General Birth-death Processes Research Articles

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.

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.

Computational methods for birth-death processes.

Many important stochastic counting models can be written as general birth-death processes (BDPs). BDPs are continuous-time Markov chains on the non-negative integers in which only jumps to adjacent states are allowed. BDPs can be used to easily parameterize a rich variety of probability distributions on the non-negative integers, and straightforward conditions guarantee that these distributions are proper. BDPs also provide a mechanistic interpretation - birth and death of actual particles or organisms - that has proven useful in evolution, ecology, physics, and chemistry. Although the theoretical properties of general BDPs are well understood, traditionally statistical work on BDPs has been limited to the simple linear (Kendall) process. Aside from a few simple cases, it remains impossible to find analytic expressions for the likelihood of a discretely-observed BDP, and computational difficulties have hindered development of tools for statistical inference. But the gap between BDP theory and practical methods for estimation has narrowed in recent years. There are now robust methods for evaluating likelihoods for realizations of BDPs: finite-time transition, first passage, equilibrium probabilities, and distributions of summary statistics that arise commonly in applications. Recent work has also exploited the connection between continuously- and discretely-observed BDPs to derive EM algorithms for maximum likelihood estimation. Likelihood-based inference for previously intractable BDPs is much easier than previously thought and regression approaches analogous to Poisson regression are straightforward to derive. In this review, we outline the basic mathematical theory for BDPs and demonstrate new tools for statistical inference using data from BDPs.

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.

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.

Regularity and reversibility results for birth-death-migration processes

A spatial process is considered in which two general birth-death processes are linked by migration of individuals. We examine conditions for weak symmetry and regularity, and develop necessary and sufficient conditions for recurrence. The results are easily extended to the k-process case.

Regularity and reversibility results for birth-death-migration processes

A spatial process is considered in which two general birth-death processes are linked by migration of individuals. We examine conditions for weak symmetry and regularity, and develop necessary and sufficient conditions for recurrence. The results are easily extended to the k-process case.

Relative extreme values and extremum-times of jump process: Upward type

Relative extreme values are defined by the supremum and minimum of a general jump process before its first time quitting from some state set, and relative extremum-times are defined by the first times reaching relative extreme values. The main objective of this paper is to find out the exact distributions and moments of them as the maximum of the set is up or equal to the process initial state. As especial cases, these results are applied to a general birth-death process and generalized birth-death processes.

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.

Path integral approach to birth-death processes on a lattice

The Fock space formalism for classical objects first introduced by Doi is cast in a path integral form and applied to general birth-death processes on a lattice. The introduction of suitable auxiliary variables allows one to formulate random walks with memory and irreversible aggregation processes in a Markovian way, which is treatable in this formalism. Existing field theories of such processes are recovered in the continuum limit. Implications of the method for their asymptotic behaviour are briefly discussed.

Joint distributions of random variables and their integrals for certain birth-death and diffusion processes

For the linear growth birth-death process with parameters λn = nλ, μn = nμ, Puri ((1966), (1968)) has investigated the joint distribution of the number X(t) of survivors in the process and the associated integral Y(t) = ∫0tX(τ)dτ. In particular, he has obtained limiting results as t → ∞. Recently one of us (McNeil (1970)) has derived the distribution of the integral functional Wx = ∫0Txg{X(τ)}dτ, where Tx is the first passage time to the origin in a general birth-death process with X(0) = x and g(·) is an arbitrary function. Functionals of the form Wx arise naturally in traffic and storage theory; for example Wx may represent the total cost of a traffic jam, or the cost of storing a commodity until expiration of the stock. Moments of such functionals were found in the case of M/G/1 and GI/M/1 queues by Gaver (1969) and Daley (1969).

Joint distributions of random variables and their integrals for certain birth-death and diffusion processes

For the linear growth birth-death process with parameters λ n = nλ, μ n = nμ, Puri ((1966), (1968)) has investigated the joint distribution of the number X(t) of survivors in the process and the associated integral Y(t) = ∫0 t X(τ)dτ. In particular, he has obtained limiting results as t → ∞. Recently one of us (McNeil (1970)) has derived the distribution of the integral functional W x = ∫0 Tx g{X(τ)}dτ, where T x is the first passage time to the origin in a general birth-death process with X(0) = x and g(·) is an arbitrary function. Functionals of the form W x arise naturally in traffic and storage theory; for example W x may represent the total cost of a traffic jam, or the cost of storing a commodity until expiration of the stock. Moments of such functionals were found in the case of M/G/1 and GI/M/1 queues by Gaver (1969) and Daley (1969).

Numerical Evaluation of Continued Fractions

Numerical Evaluation of Continued Fractions