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The paper written in 1925 by G. Udny Yule that we celebrate in this special issue introduces several novelties and results that we recall in detail. First, we discuss Yule (1925)'s main legacies over the past century, focusing on empirical frequency distributions with heavy tails and random tree models for phylogenies. We estimate the year when Yule's work was re-discovered by scientists interested in stochastic processes of population growth (1948) and the year from which it began to be cited (1951, Yule's death). We highlight overlooked aspects of Yule's work (e.g., the Yule process of Yule processes) and correct some common misattributions (e.g., the Yule tree). Second, we generalize Yule's results on the average frequency of genera of a given age and size (number of species). We show that his formula also applies to the age $A$ and size $S$ of any randomly chosen genus and that the pairs $(A_i, S_i)$ are equally distributed and independent across genera. This property extends to triples $(H_i, A_i, S_i)$, where $(H_i)$ are the coalescence times of the genus phylogeny, even when species diversification within genera follows any integer-valued process, including species extinctions. Studying $(A, S)$ in this broader context allows us to identify cases where $S$ has a power-law tail distribution, with new applications to urn schemes., Comment: 33 pages, 4 figures. To appear in theme issue of Philosophical Transactions of the Royal Society Series B on "Phylogenetic models a century beyond G. U. Yule's 'mathematical theory of evolution'"

New automated and high-throughput methods allow the manipulation and selection of numerous bacterial populations. In this manuscript we are interested in the neutral diversity patterns that emerge from such a setup in which many bacterial populations are grown in parallel serial transfers, in some cases with population-wide extinction and splitting events. We model bacterial growth by a birth-death process and use the theory of coalescent point processes. We show that there is a dilution factor that optimises the expected amount of neutral diversity for a given amount of cycles, and study the power law behaviour of the mutation frequency spectrum for different experimental regimes. We also explore how neutral variation diverges between two recently split populations by establishing a new formula for the expected number of shared and private mutations. Finally, we show the interest of such a setup to select a phenotype of interest that requires multiple mutations., Comment: 51 pages, 25 figures

Tree-child networks are a recently-described class of directed acyclic graphs that have risen to prominence in phylogenetics (the study of evolutionary trees and networks). Although these networks have a number of attractive mathematical properties, many combinatorial questions concerning them remain intractable. In this paper, we show that endowing these networks with a biologically relevant ranking structure yields mathematically tractable objects, which we term ranked tree-child networks (RTCNs). We explain how to derive exact and explicit combinatorial results concerning the enumeration and generation of these networks. We also explore probabilistic questions concerning the properties of RTCNs when they are sampled uniformly at random. These questions include the lengths of random walks between the root and leaves (both from the root to the leaves and from a leaf to the root); the distribution of the number of cherries in the network; and sampling RTCNs conditional on displaying a given tree. We also formulate a conjecture regarding the scaling limit of the process that counts the number of lineages in the ancestry of a leaf. The main idea in this paper, namely using ranking as a way to achieve combinatorial tractability, may also extend to other classes of networks.

We present a unifying, tractable approach for studying the spread of viruses causing complex diseases requiring 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, has three benefits. First, regardless of the number of types, the age distribution of the population can be described by means of a first-order, one-dimensional partial differential equation (PDE) known as the McKendrick-von Foerster equation. The frequency of type $i$ is simply obtained by integrating the probability of being in state $i$ at a given age against the age distribution. This representation induces a simple methodology based on the additional assumption of Poisson sampling to infer and forecast the epidemic. We illustrate this technique using 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, where the initial condition of the PDE emerges as the limiting age structure of an exponentially growing population starting from a single individual.

We introduce a Poissonization method to study the coalescent structure of uniform samples from branching processes. This method relies on the simple observation that a uniform sample of size $k$ taken from a random set with positive Lebesgue measure may be represented as a mixture of Poisson samples with rate $\lambda$ and mixing measure $k \mathrm{d} \lambda/ \lambda$. We develop a multitype analogue of this mixture representation, and use it to characterise the coalescent structure of multitype continuous-state branching processes in terms of random multitype forests. Thereafter we study the small time asymptotics of these random forests, establishing a correspondence between multitype continuous-state branching proesses and multitype $\Lambda$-coalescents., Comment: 31 pages

Consider the Markov process taking values in the partitions of N such that each pair of blocks merges at rate one, and each integer is eroded, i.e., becomes a singleton block, at rate d. This is a special case of exchangeable fragmentation-coalescence process called Kingman's coalescent with erosion. We provide a new construction of the stationary distribution of this process as a sample from a standard flow of bridges. This allows us to give a representation of the asymptotic frequencies of this stationary distribution in terms of a sequence of hierarchically independent diffusions. Moreover, we introduce a new process called Kingman's coalescent with immigration, where pairs of blocks coalesce at rate one, and new blocks of size one immigrate at rate d. By coupling Kingman's coalescents with erosion and with immigration, we are able to show that the size of a block chosen uniformly at random from the stationary distribution of the restriction of Kingman's coalescent with erosion to {1,...,n} converges to the total progeny of a critical binary branching process.

