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International audience; We present a unifying, tractable approach for studying the spread of viruses causing complex diseases that require to be modeled using a large number of types (e.g., infective stage, clinical state, risk factor class). We show that recording each infected individual's infection age, i.e., the time elapsed since infection, 1. The age distribution $n(t, a)$ of the population at time $t$ can be 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. We illustrate this technique using a French data from the COVID-19 epidemic. Second, our approach generalizes and simplifies standard compartmental models using high-dimensional systems of ordinary differential equations (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, we show that our population model naturally appears as a universal scaling limit of a large class of fully stochastic individual-based epidemic models, here the initial condition of the PDE emerges as the limiting age structure of an exponentially growing population starting from a single individual.

Similarly as in (Blancas et al. 2018) where nested coalescent processes are studied, we generalize the definition of partition-valued homogeneous Markov fragmentation processes to the setting of nested partitions, i.e. pairs of partitions $(\zeta,\xi)$ where $\zeta$ is finer than $\xi$. As in the classical univariate setting, under exchangeability and branching assumptions, we characterize the jump measure of nested fragmentation processes, in terms of erosion coefficients and dislocation measures. Among the possible jumps of a nested fragmentation, three forms of erosion and two forms of dislocation are identified - one of which being specific to the nested setting and relating to a bivariate paintbox process., Comment: 37 pages, 6 figures

This thesis consists of four self-contained chapters whose motivations stem from population genetics and evolutionary biology, and related to the theory of fragmentation or coalescent processes. Chapter 2 introduces an infinite random binary tree built from a so-called coalescent point process equipped with Poissonian mutations along its branches and with a finite measure on its boundary. The allelic partition -- partition of the boundary into groups carrying the same combination of mutations -- is defined for this tree and its intensity measure is described. Chapters 3 and 4 are devoted to the study ofnested -- i.e. taking values in the space of nested pairs of partitions --coalescent and fragmentation processes, respectively. These Markov processes are analogs of Λ-coalescents and homogeneous fragmentations in a nested setting-- modeling a gene tree nested within a species tree. Nested coalescents are characterized in terms of Kingman coefficients and (possibly bivariate) coagulation measures, while nested fragmentations are similarly characterized in terms of erosion coefficients and (possibly bivariate) dislocation measures. Finally Chapter 5 gives a construction of fragmentation processes with speedmarks, which are fragmentation processes where each fragment is given a mark that speeds up or slows down its rate of fragmentation, and where the marks evolve as positive self-similar Markov processes. A Lévy-Khinchin representation of these generalized fragmentation processes is given, as well as sufficient conditions for their absorption in finite time to a frozen state,and for the genealogical tree of the process to have finite total length.; Cette thèse est constituée de quatre chapitres indépendants, puisant leur origine en génétique des populations et en biologie évolutive, et liés à la théorie des processus de fragmentation ou de coalescence. Le chapitre 2 traite d'un arbre aléatoire binaire infini construit à partir d'un processus ponctuel coalescent équipé de mutations poissonniennes le long de ses branches et d'une mesure sur sa frontière. La partition allélique -- partition de la frontière en parties qui portent la même combinaison de mutations -- est définie pour cet arbre et sa mesure d'intensité est explicitée. Les chapitres 3 et 4 sont dédiés à l'étude de processus de coalescence et de fragmentation emboîtés -- plus précisément à valeurs dans les couples de partitions emboîtées --, qui sont des analogues des Λ-coalescents et des fragmentations homogènes. Ces objets visent à modéliser un arbre de gènes niché dans un arbre d'espèces. Les coalescents emboîtés sont caractérisés par leurs coefficients de Kingman et leurs mesures de coagulation (éventuellement bivariées), tandis que les fragmentations emboîtées sont caractérisées par leurs coefficients d'érosion et leurs mesures de dislocation (éventuellement bivariées). Enfin le chapitre 5 pose la construction de processus de fragmentation à vitesses aléatoires, qui sont des processus de fragmentation où chaque fragment possède une marque qui accélère ou ralentit son taux de fragmentation, et où les marques de vitesse évoluent comme des processus de Markov positifs auto-similaires. Une caractérisation de type Lévy-Khintchine de ces processus de fragmentation généralisés est donné,ainsi que des conditions suffisantes pour l'absorption dans un état gelé.

