General Birth Death Research Articles

On the Generalized Birth–Death Process and Its Linear Versions

On the Generalized Birth–Death Process and Its Linear Versions

Birth–Death Processes with Two-Type Catastrophes

This paper concentrates on the general birth–death processes with two different types of catastrophes. The Laplace transform of transition probability function for birth–death processes with two-type catastrophes is successfully expressed with the Laplace transform of transition probability function of the birth–death processes without catastrophe. The first effective catastrophe occurrence time is considered. The Laplace transform of its probability density function, expectation and variance are obtained.

Exponential ergodicity and steady-state approximations for a class of markov processes under fast regime switching

AbstractWe study ergodic properties of a class of Markov-modulated general birth–death processes under fast regime switching. The first set of results concerns the ergodic properties of the properly scaled joint Markov process with a parameter that is taken to be large. Under very weak hypotheses, we show that if the averaged process is exponentially ergodic for large values of the parameter, then the same applies to the original joint Markov process. The second set of results concerns steady-state diffusion approximations, under the assumption that the ‘averaged’ fluid limit exists. Here, we establish convergence rates for the moments of the approximating diffusion process to those of the Markov-modulated birth–death process. This is accomplished by comparing the generator of the approximating diffusion and that of the joint Markov process. We also provide several examples which demonstrate how the theory can be applied.

Extant timetrees are consistent with a myriad of diversification histories.

Time-calibrated phylogenies of extant species (referred to here as 'extant timetrees') are widely used for estimating diversification dynamics1. However, there has been considerable debate surrounding the reliability of these inferences2-5 and, to date, this critical question remains unresolved. Here we clarify the precise information that can be extracted from extant timetrees under the generalized birth-death model, which underlies most existing methods of estimation. We prove that, for any diversification scenario, there exists an infinite number of alternative diversification scenarios that are equally likely to have generated any given extant timetree. These 'congruent' scenarios cannot possibly be distinguished using extant timetrees alone, even in the presence of infinite data. Importantly, congruent diversification scenarios can exhibit markedly different and yet similarly plausible dynamics, which suggests that many previous studies may have over-interpreted phylogenetic evidence. We introduce identifiable and easily interpretable variables that contain all available information about past diversification dynamics, and demonstrate that these can be estimated from extant timetrees. We suggest that measuring and modelling these identifiable variables offers a more robust way to study historical diversification dynamics. Our findings also make it clear that palaeontological data will continue to be crucial for answering some macroevolutionary questions.

How successful are mutants in multiplayer games with fluctuating environments? Sojourn times, fixation and optimal switching.

Using a stochastic model, we investigate the probability of fixation, and the average time taken to achieve fixation, of a mutant in a population of wild-types. We do this in a context where the environment in which the competition takes place is subject to stochastic change. Our model takes into account interactions which can involve multiple participants. That is, the participants take part in multiplayer games. We find that under certain circumstances, there are environmental switching dynamics which minimize the time that it takes for the mutants to fixate. To analyse the dynamics more closely, we develop a method by which to calculate the sojourn times for general birth–death processes in fluctuating environments.

Exact, time-independent estimation of clone size distributions in normal and mutated cells.

Biological tools such as genetic lineage tracing, three-dimensional confocal microscopy and next-generation DNA sequencing are providing new ways to quantify the distribution of clones of normal and mutated cells. Understanding population-wide clone size distributions in vivo is complicated by multiple cell types within observed tissues, and overlapping birth and death processes. This has led to the increased need for mathematically informed models to understand their biological significance. Standard approaches usually require knowledge of clonal age. We show that modelling on clone size independent of time is an alternative method that offers certain analytical advantages; it can help parametrize these models, and obtain distributions for counts of mutated or proliferating cells, for example. When applied to a general birth–death process common in epithelial progenitors, this takes the form of a gambler's ruin problem, the solution of which relates to counting Motzkin lattice paths. Applying this approach to mutational processes, alternative, exact, formulations of classic Luria–Delbrück-type problems emerge. This approach can be extended beyond neutral models of mutant clonal evolution. Applications of these approaches are twofold. First, we resolve the probability of progenitor cells generating proliferating or differentiating progeny in clonal lineage tracing experiments in vivo or cell culture assays where clone age is not known. Second, we model mutation frequency distributions that deep sequencing of subclonal samples produce.

Estimation for General Birth-Death Processes

Birth-death processes (BDPs) are continuous-time Markov chains that track the number of “particles” in a system over time. While widely used in population biology, genetics, and ecology, statistical inference of the instantaneous particle birth and death rates remains largely limited to restrictive linear BDPs in which per-particle birth and death rates are constant. Researchers often observe the number of particles at discrete times, necessitating data augmentation procedures such as expectation-maximization (EM) to find maximum likelihood estimates (MLEs). For BDPs on finite state-spaces, there are powerful matrix methods for computing the conditional expectations needed for the E-step of the EM algorithm. For BDPs on infinite state-spaces, closed-form solutions for the E-step are available for some linear models, but most previous work has resorted to time-consuming simulation. Remarkably, we show that the E-step conditional expectations can be expressed as convolutions of computable transition probabilities for any general BDP with arbitrary rates. This important observation, along with a convenient continued fraction representation of the Laplace transforms of the transition probabilities, allows for novel and efficient computation of the conditional expectations for all BDPs, eliminating the need for truncation of the state-space or costly simulation. We use this insight to derive EM algorithms that yield maximum likelihood estimation for general BDPs characterized by various rate models, including generalized linear models (GLM). We show that our Laplace convolution technique outperforms competing methods when they are available and demonstrate a technique to accelerate EM algorithm convergence. We validate our approach using synthetic data and then apply our methods to cancer cell growth and estimation of mutation parameters in microsatellite evolution.

