Besançon meeting on Probability, Ecology & Evolution 2024

On 10 December 2024, we hosted a one-day meeting at the MSHE in Besançon, bringing together researchers from Austria, France, Germany, and Switzerland for a workshop on stochastic models in biology. We thank all the participants for joining us and contributing to the success of our first event.

This event was supported by LmB, RDI-BMB, Chrono-environnement and EUR EIPHI (RaySynMath project).

Detailed program.

10:00 — 10:30

Welcome coffee 🥐

10:30 — 11:15

Céline Bonnet
Site frequency spectrum of a rescued population under rare resistant mutations

11:15 — 12:00

Florin Boenkost
Dust solutions and the nested Kingman coalescent

12:00 — 13:30

Lunch Break 🍴
vegetarian / vegan buffet

13:30 — 14:15

Martin Möhle
On multi-type Cannings models and their multi-type limiting coalescents

14:15 — 15:00

Madeleine Kubasch
Epidemic dynamics on large multi-level contact networks including households and workplaces

15:00 — 15:45

Coffee break ☕

15:45 — 16:30

Nicolás Zalduendo
The multi-type bisexual Galton–Watson process

16:30 — 17:15

Vincent Bansaye
Some growing, living networks and branching processes

Participants are welcome to join for drinks after the conference, at their own expense.

A dinner will be organized for the invited speakers, more details coming soon.

Abstracts.

Site frequency spectrum of a rescued population under rare resistant mutations

Céline Bonnet, 10:30 — 11:15

The aim of this talk is to study the impact of resistance acquisition on the distribution of neutral mutations in a cell population under therapeutic pressure. The cell population is modeled by a bi-type branching process. Initially, the cells all carry type 0, associated with a negative growth rate. Mutations towards type 1 are assumed to be rare and random, and lead to the survival of cells under treatment, i.e. type 1 is associated with a positive growth rate, and thus models the acquisition of a resistance. Cells also carry neutral mutations, acquired at birth and accumulated by inheritance, that do not affect their type. We describe the expectation of the "Site Frequency Spectrum" (SFS), which is an index of neutral mutation distribution in a population, under the asymptotic of rare events of resistance acquisition and of large initial population. Precisely, we give asymptotically-equivalent expressions of the expected number of neutral mutations shared by both a small and a large number of cells. To identify the influence of relatives on the SFS, our work also lead us to study in detail subcritical binary Galton–Watson trees, where each leaf is marked with a small probability. As a by-product of this study, we thus provide the law of the generation of a randomly chosen leaf in such a Galton–Watson tree conditioned on the number of marks. Back to program ⤴️

Dust solutions and the nested Kingman coalescent

Florin Boenkost, 11:15 — 12:00

The nested Kingman Coalescent \((\mathcal{K}^n_t, t\geq 0)\) is a model for the genealogy of lineages belonging to different species, which can be thought of as (gene-)trees within a (species-)tree. We assume that the species tree is given as a Kingman tree and once species coalesce the inner gene lineages are allowed to coalesce.

Here, we focus on the empirical measure \(g^n_{t}= \frac{1}{s^n_t} \sum_{i=1}^{s_t^n} \delta_{\Pi_t^n (i)}\) of block sizes in the nested coalescent, where \(\Pi_t^n (i)\) denotes the number of gene lineages in species \(i\). As the number of species tends to infinity we show convergence towards a solution \(u(t,x)\) of the Smoluchowski coagulation-transport equation \[ \partial_t \, u = \partial_x \left( \frac{x^2}{2} u\right) + \frac{1}{t} (u \star u - u), \] where \(u \star u\) denotes the convolution. In particular, if there are fewer lineages per species than species itself, \(g_t^n\) converges to a dust solution \(u(t,x)\), meaning that \(u(t,x) \to \delta_0(dx)\) as \(t\to 0.\) Dust solutions of this equation appeared in the work of [Lambert and Schertzer 2020] and a priori there are infinitely many. We show that the solution is unique up to a scaling parameter which encodes the initial number of lineages per species.

