Stochastic Models for the Inference of Life Evolution


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.


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.


You can reach us by email (amaury.lambert - at - or (smile - at -

Light on



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.



Time Reversal Dualities for some Random Forests

We consider a random forest \$$\mathcal{F}^*\$$, defined as a sequence of i.i.d. birth-death (BD) trees, each started at time 0 from a single ancestor, stopped at the first tree having survived up to a fixed time \$$T\$$. We denote by \$$\left(\xi^*_t, 0\leq t \leq T \right)\$$ the population size process associated to this forest, and we prove that if the BD trees are supercritical, then the time-reversed process \$$\left(\xi^*_{T-t}, 0 \leq t \leq T\right)\$$, has the same distribution as \$$\left(\widetilde\xi^*_t, 0 \leq t \leq T\right)\$$, the corresponding population size process of an equally defined forest \$$\widetilde{\mathcal{F}}^*\$$, but where the underlying BD trees are subcritical, obtained by swapping birth and death rates or equivalently, conditioning on ultimate extinction. We generalize this result to splitting trees (i.e. life durations of individuals are not necessarily exponential), provided that the i.i.d. lifetimes of the ancestors have a specific explicit distribution, different from that of their descendants. The results are based on an identity between the contour of these random forests truncated up to \$$T\$$ and the duality property of L\'evy processes. This identity allows us to also derive other useful properties such as the distribution of the population size process conditional on the reconstructed tree of individuals alive at \$$T\$$, which has potential applications in epidemiology.

Upcoming seminars


Planning des salles du Collège de France.
Intranet du Collège de France.