A phylogenetic birth-and-death model is a probabilistic graphical model for a so-called phylogenetic profile, i.e., the size distribution for a homolog gene family at the terminal nodes of a phylogeny. Profile datasets are used in bioinformatics analyses for the inference of evolutionary trees, and of functional associations between gene families, as well as for the quantification of various processes guiding genome evolution. Here we describe the mathematical formalism for phylogenetic birth-and-death models. We also present an algorithm for computing the likelihood in a gain-loss-duplication model.
We propose the following simple stochastic model for phylogenetic trees. New types are born and die according to a birth and death chain. At each birth we associate a fitness to the new type sampled from a fixed distribution. At each death the type with the smallest fitness is killed. We show that if the birth (i.e. mutation) rate is subcritical we get a phylogenetic tree consistent with an influenza tree (few types at any given time and one dominating type lasting a long time). When the birth rate is supercritical we get a phylogenetic tree consistent with an HIV tree (many types at any given…
Birth-and-death models are now a common mathematical tool to describe branching patterns observed in real-world phylogenetic trees. Liggett and Schinazi (2009) is one such example. The authors propose a simple birth-and-death model that is compatible with phylogenetic trees of both influenza and HIV, depending on the birth rate parameter. An interesting special case of this model is the critical case where the birth rate equals the death rate. This is a non-trivial situation and to study its asymptotic behaviour we employed the Laplace transform. With this we correct the proof of Liggett and S…
We show that simple stochastic models of genome evolution lead to power law asymptotics of protein domain family size distribution. These models, called Birth, Death and Innovation Models (BDIM), represent a special class of balanced birth-and-death processes, in which domain duplication and deletion rates are asymptotically equal up to the second order. The simplest, linear BDIM shows an excellent fit to the observed distributions of domain family size in diverse prokaryotic and eukaryotic genomes. However, the stochastic version of the linear BDIM explored here predicts that the actual size …
Here we introduce a general class of multiple calibration birth-death tree priors for use in Bayesian phylogenetic inference. All tree priors in this class separate ancestral node heights into a set of "calibrated nodes" and "uncalibrated nodes" such that the marginal distribution of the calibrated nodes is user-specified whereas the density ratio of the birth-death prior is retained for trees with equal values for the calibrated nodes. We describe two formulations, one in which the calibration information informs the prior on ranked tree topologies, through the (conditional) prior, and the ot…
Many population genetic models have been developed for the purpose of inferring population size and growth rates from random samples of genetic data. We examine two popular approaches to this problem, the coalescent and the birth-death-sampling model, in the context of estimating population size and birth rates in a population growing exponentially according to the birth-death branching process. For sequences sampled at a single time, we found the coalescent and the birth-death-sampling model gave virtually indistinguishable results in terms of the growth rates and fraction of the population s…
The statistical estimation of phylogenies is always associated with uncertainty, and accommodating this uncertainty is an important component of modern phylogenetic comparative analysis. The birth-death polytomy resolver is a method of accounting for phylogenetic uncertainty that places missing (unsampled) taxa onto phylogenetic trees, using taxonomic information alone. Recent studies of birds and mammals have used this approach to generate pseudo-posterior distributions of phylogenetic trees that are complete at the species level, even in the absence of genetic data for many species. Many res…
The spread of an advantageous mutation through a population is of fundamental interest in population genetics. While the classical Moran model is formulated for a well-mixed population, it has long been recognized that in real-world applications, the population usually has an explicit spatial structure which can significantly influence the dynamics. In the context of cancer initiation in epithelial tissue, several recent works have analyzed the dynamics of advantageous mutant spread on integer lattices, using the biased voter model from particle systems theory. In this spatial version of the M…
We consider birth-and-death stochastic evolution of genotypes with different lengths. The genotypes might mutate that provides a stochastic changing of lengthes by a free diffusion law. The birth and death rates are length dependent which corresponds to a selection effect. We study an asymptotic behavior of a density for an infinite collection of genotypes. The cases of space homogeneous and space heterogeneous densities are considered.