The nested Kingman coalescent describes the dynamics of particles (called genes) contained in larger components (called species), where pairs of species coalesce at constant rate and pairs of genes coalesce at constant rate provided they lie within the same species. We prove that starting from $rn$ species, the empirical distribution of species masses (numbers of genes$/n$) at time $t/n$ converges as $n\to\infty$ to a solution of the deterministic coagulation-transport equation $$ \partial_t d \ = \ \partial_x ( \psi d ) \ + \ a(t)\left(d\star d - d \right), $$ where $\psi(x) = cx^2$, $\star$ denotes convolution and $a(t)= 1/(t+\delta)$ with $\delta=2/r$. The most interesting case when $\delta =0$ corresponds to an infinite initial number of species. This equation describes the evolution of the distribution of species of mass $x$, where pairs of species can coalesce and each species' mass evolves like $\dot x = -\psi(x)$. We provide two natural probabilistic solutions of the latter IPDE and address in detail the case when $\delta=0$. The first solution is expressed in terms of a branching particle system where particles carry masses behaving as independent continuous-state branching processes. The second one is the law of the solution to the following McKean-Vlasov equation $$ dx_t \ = \ - \psi(x_t) \,dt \ + \ v_t\,\Delta J_t $$ where $J$ is an inhomogeneous Poisson process with rate $1/(t+\delta)$ and $(v_t; t\geq0)$ is a sequence of independent rvs such that $ {\mathcal L}(v_t) = {\mathcal L}(x_t)$. We show that there is a unique solution to this equation and we construct this solution with the help of a marked Brownian coalescent point process. When $\psi(x)=x^\gamma$, we show the existence of a self-similar solution for the PDE which relates when $\gamma=2$ to the speed of coming down from infinity of the nested Kingman coalescent., Comment: 67 pages, 3 figures

We consider a Moran model with recombination in a haploid population of size $N$. At each birth event, with probability $1-\rho_N R$ the offspring copies one parent's chromosome, and with probability $\rho_N R$ she inherits a chromosome that is a mosaic of both parental chromosomes. We assume that at time $0$ each individual has her chromosome painted in a different color and we study the color partition of the chromosome that is asymptotically fixed in a large population, when we look at a portion of the chromosome such that $\rho := \lim_{N\to \infty} \frac{\rho_N N}{2}\to \infty$. To do so, we follow backwards in time the ancestry of the chromosome of a randomly sampled individual. This yields a Markov process valued in the color partitions of the half-line, that was introduced by Esser et al. (2016), in which blocks can merge and split, called the partitioning process. Its stationary distribution is closely related to the fixed chromosome in our Moran model with recombination. We are able to provide an approximation of this stationary distribution when $\rho \gg 1$ and an error bound. This allows us to show that the distribution of the (renormalised) length of the leftmost block of the partition (i.e. the region of the chromosome that carries the same color as 0) converges to an exponential distribution. In addition, the geometry of this block can be described in terms of a Poisson point process with an explicit intensity measure.

Kingman (1978)'s representation theorem states that any exchangeable partition of $\mathbb{N}$ can be represented as a paintbox based on a random mass-partition. Similarly, any exchangeable composition (i.e. ordered partition of $\mathbb{N}$) can be represented as a paintbox based on an interval-partition (Gnedin 1997). Our first main result is that any exchangeable coalescent process (not necessarily Markovian) can be represented as a paintbox based on a random non-decreasing process valued in interval-partitions, called nested interval-partition, generalizing the notion of comb metric space introduced by Lambert & Uribe Bravo (2017) to represent compact ultrametric spaces. As a special case, we show that any $\Lambda$-coalescent can be obtained from a paintbox based on a unique random nested interval partition called $\Lambda$-comb, which is Markovian with explicit transitions. This nested interval-partition directly relates to the flow of bridges of Bertoin & Le Gall (2003). We also display a particularly simple description of the so-called evolving coalescent (Pfaffelhuber & Wakolbinger 2006) by a comb-valued Markov process. Next, we prove that any measured ultrametric space $U$, under mild measure-theoretic assumptions on $U$, is the leaf set of a tree composed of a separable subtree called the backbone, on which are grafted additional subtrees, which act as star-trees from the standpoint of sampling. Displaying this so-called weak isometry requires us to extend the Gromov-weak topology of Greven et al (2006), that was initially designed for separable metric spaces, to non-separable ultrametric spaces. It allows us to show that for any such ultrametric space $U$, there is a nested interval-partition which is 1) indistinguishable from $U$ in the Gromov-weak topology; 2) weakly isometric to $U$ if $U$ has complete backbone; 3) isometric to $U$ if $U$ is complete and separable., Comment: 62 pages, 7 figures

