CS28: Propagation of chaos in life science models

date: 7/17/2025, time: 16:00-17:30, room: IM HS

Organizer: Dasha Loukianova (Université d’Evry-Paris-Saclay)

Chair: Dasha Loukianova (Université d’Evry-Paris-Saclay)

Conditional propagation of chaos for a neural model through the study of generators of measure-valued Markov processes

Xavier ERNY (Télécom SudParis)

We prove a conditional propagation of chaos property for a model of interacting neurons. More precisely, the convergence in distribution of the empirical measure of a particle system, as the number of particles goes to infinity. The limit measure-valued process is characterized as the directing measure of infinite-particle system with common noise. The main goal of this presentation is to present analytical results about measure-variable functions in order to compute explicitly the expressions of the generators of these Markov measure-valued processes, and to show their convergence with an explicit convergence speed.

Strong propagation of chaos for systems of interacting particles with nearly stable jumps

Elisa Marini (CEREMADE, UMR CNRS 7534, Université Paris Dauphine-PSL)

We consider a system of \(N\) interacting particles, described by SDEs driven by Poisson random measures, where the coefficients depend on the empirical measure of the system. Every particle jumps with a rate depending on its position. When this happens, all the other particles of the system receive a same random kick distributed according to a heavy tailed random variable belonging to the domain of attraction of an \(\alpha\)-stable law and scaled by \(N^{-1/\alpha}\), \(\alpha \in (0,2)\setminus \{1\}\). We call these jumps collateral jumps. Moreover, in case \(0<\alpha<1\), the jumping particle itself undergoes a macroscopic, main jump. Similar systems are employed to model families of interacting neurons and, in that context, main and collateral jumps represent respectively the hyperpolarization of a neuron after a spike and the synaptic inputs received by post-synaptic neurons from pre-synaptic ones. We prove that our system has the conditional propagation of chaos property: as \(N\to +\infty\), the finite particle system converges to an infinite exchangeable system which obeys a McKean-Vlasov SDE driven by an \(\alpha\)-stable process, and particles in the limit system are independent, conditionally on the driving \(\alpha\)-stable process.

SIR model on inhomogeneous graphs with infection-age dependent infectivity

Aurelien Velleret (LaMME, Université Evry Paris Saclay)

The prediction of epidemic spread within a population largely relies on simplified models, in which interactions are represented by means of implicit traits. These traits are characterizing the population heterogeneity, such as the age or occupation of individuals. The usual sampling methods are meant to provide estimations for the relation between the traits and the interaction levels, that is of the kernel function in the model.

Starting from a stochastic individual based description of an epidemic spreading on a random graph, we will look at the dynamics as the size \(n\) of the graph tends to infinity. What is the generality of such a model reduction beyond the case of dense graphs for which the notion of graphon (as a specific case of interaction kernel) has been originally proposed?

With the typical SIS infection process between individuals, we recover in the limit an infinite-dimensional integro-differential equation studied by Delmas, Dronnier and Zitt (2022) for an SIS epidemic propagating on a graphon. This convergence covers the cases both of dense and of sparse graphs, when the number of edges is of order \(0(n^a)\) with \(a\in(1,2]\) (the case of very sparse graphs with \(a=1\) and boundedness properties in the number of neighbors is of different nature). This provides ground for the current statistical evaluation even though the individuals are in contact with typically a reduced portion of the entire population.

These results can be extended for more elaborated infection history. When considering infectivity profiles that vary with the duration from the infection event, we recover in the limit an age-structured process that extends the description proposed in Forien, Pang and Pardoux (2022) for a finite number of individual traits. I will show how the involved propagation of chaos argument is then extended to this new graphon structure on a continuous state space.

Bibliography

\([1]\) G. Pang, E. Pardoux, and A. Velleret, SIR model on inhomogeneous graphs with infection-age dependent infectivity, https://arxiv.org/abs/2502.04225

\([2]\) J-F. Delmas, P. Frasca, F. Garin, C. Tran, A. Velleret, and P-A. Zitt, Individual based SIS models on (not so) dense large random networks, ALEA: Probab. Math. Stat. 21 (2024), 1375–1405.

\([3]\) J.-F. Delmas, D. Dronnier, and P.-A. Zitt, An infinite-dimensional metapopulation SIS model, J. Differ. Equ. 313 (2022), 1–53.

\([4]\) R. Forien, G. Pang, and E. Pardoux, Epidemic models with varying infectivity, SIAM Journal on Applied Mathematics 81 (2021), no. 5, 1893–1930.