelin Berthelot (stations Luxembourg or Saint-Michel on RER B).
Our working spaces are rooms 107, 121 and 122 on first floor of building B1 (ask us for the code). Building B1 is facing you upon exiting the traversing hall behind Champollion's statue.

Contact

You can reach us by email (amaury.lambert - at - upmc.fr) or (smile - at - listes.upmc.fr).

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Publication

2020

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


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.

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