For the random interval partition of $[0,1]$ generated by the uniform stick-breaking scheme known as GEM$(1)$, let $u_k$ be the probability that the first $k$ intervals created by the stick-breaking scheme are also the first $k$ intervals to be discovered in a process of uniform random sampling of points from $[0,1]$. Then $u_k$ is a renewal sequence. We prove that $u_k$ is a rational linear combination of the real numbers $1, \zeta(2), \ldots, \zeta(k)$ where $\zeta$ is the Riemann zeta function, and show that $u_k$ has limit $1/3$ as $k \to \infty$. Related results provide probabilistic interpretations of some multiple zeta values in terms of a Markov chain derived from the interval partition. This Markov chain has the structure of a weak record chain. Similar results are given for the GEM$(\theta)$ model, with beta$(1,\theta)$ instead of uniform stick-breaking factors, and for another more algebraic derivation of renewal sequences from the Riemann zeta function., Comment: 25 pages. This paper is published by https://www.ams.org/journals/tran/2019-371-08/S0002-9947-2018-07516-X/

We consider fragmentation processes with values in the space of marked partitions of $\mathbb{N}$, i.e. partitions where each block is decorated with a nonnegative real number. Assuming that the marks on distinct blocks evolve as independent positive self-similar Markov processes and determine the speed at which their blocks fragment, we get a natural generalization of the self-similar fragmentations of Bertoin (2002). Our main result is the characterization of these generalized fragmentation processes: a L\'evy-Khinchin representation is obtained, using techniques from positive self-similar Markov processes and from classical fragmentation processes. We then give sufficient conditions for their absorption in finite time to a frozen state, and for the genealogical tree of the process to have finite total length., Comment: 38 pages

Starting from any graph on $\{1, \ldots, n\}$, consider the Markov chain where at each time-step a uniformly chosen vertex is disconnected from all of its neighbors and reconnected to another uniformly chosen vertex. This Markov chain has a stationary distribution whose support is the set of non-empty forests on $\{1, \ldots, n\}$. The random forest corresponding to this stationary distribution has interesting connections with the uniform rooted labeled tree and the uniform attachment tree. We fully characterize its degree distribution, the distribution of its number of trees, and the limit distribution of the size of a tree sampled uniformly. We also show that the size of the largest tree is asymptotically $\alpha \log n$, where $\alpha = (1 - \log(e - 1))^{-1} \approx 2.18$, and that the degree of the most connected vertex is asymptotically $\log n / \log\log n$., Comment: Accepted version

We consider the compact space of pairs of nested partitions of $\mathbb N$, where by analogy with models used in molecular evolution, we call "gene partition" the finer partition and "species partition" the coarser one. We introduce the class of nondecreasing processes valued in nested partitions, assumed Markovian and with exchangeable semigroup. These processes are said simple when each partition only undergoes one coalescence event at a time (but possibly the same time). Simple nested exchangeable coalescent (SNEC) processes can be seen as the extension of $\Lambda$-coalescents to nested partitions. We characterize the law of SNEC processes as follows. In the absence of gene coalescences, species blocks undergo $\Lambda$-coalescent type events and in the absence of species coalescences, gene blocks lying in the same species block undergo i.i.d. $\Lambda$-coalescents. Simultaneous coalescence of the gene and species partitions are governed by an intensity measure $\nu_s$ on $(0,1]\times {\mathcal M}_1 ([0,1])$ providing the frequency of species merging and the law in which are drawn (independently) the frequencies of genes merging in each coalescing species block. As an application, we also study the conditions under which a SNEC process comes down from infinity., Comment: 37 pages

Consider a random real tree whose leaf set, or boundary, is endowed with a finite mass measure. Each element of the tree is further given a type, or allele, inherited from the most recent atom of a random point measure (infinitely-many-allele model) on the skeleton of the tree. The partition of the boundary into distinct alleles is the so-called allelic partition. In this paper, we are interested in the infinite trees generated by supercritical, possibly time-inhomogeneous, binary branching processes, and in their boundary, which is the set of particles `co-existing at infinity'. We prove that any such tree can be mapped to a random, compact ultrametric tree called coalescent point process, endowed with a `uniform' measure on its boundary which is the limit as $t\to\infty$ of the properly rescaled counting measure of the population at time $t$. We prove that the clonal (i.e., carrying the same allele as the root) part of the boundary is a regenerative set that we characterize. We then study the allelic partition of the boundary through the measures of its blocks. We also study the dynamics of the clonal subtree, which is a Markovian increasing tree process as mutations are removed., 45 pages, 6 figures