We analyze evolutionary dynamics on graphs, where the nodes represent individuals of a population. The links of a node describe which other individuals can be displaced by the offspring of the individual on that node. Amplifiers of selection are graphs for which the fixation probability is increased for advantageous mutants and decreased for disadvantageous mutants. A few examples of such amplifiers have been developed, but so far it is unclear how many such structures exist and how to construct them. Here, we show that almost any undirected random graph is an amplifier of selection for Birth-…
Evolutionary models on graphs, as an extension of the Moran process, have two major implementations: birth-death (BD) models (or the invasion process) and death- birth (DB) models (or voter models). The isothermal theorem states that the fixation probability of mutants in a large group of graph structures (known as isothermal graphs, which include regular graphs) coincides with that for the mixed population. This result has been proven by Lieberman et al (Nature 433: 312-316, 2005) in the case of BD processes, where mutants differ from the wild types by their birth rate (and not by their death…
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 exp…
This paper is an attempt to formalize analytically the question raised in "World Population Explained: Do Dead People Outnumber Living, Or Vice Versa?" Huffington Post, \cite{HJ}. We start developing simple deterministic Malthusian growth models of the problem (with birth and death rates either constant or time-dependent) before running into both linear birth and death Markov chain models and age-structured models.
Broom and Rychtář [Proc. R. Soc. A (2008) 464, 2609--2627] found an exact solution for the fixation probabilities of the Moran process for a structured population, in which the interaction structure among individuals is given by the so-called star graph, i.e. one central vertex and $n$ leaves, the leaves connecting only to the center. We generalize on their solution by allowing individuals' fitnesses to depend on the population frequency, and also by allowing a possible change in the order of reproduction and death draws. In their cited paper, Broom and Rychtář considered the birth-death (BD) …
We consider a birth-death process with the birth rates $iλ$ and death rates $iμ+i(i-1)θ$, where $i$ is the current state of the process. A positive competition rate $θ$ is assumed to be small. In the supercritical case when $λ>μ$ this process can be viewed as a demographic model for a population with a high carrying capacity around $λ-μ\overθ$. The article reports in a self-contained manner on the asymptotic properties of the time to extinction for this logistic branching process as $θ\to0$. All three reproduction regimes $λ>μ$, $λ<μ$, and $λ=μ$ are studied.
In this paper, we introduce a modified epidemic model on regular and scale-free networks respectively. We consider the birth rate $δ$, cure rate $γ$, infection rate $λ$, $α$ from the infectious disease, and death rate $β$ from other factors. Through mean-field analysis, we find that on regular network there is an epidemic threshold $λ_{c}$ dependent on the parameters $δ$, $γ$, $α$, and $β$; while for power law degree distribution network epidemic threshold is absent in the thermodynamic limit. The result is the same as that of the standard SIS model. This reminds us the structure of the networ…
The fixation probability of a single mutant invading a population of residents is among the most widely-studied quantities in evolutionary dynamics. Amplifiers of natural selection are population structures that increase the fixation probability of advantageous mutants, compared to well-mixed populations. Extensive studies have shown that many amplifiers exist for the Birth-death Moran process, some of them substantially increasing the fixation probability or even guaranteeing fixation in the limit of large population size. On the other hand, no amplifiers are known for the death-Birth Moran p…
Stochastic birth-death processes are described as continuous-time Markov processes in models of population dynamics. A system of infinite, coupled ordinary differential equations (the so-called master equation) describes the time-dependence of the probability of each system state. Using a generating function, the master equation can be transformed into a partial differential equation. In this contribution we analyze the integrability of two types of stochastic birth-death processes (with polynomial birth and death rates) using standard differential Galois theory. We discuss the integrability o…
We use a combination of analytic models and computer simulations to gain insight into the dynamics of evolution. Our results suggest that certain interesting phenomena should eventually emerge from the fossil record. For example, there should be a ``tortoise and hare effect'': Those genera with the smallest species death rate are likely to survive much longer than genera with large species birth and death rates. A complete characterization of the behavior of a branch of the phylogenetic tree corresponding to a genus and accurate mathematical representations of the various stages are obtained. …
