Discrete-time Branching Process Research Articles

Branching model with state dependent offspring distribution for Chlamydia spread

Chlamydiae are bacteria with an interesting unusual developmental cycle. Initially, a single bacterium in its infectious form (elementary body, EB) enters the host cell, where it converts into its dividing form (reticulate body, RB), and divides by binary fission. Since only the EB form is infectious, before the host cell dies, RBs start to convert into EBs. After the host cell dies RBs do not survive. We model the population growth by a 2-type discrete-time branching process, where the probability of duplication depends on the state. Maximizing the EB production leads to a stochastic optimization problem. Simulation study shows that our novel model is able to reproduce the main features of the development of the population.

Assessing epidemic curves for evidence of superspreading.

The expected number of secondary infections arising from each index case, referred to as the reproduction or number, is a vital summary statistic for understanding and managing epidemic diseases. There are many methods for estimating ; however, few explicitly model heterogeneous disease reproduction, which gives rise to superspreading within the population. We propose a parsimonious discrete-time branching process model for epidemic curves that incorporates heterogeneous individual reproduction numbers. Our Bayesian approach to inference illustrates that this heterogeneity results in less certainty on estimates of the time-varying cohort reproduction number . We apply these methods to a COVID-19 epidemic curve for the Republic of Ireland and find support for heterogeneous disease reproduction. Our analysis allows us to estimate the expected proportion of secondary infections attributable to the most infectious proportion of the population. For example, we estimate that the 20% most infectious index cases account for approximately 75%-98% of the expected secondary infections with 95% posterior probability. In addition, we highlight that heterogeneity is a vital consideration when estimating .

Moments and asymptotic properties for supercritical branching processes with immigration in random environments

We consider a supercritical discrete-time branching process with immigration Y in a stationary and ergodic environment ξ. Let mn be the mean of the reproduction distribution at time n conditioned on the environment ξ and be the natural submartingale of the model. We show sufficient conditions for the boundedness of the moments and for and discover the exponential Lp decay rates of as well as the rates of Then, as an application of the moment results, we show the exponential decay rates of and the convergence rates of the average of ratios

A branching process model of evolutionary rescue

Evolutionary rescue is the process whereby a declining population may start growing again, thus avoiding extinction, via an increase in the frequency of fitter genotypes. These genotypes may either already be present in the population in small numbers, or arise by mutation as the population declines. We present a simple two-type discrete-time branching process model and use it to obtain results such as the probability of rescue, the shape of the population growth curve of a rescued population, and the time until the first rescuing mutation occurs. Comparisons are made to existing results in the literature in cases where both the mutation rate and the selective advantage of the beneficial mutations are small.

Yaglom’s limit for critical Galton–Watson processes in varying environment: A probabilistic approach

A Galton–Watson process in varying environment is a discrete time branching process where the offspring distributions vary among generations. Based on a two-spine decomposition technique, we provide a probabilistic argument of a Yaglom-type limit for this family processes. The result states that, in the critical case, a suitable normalisation of the process conditioned on non-extinction converges in distribution to a standard exponential random variable.

Estimation of parameters in biological species with several mating and reproduction alternatives

With the purpose of modeling the demographic dynamics of biological species in which different mating and reproduction alternatives are feasible, in Molina et al. (2014) we introduced a new mathematical model based on discrete-time branching processes. Assuming that the reproduction phase is governed by probability distributions belonging to the power series family, some reproductive parameters for such species were estimated. In this work, in a more general statistical context, we generalize this research. By considering observations, until a given generation, of the number of female and male individuals in each generation, we now investigate several inferential questions about the parameters of biological interest included in the mathematical model. We study such questions using procedures based on Bayesian statistical methodology. We apply the proposed methods to describe the dynamics of salmonid species.

