Sterile versus prolific individuals pertaining to linear-fractional Bienaym{\'e}-Galton-Watson trees

In a Bienaym\'{e}-Galton-Watson process for which there is a positiveprobability for individuals of having no offspring, there is a subtlebalance and dependence between the sterile nodes (the dead nodes or leaves)and the prolific ones (the productive nodes) both at and up to the currentgeneration. We explore the many facets of this problem, especially in thecontext of an exactly solvable linear-fractional branching mechanism at allgeneration. Eased asymptotic issues are investigated. Relation of thisspecial branching process to skip-free to the left and simple random walks'excursions is then investigated. Mutual statistical information on theirshapes can be learnt from this association

Employing phylogenetic tree shape statistics to resolve the underlying host population structure

BACKGROUND: Host population structure is a key determinant of pathogen and infectious disease transmission patterns. Pathogen phylogenetic trees are useful tools to reveal the population structure underlying an epidemic. Determining whether a population is structured or not is useful in informing the type of phylogenetic methods to be used in a given study. We employ tree statistics derived from phylogenetic trees and machine learning classification techniques to reveal an underlying population structure. RESULTS: In this paper, we simulate phylogenetic trees from both structured and non-structured host populations. We compute eight statistics for the simulated trees, which are: the number of cherries; Sackin, Colless and total cophenetic indices; ladder length; maximum depth; maximum width, and width-to-depth ratio. Based on the estimated tree statistics, we classify the simulated trees as from either a non-structured or a structured population using the decision tree (DT), K-nearest neighbor (KNN) and support vector machine (SVM). We incorporate the basic reproductive number ([Formula: see text]) in our tree simulation procedure. Sensitivity analysis is done to investigate whether the classifiers are robust to different choice of model parameters and to size of trees. Cross-validated results for area under the curve (AUC) for receiver operating characteristic (ROC) curves yield mean values of over 0.9 for most of the classification models. CONCLUSIONS: Our classification procedure distinguishes well between trees from structured and non-structured populations using the classifiers, the two-sample Kolmogorov-Smirnov, Cucconi and Podgor-Gastwirth tests and the box plots. SVM models were more robust to changes in model parameters and tree size compared to KNN and DT classifiers. Our classification procedure was applied to real -world data and the structured population was revealed with high accuracy of [Formula: see text] using SVM-polynomial classifier

Coagulation and Fragmentation Models

Analysis of coagulation and fragmentation is crucial to understanding many processes of scientific and industrial importance. In recent years this has led to intensified research activities in the areas of differential equations, probability theory, and combinatorics. The purpose of the workshop was to bring together people from these different areas working on various aspects of coagulation and fragmentation. We believe that the insights resulting from the interactions which have been stimulated that week should lead to further advances both in the development of mathematical techniques and in new applications

On joint subtree distributions under two evolutionary models

In population and evolutionary biology, hypotheses about micro-evolutionary and macroevolutionary processes are commonly tested by comparing the shape indices of empirical evolutionary trees with those predicted by neutral models. A key ingredient in this approach is the ability to compute and quantify distributions of various tree shape indices under random models of interest. As a step to meet this challenge, in this paper we investigate the joint distribution of cherries and pitchforks (that is, subtrees with two and three leaves) under two widely used null models: the Yule-Harding-Kingman (YHK) model and the proportional to distinguishable arrangements (PDA) model. Based on two novel recursive formulae, we propose a dynamic approach to numerically compute the exact joint distribution (and hence the marginal distributions) for trees of any size. We also obtained insights into the statistical properties of trees generated under these two models, including a constant correlation between the cherry and the pitchfork distributions under the YHK model, and the log-concavity and unimodality of the cherry distributions under both models. In addition, we show that there exists a unique change point for the cherry distributions between these two models

Field theories for stochastic processes

This thesis is a collection of collaborative research work which uses field-theoretic techniques to approach three different areas of stochastic dynamics: Branching Processes, First-passage times of processes with are subject to both white and coloured noise, and numerical and analytical aspects of first-passage times in fractional Brownian Motion. Chapter 1 (joint work with Rosalba Garcia Millan, Johannes Pausch, and Gunnar Pruessner, appeared in Phys. Rev. E 98 (6):062107) contains an analysis of non-spatial branching processes with arbitrary offspring distribution. Here our focus lies on the statistics of the number of particles in the system at any given time. We calculate a host of observables using Doi-Peliti field theory and find that close to criticality these observables no longer depend on the details of the offspring distribution, and are thus universal. In Chapter 2 (joint work with Ignacio Bordeu, Saoirse Amarteifio, Rosalba Garcia Millan, Nanxin Wei, and Gunnar Pruessner, appeared in Sci. Rep. 9:15590) we study the number of sites visited by a branching random walk on general graphs. To do so, we introduce a fieldtheoretic tracing mechanism which keeps track of all already visited sites. We find the scaling laws of the moments of the distribution near the critical point. Chapter 3 (joint work with Gunnar Pruessner and Guillaume Salbreux, submitted, arXiv: 2006.00116) provides an analysis of the first-passage time problem for stochastic processes subject to white and coloured noise. By way of a perturbation theory, I give a systematic and controlled expansion of the moment generating function of first-passage times. In Chapter 4, we revise the tracing mechanism found earlier and use it to characterise three different extreme values, first-passage times, running maxima, and mean volume explored. By formulating these in field-theoretic language, we are able to derive new results for a class of non-Markovian stochastic processes. Chapter 5 and 6 are concerned with the first-passage time distribution of fractional Brownian Motion. Chapter 5 (joint work with Kay Wiese, appeared in Phys. Rev. E 101 (4):043312) introduces a new algorithm to sample them efficiently. Chapter 6 (joint work with Maxence Arutkin and Kay Wiese, submitted, arXiv:1908.10801) gives a field-theoretically obtained perturbative result of the first-passage time distribution in the presence of linear and non-linear drift.Open Acces

Probabilités et biologie

Cette session présente divers aspects de la modélisation probabiliste des populations, en mettant l’accent sur le branchement : processus de branchement à taille de population aléatoire en milieux aléatoires; coexistence de la diversité dans des processus de branchement à taille de population constante; étude des longueurs de branches externes dans les arbres généalogiques de population à taille constante; modélisation de la taille d’une tumeur (population à taille aléatoire) traitée par radiothérapie

On asymptotic joint distributions of cherries and pitchforks for random phylogenetic trees

Tree shape statistics provide valuable quantitative insights into evolutionary mechanisms underpinning phylogenetic trees, a commonly used graph representation of evolutionary relationships among taxonomic units ranging from viruses to species. We study two subtree counting statistics, the number of cherries and the number of pitchforks, for random phylogenetic trees generated by two widely used null tree models: the proportional to distinguishable arrangements (PDA) and the Yule-Harding-Kingman (YHK) models. By developing limit theorems for a version of extended Pólya urn models in which negative entries are permitted for their replacement matrices, we deduce the strong laws of large numbers and the central limit theorems for the joint distributions of these two counting statistics for the PDA and the YHK models. Our results indicate that the limiting behaviour of these two statistics, when appropriately scaled using the number of leaves in the underlying trees, is independent of the initial tree used in the tree generating process