The first part of this paper ( arXiv:1607.02114 ) introduced splitting trees, those chronological trees admitting the self-similarity property where individuals give birth, at constant rate, to iid copies of themselves. It also established the intimate relationship between splitting trees and L\'evy processes. The chronological trees involved were formalized as Totally Ordered Measured (TOM) trees. The aim of this paper is to continue this line of research in two directions: we first decompose locally compact TOM trees in terms of their prolific skeleton (consisting of its infinite lines of descent). When applied to splitting trees, this implies the construction of the supercritical ones (which are locally compact) in terms of the subcritical ones (which are compact) grafted onto a Yule tree (which corresponds to the prolific skeleton). As a second (related) direction, we study the genealogical tree associated to our chronological construction. This is done through the technology of the height process introduced by Duquesne and Le Gall. In particular we prove a Ray-Knight type theorem which extends the one for (sub)critical L\'evy trees to the supercritical case., Comment: 32 pages, 4 figures

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

This paper gives a new flavor of what Peter Jagers and his co-authors call `the path to extinction'. In a neutral population with constant size $N$, we assume that each individual at time $0$ carries a distinct type, or allele. We consider the joint dynamics of these $N$ alleles, for example the dynamics of their respective frequencies and more plainly the nonincreasing process counting the number of alleles remaining by time $t$. We call this process the extinction process. We show that in the Moran model, the extinction process is distributed as the process counting (in backward time) the number of common ancestors to the whole population, also known as the block counting process of the $N$-Kingman coalescent. Stimulated by this result, we investigate: (1) whether it extends to an identity between the frequencies of blocks in the Kingman coalescent and the frequencies of alleles in the extinction process, both evaluated at jump times; (2) whether it extends to the general case of $\Lambda$-Fleming-Viot processes., Comment: Submitted to appear in the Festschrift in honor of Peter Jagers

The theory of life history evolution provides a powerful framework to understand the evolutionary dynamics of pathogens in both epidemic and endemic situations. This framework, however, relies on the assumption that pathogen populations are very large and that one can neglect the effects of demographic stochasticity. Here we expand the theory of life history evolution to account for the effects of finite population size on the evolution of pathogen virulence. We show that demographic stochasticity introduces additional evolutionary forces that can qualitatively affect the dynamics and the evolutionary outcome. We discuss the importance of the shape of pathogen fitness landscape and host heterogeneity on the balance between mutation, selection and genetic drift. In particular, we discuss scenarios where finite population size can dramatically affect classical predictions of deterministic models. This analysis reconciles Adaptive Dynamics with population genetics in finite populations and thus provides a new theoretical toolbox to study life-history evolution in realistic ecological scenarios., Comment: 13 pages, 5 figures, Supplementary Information

At time 0, start a time-continuous binary branching process, where particles give birth to a single particle independently (at a possibly time-dependent rate) and die independently (at a possibly time-dependent and age-dependent rate). A particular case is the classical birth--death process. Stop this process at time $T>0$. It is known that the tree spanned by the $N$ tips alive at time $T$ of the tree thus obtained (called reduced tree or coalescent tree) is a coalescent point process (CPP), which basically means that the depths of interior nodes are iid. Now select each of the $N$ tips independently with probability $y$ (Bernoulli sample). It is known that the tree generated by the selected tips, which we will call Bernoulli sampled CPP, is again a CPP. Now instead, select exactly $k$ tips uniformly at random among the $N$ tips ($k$-sample). We show that the tree generated by the selected tips is a mixture of Bernoulli sampled CPPs with the same parent CPP, over some explicit distribution of the sampling probability $y$. An immediate consequence is that the genealogy of a $k$-sample can be obtained by the realization of $k$ random variables, first the random sampling probability $Y$ and then the $k-1$ node depths which are iid conditional on $Y=y$., Comment: 19 pages, 1 figure, submitted to a special issue of Theoretical Population Biology in memory of Paul Joyce

We introduce a new random graph model motivated by biological questions relating to speciation. This random graph is defined as the stationary distribution of a Markov chain on the space of graphs on $\{1, \ldots, n\}$. The dynamics of this Markov chain is governed by two types of events: vertex duplication, where at constant rate a pair of vertices is sampled uniformly and one of these vertices loses its incident edges and is rewired to the other vertex and its neighbors; and edge removal, where each edge disappears at constant rate. Besides the number of vertices $n$, the model has a single parameter $r_n$. Using a coalescent approach, we obtain explicit formulas for the first moments of several graph invariants such as the number of edges or the number of complete subgraphs of order $k$. These are then used to identify five non-trivial regimes depending on the asymptotics of the parameter $r_n$. We derive an explicit expression for the degree distribution, and show that under appropriate rescaling it converges to classical distributions when the number of vertices goes to infinity. Finally, we give asymptotic bounds for the number of connected components, and show that in the sparse regime the number of edges is Poissonian., Comment: added Proposition 2.4 and formal proofs of Proposition 2.3 and 2.6

Ultrametric trees are trees whose leaves lie at the same distance from the root. They are used to model the genealogy of a population of particles co-existing at the same point in time. We show how the boundary of an ultrametric tree, like any compact ultrametric space, can be represented in a simple way via the so-called comb metric. We display a variety of examples of random combs and explain how they can be used in applications. In particular, we review some old and recent results regarding the genetic structure of the population when throwing neutral mutations on the skeleton of the tree., Comment: 20 pages, 7 figures, proceedings of MAS 2016, Grenoble, France (Stochastic modeling and Statistics Conference, French Society for Applied and Industrial Math, SMAI)