Distinguishing successive ancient polyploidy levels based on genome-internal syntenic alignment

BackgroundA basic tool for studying the polyploidization history of a genome, especially in plants, is the distribution of duplicate gene similarities in syntenically aligned regions of a genome. This distribution can usually be decomposed into two or more components identifiable by peaks, or local maxima, each representing a different polyploidization event. The distributions may be generated by means of a discrete time branching process, followed by a sequence divergence model. The branching process, as well as the inference of fractionation rates based on it, requires knowledge of the ploidy level of each event, which cannot be directly inferred from the pair similarity distribution.ResultsFor a sequence of two events of unknown ploidy, either tetraploid, giving rise to whole genome doubling (WGD), or hexaploid, giving rise to whole genome tripling (WGT), we base our analysis on triples of similar genes. We calculate the probability of the four triplet types with origins in one or the other event, or both, and impose a mutational model so that the distribution resembles the original data. Using a ML transition point in the similarities between the two events as a discriminator for the hypothesized origin of each similarity, we calculate the predicted number of triplets of each type for each model combining WGT and/or WGD. This yields a predicted profile of triplet types for each model. We compare the observed and predicted triplet profiles for each model to confirm the polyploidization history of durian, poplar and cabbage.ConclusionsWe have developed a way of inferring the ploidy of up to three successive WGD and/or WGT events by estimating the time of origin of each of the similarities in triples of genes. This may be generalized to a larger number of events and to higher ploidies.

Population-dependent two-sex branching processes with random mating: rates of growth

In Jacob et al. [A General Class of Population-Dependent Two-Sex Processes with Random Mating. Bernoulli 2017, 23, 1737–1758], a new class of two-sex branching processes in discrete time was introduced. These processes present the novelty that, in each generation, mating between females and males is randomly governed by a set of Bernoulli distributions allowing polygamous behavior with only perfect fidelity on the part of female individuals. Moreover, mating as well as reproduction can be influenced by the number of females and males in the population. In that article, the authors study conditions leading to the almost sure extinction or to the possible survival with positive probability of such processes. In this work, we continue the research about this class of two-sex processes by investigating the rate of growth of the population in case of survival. In particular, we provide conditions for a simultaneous geometric growth of the number of females and males and conditions for the geometric growth of females with a stable population of males.

An accurate approximation for the expected site frequency spectrum in a Galton–Watson process under an infinite sites mutation model

If viruses or other pathogens infect a single host, the outcome of infection often hinges on the fate of the initial invaders. The initial basic reproduction number R0, the expected number of cells infected by a single infected cell, helps determine whether the initial viruses can establish a successful beachhead. To determine R0, the Kingman coalescent or continuous-time birth-and-death process can be used to infer the rate of exponential growth in an historical population. Given M sequences sampled in the present, the two models can make the inference from the site frequency spectrum (SFS), the count of mutations that appear in exactly k sequences (k=1,2,…,M). In the case of viruses, however, if R0 is large and an infected cell bursts while propagating virus, the two models are suspect, because they are Markovian with only binary branching. Accordingly, this article develops an approximation for the SFS of a discrete-time branching process with synchronous generations (i.e., a Galton–Watson process). When evaluated in simulations with an asynchronous, non-Markovian model (a Bellman–Harris process) with parameters intended to mimic the bursting viral reproduction of HIV, the approximation proved superior to approximations derived from the Kingman coalescent or continuous-time birth-and-death process. This article demonstrates that in analogy to methods in human genetics, the SFS of viral sequences sampled well after latent infection can remain informative about the initial R0. Thus, it suggests the utility of analyzing the SFS of sequences derived from patient and animal trials of viral therapies, because in some cases, the initial R0 may be able to indicate subtle therapeutic progress, even in the absence of statistically significant differences in the infection of treatment and control groups.

The genealogy of an exactly solvable Ornstein–Uhlenbeck type branching process with selection

We study the genealogy of a solvable population model with $N$ particles on the real line which evolves according to a discrete-time branching process with selection. At each time step, every particle gives birth to children around $a$ times its current position, where $a>0$ is a parameter of the model. Then, the $N$ rightmost new-born children are selected to form the next generation. We show that the genealogical trees of the process converge to those of a Beta coalescent as $N \to \infty$. The process we consider can be seen as a toy-model version of a continuous-time branching process with selection, in which particles move according to independent Ornstein-Uhlenbeck processes. The parameter $a$ is akin to the pulling strength of the Ornstein-Uhlenbeck motion.