A generalized birth–death stochastic model for high-frequency order book dynamics

We use a generalized birth–death stochastic process to model the high-frequency dynamics of the limit order book, and illustrate it using parameters estimated from Level II data for a stock on the London Stock Exchange. A new feature of this model is that limit orders are allowed to arrive in multiple sizes, an important empirical feature of the order book. We can compute various quantities of interest without resorting to simulation, conditional on the state of the order book, such as the probability that the next move of the mid-price will be upward, or the probability, as a function of order size, that a limit ask order will be executed before a downward move in the mid-price. This generalizes the successful model of Cont et al. [Oper. Res., 2010, 58, 549–563] by means of a new technical approach to computing the distribution of first passage times.

Transition probabilities for general birth–death processes with applications in ecology, genetics, and evolution

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 explicit an important derivation connecting transition probabilities and continued fractions. We then develop an efficient algorithm for computing these probabilities that analyzes the error associated with approximations in the method. We demonstrate that this error-controlled method agrees with known solutions and outperforms previous approaches to computing these probabilities. Finally, we apply our novel method to several important problems in ecology, evolution, and genetics.

Moments of first passage times in general birth–death processes

We consider ordinary and conditional first passage times in a general birth–death process. Under existence conditions, we derive closed-form expressions for the kth order moment of the defined random variables, k ≥ 1. We also give an explicit condition for a birth–death process to be ergodic degree 3. Based on the obtained results, we analyze some applications for Markovian queueing systems. In particular, we compute for some non-standard Markovian queues, the moments of the busy period duration, the busy cycle duration, and the state-dependent waiting time in queue.

A theory on the origin of cooperativity in DNA renaturation kinetics

Theoretical proof for the existence of cooperativity (experimental evidences have already been reported in our earlier work: Biochem. Biophys. Res. Commun., 293 (2002) 870) in the phenomenon of DNA renaturation has been obtained using a generalized birth–death master equation approach. Results have shown that no special searching mechanism (i.e. a facilitated mechanism) was necessary for complementary DNA strands to find their correct-contacts but searching by a ‘random jump’ method was enough to explain not only the observed cooperativity phenomenon but also the magnitude of correct-contact-forming rate constant. Impact of cooperativity phenomenon on length of primers and their specificity in polymerase chain reaction has also been discussed.

The combinatorics of birth–death processes and applications to queues

In this paper we present a combinatorial technique which allows the derivation of the transition functions of general birth–death processes. This method provides a flexible tool for the transient analysis of Markovian queueing systems with state dependent transition rates, like M/M/c models or systems with balking and reneging.

Traffic overflow in closed queueing models

We consider a generalized birth death process that represents a multiserver (closed or open) overflow queueing system with n primary servers and c-n secondary servers. An arrival to the system joins a server of the primary group, if available, otherwise it overflows to the secondary group. If all servers are busy, arrivals are queued, provided the queue buffer is not full, and served as servers become free. Arrivals that find the queue buffer full are lost. Our overflow model differs from models in the open literature in that it combines state dependent arrival rates with group dependent service rates. We present a general formulation that allows a simple derivation of the joint stationary probabilities of i busy primary servers and j busy secondary severs. This main result easily lends itself to an efficient iterative algorithm to evaluate the joint probabilities. We apply our basic theorem to produce new results for several overflow models. A distinctive feature of our approach is that it uses transform-free analysis.

Dependencies in Markovian networks

Monotonicity and correlation results for queueing network processes, generalized birth–death processes and generalized migration processes are obtained with respect to various orderings of the state space. We prove positive (e.g. association) and negative (e.g. negative association) correlations in space and positive correlations in time for different situations, in steady state as well as in the transient phase of the system. This yields exact bounds for joint probabilities in terms of their independent versions.

Dependencies in Markovian networks

Monotonicity and correlation results for queueing network processes, generalized birth–death processes and generalized migration processes are obtained with respect to various orderings of the state space. We prove positive (e.g. association) and negative (e.g. negative association) correlations in space and positive correlations in time for different situations, in steady state as well as in the transient phase of the system. This yields exact bounds for joint probabilities in terms of their independent versions.

Numerical Evaluation of Continued Fractions

Numerical Evaluation of Continued Fractions

Further Models for Phage Reproduction in a Bacterium

Several models for phage reproduction have recently been presented by Gani [1962, 1963]. In this paper we consider two similar, but slightly more general models which can be made to fit some of the available experimental data. Virus, by Weidel [1958], gives a more complete description of the following biological process with which we are concerned. Outside the bacterium, a phage exists as a core enclosed in a protein coat. Although more than one phage may attack the bacterium, only the case of a single infection is considered here. When the phage penetrates the bacterium the coat remains outside and only the core enters the cell, where it is replicated in great numbers. Since the core is actually a strand of DNA, the reproduction is thought to occur by means of a template mechanism. For our purpose this process can be considered to be birth by division. The protein coats required by mature phages are produced simultaneously with the DNA cores. Since a phage particle can no longer reproduce after being coated, this event corresponds to a death. Reproduction continues until the cell bursts releasing both infectious phages and uncoated particles which are incapable of attacking further bacteria. In section 2 we consider the intuitively appropriate model, a generalized birth-death process. This case unfortunately proves to be somewhat unmanageable. Accordingly, section 3 deals with a second process designed to approximate the first; in this case all quantities of interest are easily obtained. In the final section we discuss eclipse and show how both processes can be modified to account for this phenomenon. 2. APPLICATION OF THE GENERALIZED BIRTH-DEATH PROCESS