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., Comment: 45 pages, 6 figures

We consider a continuous population whose dynamics is described by the standard stationary Fleming-Viot process, so that the genealogy of $n$ uniformly sampled individuals is distributed as the Kingman $n$-coalescent. In this note, we study some genealogical properties of this population when the sample is conditioned to fall entirely into a subpopulation with most recent common ancestor (MRCA) shorter than $\varepsilon$. First, using the comb representation of the total genealogy (Lambert & Uribe Bravo 2016), we show that the genealogy of the descendance of the MRCA of the sample on the timescale $\varepsilon$ converges as $\varepsilon\to 0$. The limit is the so-called Brownian coalescent point process (CPP) stopped at an independent Gamma random variable with parameter $n$, which can be seen as the genealogy at a large time of the total population of a rescaled critical birth-death process, biased by the $n$-th power of its size. Secondly, we show that in this limit the coalescence times of the $n$ sampled individuals are i.i.d. uniform random variables in $(0,1)$. These results provide a coupling between two standard models for the genealogy of a random exchangeable population: the Kingman coalescent and the Brownian CPP., Comment: 22 pages, 1 figure

We consider a spectrally positive L\'evy process $X$ that does not drift to $+\infty$, viewed as coding for the genealogical structure of a (sub)critical branching process, in the sense of a contour or exploration process \cite{GaJa98,Lam10}. We denote by $I$ the past infimum process defined for each $t\geq 0$ by $I_t:= \inf_{[0,t]} X$ and we let $\gamma$ be the unique time at which the excursion of the reflected process $X-I$ away from 0 attains its supremum. We prove that the pre-$\gamma$ and the post-$\gamma$ subpaths of this excursion are invariant under space-time reversal, which has several other consequences in terms of duality for excursions of L\'evy processes. It implies in particular that the local time process of this excursion is also invariant when seen backward from its height. As a corollary, we obtain that some (sub)critical branching processes such as the (sub)critical Crump-Mode-Jagers (CMJ) processes and the excursion away from 0 of the critical Feller diffusion, which is the width process of the continuum random tree, are invariant under time reversal from their extinction time.

Combinatorial trees can be used to represent genealogies of asexual individuals. These individuals can be endowed with birth and death times, to obtain a so-called `chronological tree'. In this work, we are interested in the continuum analogue of chronological trees in the setting of real trees. This leads us to consider totally ordered and measured trees, abbreviated as TOM trees. First, we define an adequate space of TOM trees and prove that under some mild conditions, every compact TOM tree can be represented in a unique way by a so-called contour function, which is right-continuous, admits limits from the left and has non-negative jumps. The appropriate notion of contour function is also studied in the case of locally compact TOM trees. Then we study the splitting property of (measures on) TOM trees which extends the notion of `splitting tree' studied in \cite{MR2599603}, where during her lifetime, each individual gives birth at constant rate to independent and identically distributed copies of herself. We prove that the contour function of a TOM tree satisfying the splitting property is associated to a spectrally positive L\'evy process that is not a subordinator, both in the critical and subcritical cases of compact trees as well as in the supercritical case of locally compact trees., Comment: 34 pages, 2 figures; minor reorganization; some clarifications are added

The goal of these lectures is to review some mathematical aspects of random tree models used in evolutionary biology to model gene trees or species trees. We start with stochastic models of tree shapes (finite trees without edge lengths), culminating in the $\beta$-family of Aldous' branching models. We next introduce real trees (trees as metric spaces) and show how to study them through their contour, provided they are properly measured and ordered. We then focus on the reduced tree, or coalescent tree, which is the tree spanned by individuals/species alive at the same fixed time. We show how reduced trees, like any compact ultrametric space, can be represented in a simple way via the so-called comb metric. Beautiful examples of random combs include the Kingman coalescent and coalescent point processes. We end up displaying some recent biological applications of coalescent point processes to the inference of species diversification, to conservation biology and to epidemiology., Comment: 72 pages, 15 figures. Lecture notes of a mini-course given at the XIX Brazilian School of Probability (aug 2015), random tree, tree shape, real tree, reduced tree, branching process, coalescent, comb, phylogenetics, population dynamics, population genetics. To appear in the Brazilian Journal of Probability and Statistics

We call a \emph{comb} a map $f:I\to [0,\infty)$, where $I$ is a compact interval, such that $\{f\ge \varepsilon\}$ is finite for any $\varepsilon$. A comb induces a (pseudo)-distance $\dtf$ on $\{f=0\}$ defined by $\dtf(s,t) = \max_{(s\wedge t, s\vee t)} f$. We describe the completion $\bar I$ of $\{f=0\}$ for this metric, which is a compact ultrametric space called \emph{comb metric space}. Conversely, we prove that any compact, ultrametric space $(U,d)$ without isolated points is isometric to a comb metric space. We show various examples of the comb representation of well-known ultrametric spaces: the Kingman coalescent, infinite sequences of a finite alphabet, the $p$-adic field and spheres of locally compact real trees. In particular, for a rooted, locally compact real tree defined from its contour process $h$, the comb isometric to the sphere of radius $T$ centered at the root can be extracted from $h$ as the depths of its excursions away from $T$., Comment: 24 pages, 3 figures