A general class of population-dependent two-sex processes with random mating

We introduce a class of two-sex branching processes in discrete time where, in each generation, mating between females and males is randomly governed by a set of Bernoulli distributions allowing polygamous behaviour with only perfect fidelity on the part of female. Moreover, mating as well as reproduction can be influenced by the number of females and males in the population. We study here, for any population whose dynamics is modeled by such processes, conditions leading to its extinction or to a possible persistence. Moreover, the behaviours of the female and male populations are analyzed more finely in case of persistence.

A Branching Process for Convergent Product Optimization

<p><em>We consider a discrete time branching process where the population consists of k types of convergent products, an action is chosen for that which affects the lifetime, the number and types of its functions, and the profit received. The problem of maximizing the expected profit is shown to be equivalent to a generalized Markov decision problem of maximizing the expected profit is shown to be equivalent to a generalized Markov decision problem where the </em><em> </em><em>transition matrices are non-negative but not necessarily sub stochastic.</em></p>

Abstract 3939: Evolution and drug resistance of primary and relapse tumors in diffuse large B-cell lymphoma

Abstract Our integrated multi-omics analysis of paired primary and relapse DLBCL tumors provides comprehensive view to tumor evolution, drug resistance and disease heterogeneity. DLBCL is an aggressive systemic disease in which tumors can arise in virtually any part of the body. Activated B-cell (ABC) and germinal center B-cell (GCB)-like subtypes differ molecularly and clinically. Three-year progression-free survival (PFS) in ABC is around 40% vs over 80% in GCB. Whole genome (WGS, 7 patients), transcriptome (total RNA, 7 patients) and methylome sequencing data produced using the Illumina HiSeq platform was complemented with Illumina TruSeq amplicon sequencing covering 102,571 bp of the genome on 788 custom selected targets. Altogether 23 fresh frozen and 28 paraffin embedded tumor samples from 24 patients with an advanced disease and at least one relapse event were included in the study. To investigate the clonal hierarchy of the primary and relapsed tumors we employed dirichlet process clustering using cancer cell fraction (CCF) estimates or variant allele frequencies from somatic point mutations. Divergent evolution was prominent in the GCB subtype whereas in ABC subtype subclonal mutations are rare and only few genomic differences were detected between primary and relapse tumors (p-value=0.047, Pearson's Chi-squared test on 7 patients with WGS data) suggesting pre-treatment drug resistance which is probably triggered by global hypo- or hypermethylation at relapse stage. In the GCB subtype primary and relapse branches have diverged much before diagnosis and subsequently developed independently. Number of emerging primary or relapse specific clonal and subclonal mutations correlate with PSF and overall survival (OS). Mutational signatures were concordant with the inferred evolutionary models in four out of five GCB cases. Contribution of the somatic hypermutation signature was higher in resistant tumors and correlated with time to progression of the disease. We identified 72 relapse tumor related genes including many known B-cell lymphoma genes such as CD79B, IGLL5, PIM1, SGK1. These were used in survival analysis in the CGCI RNA data set derived from 92 diagnostic DLBCL tumors. Patients with subclonal mutations in these genes had shorter PSF and OS (Kaplan-Meier p-values 0.0122 and 0.0071, respectively). To validate and further define our hypothesis we are simulating DLBCL evolution as a discrete time branching process model (Galton-Watson branching processes). The goal is to reconstruct phylogenetic trees from a simulated tumor and explain different types of evolutionary models inferred from the patient data. Despite of remarkable heterogeneity in DLBCL we observed two main evolutionary models which are concordant with disease subtype, patient’s survival, treatment response at diagnosis and relapse, and molecular features of a tumor. Citation Format: Rainer Lehtonen, Amjad Alkodsi, Kaiyang Zhang, Mikko Kivikoski, Emilia Kozlowska, Alejandra Cervera, Annika Pasanen, Harald Holte, Sirpa Leppä, Sampsa Hautaniemi. Evolution and drug resistance of primary and relapse tumors in diffuse large B-cell lymphoma [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2017; 2017 Apr 1-5; Washington, DC. Philadelphia (PA): AACR; Cancer Res 2017;77(13 Suppl):Abstract nr 3939. doi:10.1158/1538-7445.AM2017-3939

