Stochastic Models for the Inference of Life Evolution

SMILE | Stochastic Models for the Inference of Life Evolution | Collège de France


SMILE is an interdisciplinary research group gathering mathematicians, bio-informaticians and biologists.
SMILE is affiliated to the Institut de Biologie de l'ENS, in Paris.
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.


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 - ; (guillaume.achaz - at - or (smile - at -

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The speed of vaccination rollout and the risk of pathogen adaptation

Vaccination is expected to reduce disease prevalence and to halt the spread of epidemics. But pathogen adaptation may erode the efficacy of vaccination and challenge our ability to control disease spread. Here we examine the influence of the speed of vaccination rollout on the overall risk of pathogen adaptation to vaccination. We extend the framework of evolutionary epidemiology theory to account for the different steps leading to adaptation to vaccines: (1) introduction of a vaccine-escape variant by mutation from an endemic wild-type pathogen, (2) invasion of this vaccine-escape variant in spite of the risk of early extinction, (3) spread and, eventually, fixation of the vaccine-escape variant in the pathogen population. We show that the risk of pathogen adaptation is maximal for intermediate speed of vaccination rollout. On the one hand, slower rollout decreases pathogen adaptation because selection is too weak to avoid early extinction of the new variant. On the other hand, faster rollout decreases pathogen adaptation because it reduces the influx of adaptive mutations. Hence, vaccinating faster is recommended to decrease both the number of cases and the likelihood of pathogen adaptation. We also show that pathogen adaptation is driven by its basic reproduction ratio, the efficacy of the vaccine and the effects of the vaccine-escape mutations on pathogen life-history traits. Accounting for the interplay between epidemiology, selection and genetic drift, our work clarifies the influence of vaccination policies on different steps of pathogen adaptation and allows us to anticipate the effects of public-health interventions on pathogen evolution.Significance statement Pathogen adaptation to host immunity challenges the efficacy of vaccination against infectious diseases. Are there vaccination strategies that limit the emergence and the spread of vaccine-escape variants? Our theoretical model clarifies the interplay between the timing of vaccine escape mutation events and the transient epidemiological dynamics following the start of a vaccination campaign on pathogen adaptation. We show that the risk of adaptation is maximized for intermediate vaccination coverage but can be reduced by a combination of non pharmaceutical interventions and maximizing the speed of the vaccination rollout. These recommendations may have important implications for the choice of vaccination strategies against the ongoing SARS-CoV-2 pandemic.Competing Interest StatementThe authors have declared no competing interest.Funding StatementThis study was funded by a grant from CNRS MITI to SG.Author DeclarationsI confirm all relevant ethical guidelines have been followed, and any necessary IRB and/or ethics committee approvals have been obtained.YesI confirm that all necessary patient/participant consent has been obtained and the appropriate institutional forms have been archived, and that any patient/participant/sample identifiers included were not known to anyone (e.g., hospital staff, patients or participants themselves) outside the research group so cannot be used to identify individuals.YesI understand that all clinical trials and any other prospective interventional studies must be registered with an ICMJE-approved registry, such as I confirm that any such study reported in the manuscript has been registered and the trial registration ID is provided (note: if posting a prospective study registered retrospectively, please provide a statement in the trial ID field explaining why the study was not registered in advance).YesI have followed all appropriate research reporting guidelines and uploaded the relevant EQUATOR Network research reporting checklist(s) and other pertinent material as supplementary files, if applicable.YesThis is a theoretical study.



Testing for Independence between Evolutionary Processes

Evolutionary events co-occurring along phylogenetic trees usually point to complex adaptive phenomena, sometimes implicating epistasis. While a number of methods have been developed to account for co-occurrence of events on the same internal or external branch of an evolutionary tree, there is a need to account for the larger diversity of possible relative positions of events in a tree. Here we propose a method to quantify to what extent two or more evolutionary events are associated on a phylogenetic tree. The method is applicable to any discrete character, like substitutions within a coding sequence or gains/losses of a biological function. Our method uses a general approach to statistically test for significant associations between events along the tree, which encompasses both events inseparable on the same branch, and events genealogically ordered on different branches. It assumes that the phylogeny and themapping of branches is known without errors. We address this problem from the statistical viewpoint by a linear algebra representation of the localization of the evolutionary events on the tree.We compute the full probability distribution of the number of paired events occurring in the same branch or in different branches of the tree, under a null model of independence where each type of event occurs at a constant rate uniformly inthephylogenetic tree. The strengths and weaknesses of themethodare assessed via simulations; we then apply the method to explore the loss of cell motility in intracellular pathogens.



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.



Predicted success of prophylactic antiviral therapy to block or delay SARS-CoV-2 infection depends on the targeted mechanism

Repurposed drugs that are immediately available and have a good safety profile constitute a first line of defense against new viral infections. Despite a limited antiviral activity against SARS-CoV-2, several drugs serve as candidates for application, not only in infected individuals but also as prophylaxis to prevent infection establishment. Here we use a stochastic model to describe the early phase of a viral infection. We find that the critical efficacy needed to block viral establishment is typically above 80\%. This value can be improved by combination therapy. Below the critical efficacy, establishment can still sometimes be prevented; for that purpose, drugs blocking viral entry into target cells (or equivalently enhancing viral clearance) are more effective than drugs reducing viral production or enhancing infected cell death. When a viral infection cannot be prevented because of high exposure or low drug efficacy, antivirals can still delay the time to reach detectable viral loads from 4 days when untreated to up to 30 days. This delay flattens the within-host epidemic curve, and possibly reduces transmission and symptom severity. These results suggest that antiviral prophylaxis, even with reduced efficacy, could be efficiently used to prevent or alleviate infection in people at high risk. It could thus be an important component of the strategy to combat the SARS-CoV-2 pandemic in the months or years to come.

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