A birth-death process is a continuous-time Markov chain that counts the number of particles in a system over time. In the general process with $n$ current particles, a new particle is born with instantaneous rate $λ_n$ and a particle dies with instantaneous rate $μ_n$. Currently no robust and efficient method exists to evaluate the finite-time transition probabilities in a general birth-death process with arbitrary birth and death rates. In this paper, we first revisit the theory of continued fractions to obtain expressions for the Laplace transforms of these transition probabilities and make …
Individual-based models of chemical or biological dynamics usually consider individual entities diffusing in space and performing a birth-death type dynamics. In this work we study the properties of a model in this class where the birth dynamics is mediated by the local, within a given distance, density of particles. Groups of individuals are formed in the system and in this paper we concentrate on the study of the properties of these clusters (lifetime, size, and collective diffusion). In particular, in the limit of the interaction distance approaching the system size, a unique cluster appear…
The dynamics of populations is frequently subject to intrinsic noise. At the same time unknown interaction networks or rate constants can present quenched uncertainty. Existing approaches often involve repeated sampling of the quenched disorder and then running the stochastic birth-death dynamics on these samples. In this paper we take a different view, and formulate an effective jump process, representative of the ensemble of quenched interactions as a whole. Using evolutionary games with random payoff matrices as an example, we develop an algorithm to simulate this process, and we discuss di…
First, we revisit the stochastic Luria-Delbrück model: a classic two-type branching process which describes cell proliferation and mutation. We prove limit theorems and exact results for the mutation times, clone sizes, and number of mutants. Second, we extend the framework to consider mutations at multiple sites along the genome. The number of mutants in the two-type model characterises the mean site frequency spectrum in the multiple-site model. Our predictions are consistent with previously published cancer genomic data.
We suggest a general spectral method for calculating statistics of multi-step birth-death processes and chemical reactions of the type mA->nA (m and n are positive integers) which possess an absorbing state. The method yields accurate results for the extinction statistics, and for the quasi-stationary probability distribution, including large deviations, of the metastable state. The power of the method is demonstrated on the example of binary annihilation and triple branching 2A->0 and A->3A, representative of the rather general class of dissociation-recombination reactions.
We propose a new deterministic growth model which captures certain features of both the Gompertz and Korf laws. We investigate its main properties, with special attention to the correction factor, the relative growth rate, the inflection point, the maximum specific growth rate, the lag time and the threshold crossing problem. Some data analytic examples and their performance are also considered. Furthermore, we study a stochastic counterpart of the proposed model, that is a linear time-inhomogeneous birth-death process whose mean behaves as the deterministic one. We obtain the transition proba…
In this paper we study the long term evolution of a continuous time Markov chain formed by two interacting birth-and-death processes. The interaction between the processes is modelled by transition rates which are functions with suitable monotonicity properties. This is in line with the approach proposed by Gauss G.F. and Kolmogorov A.N. for modelling interaction between species in ecology. We obtain conditions for transience/recurrence of the Markov chain and describe in detail its asymptotic behaviour in special transient cases. In particular, we find that in some of these cases the Markov c…
We study mathematical models describing the evolution of stochastic age-structured populations. After reviewing existing approaches, we present a full kinetic framework for age-structured interacting populations undergoing birth, death and fission processes, in spatially dependent environments. We define the complete probability density for the population-size-age-chart and find results under specific conditions. Connections with more classical models are also explicitly derived. In particular, we show that factorial moments for non-interacting processes are described by a natural generalizati…