We consider a random forest $\mathcal{F}^*$, defined as a sequence of i.i.d. birth-death (BD) trees, each started at time 0 from a single ancestor, stopped at the first tree having survived up to a fixed time $T$. We denote by $\left(\xi^*_t,\ 0\leq t\leq T\right)$ the population size process associated to this forest, and we prove that if the BD trees are supercritical, then the time-reversed process $\left(\xi^*_{T-t},\ 0\leq t\leq T\right)$, has the same distribution as $\left(\widetilde\xi^*_t,\ 0\leq t\leq T\right)$, the corresponding population size process of an equally defined forest $\widetilde{\mathcal{F}}^*$, but where the underlying BD trees are subcritical, obtained by swapping birth and death rates or equivalently, conditioning on ultimate extinction. We generalize this result to splitting trees (i.e. life durations of individuals are not necessarily exponential), provided that the i.i.d. lifetimes of the ancestors have a specific explicit distribution, different from that of their descendants. The results are based on an identity between the contour of these random forests truncated up to $T$ and the duality property of L\'evy processes. This identity allows us to also derive other useful properties such as the distribution of the population size process conditional on the reconstructed tree of individuals alive at $T$, which has potential applications in epidemiology., Comment: 28 pages, 3 figures

Forward-time models of diversification (i.e., speciation and extinction) produce phylogenetic trees that grow "vertically" as time goes by. Pruning the extinct lineages out of such trees leads to natural models for reconstructed trees (i.e., phylogenies of extant species). Alternatively, reconstructed trees can be modelled by coalescent point processes (CPP), where trees grow "horizontally" by the sequential addition of vertical edges. Each new edge starts at some random speciation time and ends at the present time; speciation times are drawn from the same distribution independently. CPP lead to extremely fast computation of tree likelihoods and simulation of reconstructed trees. Their topology always follows the uniform distribution on ranked tree shapes (URT). We characterize which forward-time models lead to URT reconstructed trees and among these, which lead to CPP reconstructed trees. We show that for any "asymmetric" diversification model in which speciation rates only depend on time and extinction rates only depend on time and on a non-heritable trait (e.g., age), the reconstructed tree is CPP, even if extant species are incompletely sampled. If rates additionally depend on the number of species, the reconstructed tree is (only) URT (but not CPP). We characterize the common distribution of speciation times in the CPP description, and discuss incomplete species sampling as well as three special model cases in detail: 1) extinction rate does not depend on a trait; 2) rates do not depend on time; 3) mass extinctions may happen additionally at certain points in the past.

The reconstruction of phylogenetic trees based on viral genetic sequence data sequentially sampled from an epidemic provides estimates of the past transmission dynamics, by fitting epidemiological models to these trees. To our knowledge, none of the epidemiological models currently used in phylogenetics can account for recovery rates and sampling rates dependent on the time elapsed since transmission. Here we introduce an epidemiological model where infectives leave the epidemic, either by recovery or sampling, after some random time which may follow an arbitrary distribution. We derive an expression for the likelihood of the phylogenetic tree of sampled infectives under our general epidemiological model. The analytic concept developed in this paper will facilitate inference of past epidemiological dynamics and provide an analytical framework for performing very efficient simulations of phylogenetic trees under our model. The main idea of our analytic study is that the non-Markovian epidemiological model giving rise to phylogenetic trees growing vertically as time goes by, can be represented by a Markovian "coalescent point process" growing horizontally by the sequential addition of pairs of coalescence and sampling times. As examples, we discuss two special cases of our general model, namely an application to influenza and an application to HIV. Though phrased in epidemiological terms, our framework can also be used for instance to fit macroevolutionary models to phylogenies of extant and extinct species, accounting for general species lifetime distributions., Comment: 30 pages, 2 figures

For many taxa, the current high rates of extinction are likely to result in a significant loss of biodiversity. The evolutionary heritage of biodiversity is frequently quantified by a measure called phylogenetic diversity (PD). We predict the loss of PD under a wide class of phylogenetic tree models, where speciation rates and extinction rates may be time-dependent, and assuming independent random species extinctions at the present. We study the loss of PD when $K$ contemporary species are selected uniformly at random from the $N$ extant species as the surviving taxa, while the remaining $N-K$ become extinct. We consider two models of species sampling, the so-called field of bullets model, where each species independently survives the extinction event at the present with probability $p$, and a model for which the number of surviving species is fixed. We provide explicit formulae for the expected remaining PD in both models, conditional on $N=n$, conditional on $K=k$, or conditional on both events. When $N=n$ is fixed, we show the convergence to an explicit deterministic limit of the ratio of new to initial PD, as $n\to\infty$, both under the field of bullets model, and when $K=k_n$ is fixed and depends on $n$ in such a way that $k_n/n$ converges to $p$. We also prove the convergence of this ratio as $T\to\infty$ in the supercritical, time-homogeneous case, where $N$ simultaneously goes to $\infty$, thereby strengthening previous results of Mooers et al. (2012)., Comment: 58 pages 3 figures