Statistical inference for partially observed branching processes with immigration

AbstractIn the paper we consider the following modification of a discrete-time branching process with stationary immigration. In each generation a binomially distributed subset of the population will be observed. The number of observed individuals constitute a partially observed branching process. After inspection both observed and unobserved individuals may change their offspring distributions. In the subcritical case we investigate the possibility of using the known estimators for the offspring mean and for the mean of the stationary-limiting distribution of the process when the observation of the population sizes is restricted. We prove that, if both the population and the number of immigrants are partially observed, the estimators are still strongly consistent. We also prove that the `skipped' version of the estimator for the offspring mean is asymptotically normal and the estimator of the stationary distribution's mean is asymptotically normal under additional assumptions.

The renewed limit theorems for the discrete-time branching process and its conditioned limiting law interpretation

The renewed limit theorems for the discrete-time branching process and its conditioned limiting law interpretation

A Galton–Watson process with a threshold

Abstract In this paper we study a special class of size dependent branching processes. We assume that for some positive integer K as long as the population size does not exceed level K, the process evolves as a discrete-time supercritical branching process, and when the population size exceeds level K, it evolves as a subcritical or critical branching process. It is shown that this process does die out in finite time T. The question of when the mean value E(T) is finite or infinite is also addressed.

An inhomogeneous controlled branching process∗

We consider a discrete-time branching process in which the offspring distribution is generation-dependent and the number of reproductive individuals is controlled by a random mechanism. This model is a Markov chain, but, in general, the transition probabilities are nonstationary. Under not too restrictive hypotheses, this model presents the classical duality of branching processes: it either becomes extinct or grows to infinity. Sufficient conditions for the almost sure extinction and for a positive probability of indefinite growth are given. Finally, the rates of growth of the process are studied, provided that there is no extinction.

A Discrete-Time Branching Process Model of Yeast Prion Curing Curves*

The infectious agent of many neurodegenerative disorders is thought to be aggregates of prion protein, which are transmitted between cells. Recent work in yeast supports this hypothesis but also suggests that only aggregates below a critical size are transmitted efficiently. The total number of transmissible aggregates in a typical cell is a key determinant of strain infectivity. In a discrete-time branching process model of a yeast colony with prions, prion aggregates increase in size according to a Poisson process and only aggregates below a threshold size are transmitted during cell division. The total number of cells with aggregates in a growing population of yeast is expressed.

Stochastic Methodology for the Study of an Epidemic Decay Phase, Based on a Branching Model

We present a stochastic methodology to study the decay phase of an epidemic. It is based on a general stochastic epidemic process with memory, suitable to model the spread in a large open population with births of any rare transmissible disease with a random incubation period and a Reed-Frost type infection. This model, which belongs to the class of multitype branching processes in discrete time, enables us to predict the incidences of cases and to derive the probability distributions of the extinction time and of the future epidemic size. We also study the epidemic evolution in the worst-case scenario of a very late extinction time, making use of the Q-process. We provide in addition an estimator of the key parameter of the epidemic model quantifying the infection and finally illustrate this methodology with the study of the Bovine Spongiform Encephalopathy epidemic in Great Britain after the 1988 feed ban law.

A new approach on discrete time and period dependent branching process

In this paper, we consider discrete time branching process with overlapping generations. Distribution of the number of offspring depends on the period that parents particles may generate their offspring in next two or more periods of generation. The introduced model in this paper differs with Galton-Watson branching process at age of particles where live more than one unit of time. In this model, particles also allow to have generation for several times during their life. Finally, we obtain the mean and variance of the number of patients in period n and estimate the population mean by the Monte Carlo simulation.