Probabilities
in the service of structural,
dynamic and efficient modeling
of interactions in large populations

We establish sufficient conditions for exponential convergence to a unique quasi-stationary distribution in the total variation norm. These conditions also ensure the existence and exponential ergodicity of the Q-process, the process conditioned upon never being absorbed. The technique relies on a coupling procedure that is related to Harris recurrence (for Markov Chains). It applies to general continuous-time and continuous-space Markov processes. The main novelty is that we modulate each coupling step depending both on a final horizon of time (for survival) and on the initial distribution. By this way, we could notably include in the convergence a dependency on the initial condition. As an illustration, we consider a continuous-time birth-death process with catastrophes and a diffusion process describing a (localized) population adapting to its environment.

This paper tackles the issue of comparing asymptotically the survival of a Markov process with extinction, depending on its initial condition. Such a comparison is a crucial step in recent techniques for proving exponential convergence to a quasi-stationary distribution. We introduce a weak form of the Harnack inequality as the essential ingredient for such a comparison. This property is actually a consequence of the property of convergence we intend to prove. Although its flexibility makes it difficult to grasp its full potential, we show in our illustrations how simply and efficiently it can be used. This makes it possible to treat such an exponential convergence for new classes of càdlàg strong Markov processes. As illustrations, we consider two continuous-time processes on Rd that do not satisfy the classical Harnack inequalities, even in a local version. The first one is a piecewise deterministic process while the second is a pure jump process with restrictions on the directions of its jumps. Although newly introduced, these examples are yet among the simplest ones for which our techniques are fruitful.

In a view for a simple model where natural selection at the individual level is confronted to selection effects at the group level, we consider some individual-based models of some large population subdivided in a large number of groups. We then obtain the convergence to the law of a stochastic process with some Feynman-Kac penalization. To analyze the limiting behavior of this law, we use a recent approach, designed for the convergence to quasistationary distributions, that generalizes the principles of Harris recurrence. We are able to deal with the fixation of the stochastic process and relate the convergence to equilibrium to the one where fixation implies extinction. We notably establish different regimes of convergence. Besides the case of an exponential rate (the rate being uniform over the initial condition), critical regimes with convergence in 1/t are also to notice. We finally address the relevance of such limiting behaviors to predict the long-time behavior of the individual-based model.

Starting from a stochastic individual based description of an SIS epidemic spreading on a random network, we study the dynamics when the size of the network tends to infinity. 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. Our work covers the case of dense and sparse graphs, when the number of edges is of order Θ(na) with a>1, but not the case of very sparse graphs with a=1. In order to establish our limit theorem, we have to deal with both the convergence of the graph to the graphon and the convergence of the stochastic process spreading on top of these random structures: in particular, we propose a coupling between the process of interest and an epidemic that spreads on the complete graph but with a modified infectivity.

Comments:
Keywords: random graph, mathematical model for epidemiology, measure-valued process, large network limit, limit theorem, dense graph, sparse graph.
Acknowledgments: This work was financed by the Labex Bézout (ANR-10-LABX-58) and the platform MODCOV19 of the National Institute of Mathematical Sciences and their Interactions of CNRS. 30 pages, including an appendix of 5 pages

This article is a presentation of specific recent results describing scaling limits of individual-based models. Thanks to them, we wish to relate the time-scales typical of demographic dynamics and natural selection to the parameters of the individual-based models. Although these results are by no means exhaustive, both on the mathematical and the biological level, they complement each other. Indeed, they provide a viewpoint for many classical time-scales. Namely, they encompass the timescale typical of the life-expectancy of a single individual, the longer one wherein a population can be characterized through its demographic dynamics, and at least four interconnected ones wherein selection occurs. The limiting behavior is generally deterministic. Yet, since there are selective effects on randomness in the history of lineages, probability theory is shown to be a key factor in understanding the results. Besides, randomness can be maintained in the limiting dynamics, for instance to model rare mutations fixing in the population.

We analyze the long-term stability of a stochastic model designed to illustrate the adaptation of a population to variation in its environment. A piecewise-deterministic process modeling adaptation is coupled to a Feller logistic diffusion modeling population size. As the individual features in the population become further away from the optimal ones, the growth rate declines, making population extinction more likely. Assuming that the environment changes deterministically and steadily in a constant direction, we obtain the existence and uniqueness of the quasi-stationary distribution, the associated survival capacity and the Q-process. Our approach also provides several exponential convergence results (in total variation for the measures). From this synthetic information, we can characterize the efficiency of internal adaptation (i.e. population turnover from mutant invasions). When the latter is lacking, there is still stability, but because of the high level of population extinction. Therefore, such a characterization must be based on specific features of this quasi-ergodic regime.

Müller's ratchet model enables to analyse the adaptation stability of a population when deleterious mutations regularly affects new individuals. Here, we condition on the fact that the fittest individuals are still present in the population. Our purpose is to ensure the exponential quasi-ergodicity of the processes (with or without an upper-bound on the number of accumulating deleterious mutations). We also expect to ensure that individuals with too many deleterious mutations play a negligible role both for the survival of the fittest subpopulation and for the quasi-ergodicity of the population on a whole.