We consider a dynamic metapopulation involving one large population of size N surrounded by colonies of size \varepsilon_NN, usually called peripheral isolates in ecology, where N\to\infty and \varepsilon_N\to 0 in such a way that \varepsilon_NN\to\infty. The main population periodically sends propagules to found new colonies (emigration), and each colony eventually merges with the main population (fusion). Our aim is to study the genealogical history of a finite number of lineages sampled at stationarity in such a metapopulation. We make assumptions on model parameters ensuring that the total outer population has size of the order of N and that each colony has a lifetime of the same order. We prove that under these assumptions, the scaling limit of the genealogical process of a finite sample is a censored coalescent where each lineage can be in one of two states: an inner lineage (belonging to the main population) or an outer lineage (belonging to some peripheral isolate). Lineages change state at constant rate and inner lineages (only) coalesce at constant rate per pair. This two-state censored coalescent is also shown to converge weakly, as the landscape dynamics accelerate, to a time-changed Kingman coalescent., Comment: 17 pages,1 figure

A popular line of research in evolutionary biology is the use of time-calibrated phylogenies for the inference of diversification processes. This requires computing the likelihood of a given ultrametric tree as the reconstructed tree produced by a given model of diversification. Etienne & Rosindell (2012) proposed a lineage-based model of diversification, called protracted speciation, where species remain incipient during a random duration before turning good species, and showed that this can explain the slowdown in lineage accumulation observed in real phylogenies. However, they were unable to provide a general likelihood formula. Here, we present a likelihood formula for protracted speciation models, where rates at which species turn good or become extinct can depend both on their age and on time. Our only restrictive assumption is that speciation rate does not depend on species status. Our likelihood formula utilizes a new technique, based on the contour of the phylogenetic tree and first developed in Lambert (2010). We consider the reconstructed trees spanned by all extant species, by all good extant species, or by all representative species, which are either good extant species or incipient species representative of some good extinct species. Specifically, we prove that each of these trees is a coalescent point process, that is, a planar, ultrametric tree where the coalescence times between two consecutive tips are independent, identically distributed random variables. We characterize the common distribution of these coalescence times in some, biologically meaningful, special cases for which the likelihood reduces to an elegant analytical formula or becomes numerically tractable., Comment: 27 pages, 5 figures

Source-sink systems are metapopulations of habitat patches with different, and possibly temporally varying, habitat qualities, which are commonly used in ecology to study the fate of spatially extended natural populations. We propose new techniques that allow to disentangle the respective contributions of demography and dispersal to the dynamics and fate of a single species in a source-sink metapopulation. Our approach is valid for a general class of stochastic, individual-based, stepping-stone models, with density-independent demography and dispersal, provided the metapopulation is finite or else enjoys some transitivity property. We provide 1) a simple criterion of persistence, by studying the motion of a single random disperser until it returns to its initial position; 2) a joint characterization of the long-term growth rate and of the asymptotic occupancy frequencies of the ancestral lineage of a random survivor, by using large deviations theory. Both techniques yield formulae decoupling demography and dispersal, and can be adapted to the case of periodic or random environments, where habitat qualities are autocorrelated in space and possibly in time. In this last case, we display examples of coupled time-averaged sinks allowing survival, as was previously known in the absence of demographic stochasticity for fully mixing (Jansen and Yoshimura, 1998) and even partially mixing (Evans et al., 2012; Schreiber, 2010) metapopulations., Comment: arXiv admin note: substantial text overlap with arXiv:1111.2535

In this paper, we review recent results of ours concerning branching processes with general lifetimes and neutral mutations, under the infinitely many alleles model, where mutations can occur either at birth of individuals or at a constant rate during their lives. In both models, we study the allelic partition of the population at time t. We give closed formulae for the expected frequency spectrum at t and prove pathwise convergence to an explicit limit, as t goes to infinity, of the relative numbers of types younger than some given age and carried by a given number of individuals (small families). We also provide convergences in distribution of the sizes or ages of the largest families and of the oldest families. In the case of exponential lifetimes, population dynamics are given by linear birth and death processes, and we can most of the time provide general formulations of our results unifying both models., Comment: 20 pages, 2 figures

Consider compound Poisson processes with negative drift and no negative jumps, which converge to some spectrally positive L\'evy process with non-zero L\'evy measure. In this paper we study the asymptotic behavior of the local time process, in the spatial variable, of these processes killed at two different random times: either at the time of the first visit of the L\'evy process to 0, in which case we prove results at the excursion level under suitable conditionings; or at the time when the local time at 0 exceeds some fixed level. We prove that finite-dimensional distributions converge under general assumptions, even if the limiting process is not c\`adl\`ag. Making an assumption on the distribution of the jumps of the compound Poisson processes, we strengthen this to get weak convergence. Our assumption allows for the limiting process to be a stable L\'evy process with drift. These results have implications on branching processes and in queueing theory, namely, on the scaling limit of binary, homogeneous Crump-Mode-Jagers processes and on the scaling limit of the Processor-Sharing queue length process., Comment: Final version accepted for publication in Journal of Theoretical Probability