This is accomplished through a probabilistic representation of the solution, which in turn can be extended to provide a stochastic representation of all dust solutions in terms of a specific martingale. Lastly, we discuss an application of this result and provide a short glimpse onto other species trees (in particular \(\Lambda\)-coalescents). This talk is based on joint work in progress with Emmanuel Schertzer. Back to program ⤴️

On multi-type Cannings models and their multi-type limiting coalescents

Martin Möhle, 13:30 — 14:15

A multi-type neutral Cannings population model with mutation and fixed subpopulation sizes is analyzed. Under appropriate conditions, as all subpopulation sizes tend to infinity, the ancestral process, properly time-scaled, converges to a multi-type coalescent with mutation allowing for simultaneous multiple collisions of ancestral lineages. The limiting coalescent shares the exchangeability and consistency property. The proof gains from coalescent theory for single-type Cannings models and from decompositions into reproductive and mutational parts. Back to program ⤴️

Epidemic dynamics on large multi-level contact networks including households and workplaces

Madeleine Kubasch, 14:15 — 15:00

The spread of an epidemic is naturally influenced by the way individuals encounter one another. In particular, human contacts are structured by different social contexts in which they occur, such as at home, in school or at the office. In addition, control measures like teleworking and school closures act by reducing specific contacts, further motivating the study of models which distinguish different types contacts.

Here, we study a multilayer SIR model with two levels of mixing, namely a global level which is uniformly mixing, and a local level with two layers distinguishing household and workplace contacts, respectively. We establish the large population convergence of the corresponding stochastic process. For this purpose, we use an individual-based model whose state space specifies the remaining infectious period length for each infected. This allows to deal with the natural correlation of the epidemic states of individuals whose household and workplace share a common infected. We start by establishing the convergence to the unique deterministic solution of a measure-valued equation. In the particular case of exponentially distributed infectious periods, we show that it is possible to further reduce the obtained deterministic limit, leading to a closed, finite dimensional dynamical system capturing the epidemic dynamics.

Finally, through numerical explorations, we show that the large population limit of the household-workplace model can approximate well the epidemic, even if some assumptions on the contact network are relaxed. Back to program ⤴️

The multi-type bisexual Galton–Watson process

Nicolás Zalduendo, 15:45 — 16:30

The bisexual Galton–Watson process is an extension of the classical Galton–Watson process, but taking into account the mating of females and males, which form couples that can accomplish reproduction. Properties such as extinction conditions and asymptotic behaviour have been studied in the past years, but multi-type versions have only been treated in some particular cases. In this work we deal with a general multi-dimensional version of the model, where we consider different types of females and males, which mate according to a "mating function". We consider that this function is superadditive, which in simple words implies that two groups of females and males will form a larger number of couples together rather than separate. One of the main difficulties in the study of this process is the absence of a linear operator that is the key to understand its behavior in its asexual counterpart, but in our case it turns out to be only concave. To overcome this issue, we use a concave Perron–Frobenius theory which ensures the existence of eigen-elements for some concave operators. Using this tool, we find a necessary and sufficient condition for almost sure extinction as well as a law of large numbers. Finally, we study the convergence of the process in the long-time through the identification of a supermartingale. Back to program ⤴️

Some growing, living networks and branching processes

Vincent Bansaye, 16:30 — 17:15

We will study fragmentation growth models to describe the expansion of biological networks, particularly in the growth of filamentous fungi or morphogenesis. The models combine linear growth of fragments, several types of fragmentation, and the possibility of fusion when two fragments meet. We will first examine the branching process without fusion and establish the long-term behavior of the structured population of fragments. This will lead us to general results for describing the renormalized empirical measure, that is, to an infinite-dimensional version of the Kesten–Stigum theorem, relaxing the uniformity of the semigroup estimates from the work of Asmussen and Hering. We will then focus on a highly simplified spatial model incorporating fusion events, and we will explore how to study it in detail using a new structured branching process. This talk is linked to a previous work with M. Tomasevic and A. Veber and two works in progress respectively with T. Berah and B. Cloez, and G. Raoul and M. Tomasevic. Back to program ⤴️