We consider regenerative processes with values in some Polish space. We define their \epsilon-big excursions as excursions e such that f(e)>\epsilon, where f is some given functional on the space of excursions which can be thought of as, e.g., the length or the height of e. We establish a general condition that guarantees the convergence of a sequence of regenerative processes involving the convergence of \epsilon-big excursions and of their endpoints, for all \epsilon in a countable set whose closure contains 0. Finally, we provide various sufficient conditions on the excursion measures of this sequence for this general condition to hold and discuss possible generalizations of our approach to processes that can be written as the concatenation of i.i.d. paths., Comment: Final version accepted for publication in Annales de l'Institut Henri Poincar\'e

Source-sink systems are metapopulations of patches that can be of variable habitat quality. They can be seen as graphs, where vertices represent the patches, and the weighted oriented edges give the probability of dispersal from one patch to another. We consider either finite or source-transitive graphs, i.e., graphs that are identical when viewed from a(ny) source. We assume stochastic, individual-based, density-independent reproduction and dispersal. By studying the path of a single random disperser, we are able to display simple criteria for persistence, either necessary and sufficient, or just sufficient. In case of persistence, we characterize the growth rate of the population as well as the asymptotic occupancy frequencies of the line of ascent of a random survivor. Our method allows to decouple the roles of reproduction and dispersal. Finally, we extend our results to the case of periodic or random environments, where some habitats can have variable growth rates, autocorrelated in space and possibly in time. In the whole manuscript, special attention is given to the example of regular graphs where each pair of adjacent sources is separated by the same number of identical sinks. In the case of a periodic and random environment, we also display examples where all patches are sinks when forbidding dispersal but the metapopulation survives with positive probability in the presence of dispersal, as previously known for a two-patch mean-field model with parent-independent dispersal., Comment: 30 pages

We consider a branching population where individuals have i.i.d.\ life lengths (not necessarily exponential) and constant birth rate. We let $N_t$ denote the population size at time $t$. %(called homogeneous, binary Crump--Mode--Jagers process). We further assume that all individuals, at birth time, are equipped with independent exponential clocks with parameter $\delta$. We are interested in the genealogical tree stopped at the first time $T$ when one of those clocks rings. This question has applications in epidemiology, in population genetics, in ecology and in queuing theory. We show that conditional on $\{T<\infty\}$, the joint law of $(N_T, T, X^{(T)})$, where $X^{(T)}$ is the jumping contour process of the tree truncated at time $T$, is equal to that of $(M, -I_M, Y_M')$ conditional on $\{M\not=0\}$, where : $M+1$ is the number of visits of 0, before some single independent exponential clock $\mathbf{e}$ with parameter $\delta$ rings, by some specified L{\'e}vy process $Y$ without negative jumps reflected below its supremum; $I_M$ is the infimum of the path $Y_M$ defined as $Y$ killed at its last 0 before $\mathbf{e}$; $Y_M'$ is the Vervaat transform of $Y_M$. This identity yields an explanation for the geometric distribution of $N_T$ \cite{K,T} and has numerous other applications. In particular, conditional on $\{N_T=n\}$, and also on $\{N_T=n, T

We consider a supercritical branching population, where individuals have i.i.d. lifetime durations (which are not necessarily exponentially distributed) and give birth (singly) at constant rate. We assume that individuals independently experience neutral mutations, at constant rate $\theta$ during their lifetimes, under the infinite-alleles assumption: each mutation instantaneously confers a brand new type, called allele or haplotype, to its carrier. The type carried by a mother at the time when she gives birth is transmitted to the newborn. We are interested in the sizes and ages at time $t$ of the clonal families carrying the most abundant alleles or the oldest ones, as $t\to\infty$, on the survival event. Intuitively, the results must depend on how the mutation rate $\theta$ and the Malthusian parameter $\alpha>0$ compare. Hereafter, $N\equiv N_t$ is the population size at time $t$, constants $a,c$ are scaling constants, whereas $k,k'$ are explicit positive constants which depend on the parameters of the model. When $\alpha>\theta$, the most abundant families are also the oldest ones, they have size $cN^{1-\theta/\alpha}$ and age $t-a$. When $\alpha<\theta$, the oldest families have age $(\alpha /\theta)t+a$ and tight sizes; the most abundant families have sizes $k\log(N)-k'\log\log(N)+c$ and all have age $(\theta-\alpha)^{-1}\log(t)$. When $\alpha=\theta$, the oldest families have age $kt-k'\log(t)+a$ and tight sizes; the most abundant families have sizes $(k\log(N)-k'\log\log(N)+c)^2$ and all have age $t/2$. Those informal results can be stated rigorously in expectation. Relying heavily on the theory of coalescent point processes, we are also able, when $\alpha\leq\theta$, to show convergence in distribution of the joint, properly scaled ages and sizes of the most abundant/oldest families and to specify the limits as some explicit Cox processes.

We study the convergence of the $M/G/1$ processor-sharing, queue length process in the heavy traffic regime, in the finite variance case. To do so, we combine results pertaining to L\'{e}vy processes, branching processes and queuing theory. These results yield the convergence of long excursions of the queue length processes, toward excursions obtained from those of some reflected Brownian motion with drift, after taking the image of their local time process by the Lamperti transformation. We also show, via excursion theoretic arguments, that this entails the convergence of the entire processes to some (other) reflected Brownian motion with drift. Along the way, we prove various invariance principles for homogeneous, binary Crump-Mode-Jagers processes. In the last section we discuss potential implications of the state space collapse property, well known in the queuing literature, to branching processes., Comment: Published in at http://dx.doi.org/10.1214/12-AAP904 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We define a doubly infinite, monotone labeling of Bienayme-Galton-Watson (BGW) genealogies. The genealogy of the current generation backwards in time is uniquely determined by the coalescent point process $(A_i; i\ge 1)$, where $A_i$ is the coalescence time between individuals i and i+1. There is a Markov process of point measures $(B_i; i\ge 1)$ keeping track of more ancestral relationships, such that $A_i$ is also the first point mass of $B_i$. This process of point measures is also closely related to an inhomogeneous spine decomposition of the lineage of the first surviving particle in generation h in a planar BGW tree conditioned to survive h generations. The decomposition involves a point measure $\rho$ storing the number of subtrees on the right-hand side of the spine. Under appropriate conditions, we prove convergence of this point measure to a point measure on $\mathbb{R}_+$ associated with the limiting continuous-state branching (CSB) process. We prove the associated invariance principle for the coalescent point process, after we discretize the limiting CSB population by considering only points with coalescence times greater than $\varepsilon$., Comment: Published in at http://dx.doi.org/10.1214/11-AAP820 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We consider a neutral dynamical model of biological diversity, where individuals live and reproduce independently. They have i.i.d. lifetime durations (which are not necessarily exponentially distributed) and give birth (singly) at constant rate b. Such a genealogical tree is usually called a splitting tree, and the population counting process (N_t;t\ge 0) is a homogeneous, binary Crump--Mode--Jagers process. We assume that individuals independently experience mutations at constant rate \theta during their lifetimes, under the infinite-alleles assumption: each mutation instantaneously confers a brand new type, called allele, to its carrier. We are interested in the allele frequency spectrum at time t, i.e., the number A(t) of distinct alleles represented in the population at time t, and more specifically, the numbers A(k,t) of alleles represented by k individuals at time t, k=1,2,...,N_t. We mainly use two classes of tools: coalescent point processes and branching processes counted by random characteristics. We provide explicit formulae for the expectation of A(k,t) in a coalescent point process conditional on population size, which apply to the special case of splitting trees. We separately derive the a.s. limits of A(k,t)/N_t and of A(t)/N_t thanks to random characteristics. Last, we separately compute the expected homozygosity by applying a method characterizing the dynamics of the tree distribution as the origination time of the tree moves back in time, in the spirit of backward Kolmogorov equations., Comment: 32 pages, 2 figures. Companion paper in preparation "Splitting trees with neutral Poissonian mutations II: Large or old families"

We consider a general, neutral, dynamical model of biodiversity. Individuals have i.i.d. lifetime durations, which are not necessarily exponentially distributed, and each individual gives birth independently at constant rate \lambda. We assume that types are clonally inherited. We consider two classes of speciation models in this setting. In the immigration model, new individuals of an entirely new species singly enter the population at constant rate \mu (e.g., from the mainland into the island). In the mutation model, each individual independently experiences point mutations in its germ line, at constant rate \theta. We are interested in the species abundance distribution, i.e., in the numbers, denoted I_n(k) in the immigration model and A_n(k) in the mutation model, of species represented by k individuals, k=1,2,...,n, when there are n individuals in the total population. In the immigration model, we prove that the numbers (I_t(k);k\ge 1) of species represented by k individuals at time t, are independent Poisson variables with parameters as in Fisher's log-series. When conditioning on the total size of the population to equal n, this results in species abundance distributions given by Ewens' sampling formula. In particular, I_n(k) converges as n\to\infty to a Poisson r.v. with mean \gamma /k, where \gamma:=\mu/\lambda. In the mutation model, as n\to\infty, we obtain the almost sure convergence of n^{-1}A_n(k) to a nonrandom explicit constant. In the case of a critical, linear birth--death process, this constant is given by Fisher's log-series, namely n^{-1}A_n(k) converges to \alpha^{k}/k, where \alpha :=\lambda/(\lambda+\theta). In both models, the abundances of the most abundant species are briefly discussed., Comment: 16 pages, 4 figures. To appear in Journal of Mathematical Biology. The final publication is available at http://www.springerlink.com