scholarly journals Integrability of stochastic birth-death processes via differential Galois theory

2020 ◽  
Vol 15 ◽  
pp. 70
Author(s):  
Primitivo B. Acosta-Humánez ◽  
José A. Capitán ◽  
Juan J. Morales-Ruiz
Keyword(s):  
Master Equation ◽  
Continuous Time ◽  
Galois Theory ◽  
Linear Functions ◽  
System State ◽  
Differential Galois Theory ◽  
Death Rates ◽  
Number Of Individuals ◽  
Temporal Variable ◽  
Birth Death

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 of the PDE via a Laplace transform acting over the temporal variable. We show that the PDE is not integrable except for the case in which rates are linear functions of the number of individuals.

A linear model of population dynamics

2016 ◽  
Vol 30 (15) ◽  
pp. 1541008 ◽  
Author(s):  
A. A. Lushnikov ◽  
A. I. Kagan
Keyword(s):  
Population Dynamics ◽  
Asymptotic Analysis ◽  
Population Growth ◽  
Master Equation ◽  
Bessel Function ◽  
Population Size ◽  
Linear Model ◽  
Linear Functions ◽  
Modified Bessel Function ◽  
Death Rates

The Malthus process of population growth is reformulated in terms of the probability [Formula: see text] to find exactly [Formula: see text] individuals at time [Formula: see text] assuming that both the birth and the death rates are linear functions of the population size. The master equation for [Formula: see text] is solved exactly. It is shown that [Formula: see text] strongly deviates from the Poisson distribution and is expressed in terms either of Laguerre’s polynomials or a modified Bessel function. The latter expression allows for considerable simplifications of the asymptotic analysis of [Formula: see text].

scholarly journals A Numerical Approach for Evaluating the Time-Dependent Distribution of a Quasi Birth-Death Process

2021 ◽  
Author(s):  
Michel Mandjes ◽  
Birgit Sollie
Keyword(s):  
Continuous Time ◽  
Real Life ◽  
Numerical Approach ◽  
Time Dependent ◽  
Death Process ◽  
Continuous Time Markov Chain ◽  
Likelihood Estimator ◽  
Real Life Data ◽  
The Matrix ◽  
Birth Death

AbstractThis paper considers a continuous-time quasi birth-death (qbd) process, which informally can be seen as a birth-death process of which the parameters are modulated by an external continuous-time Markov chain. The aim is to numerically approximate the time-dependent distribution of the resulting bivariate Markov process in an accurate and efficient way. An approach based on the Erlangization principle is proposed and formally justified. Its performance is investigated and compared with two existing approaches: one based on numerical evaluation of the matrix exponential underlying the qbd process, and one based on the uniformization technique. It is shown that in many settings the approach based on Erlangization is faster than the other approaches, while still being highly accurate. In the last part of the paper, we demonstrate the use of the developed technique in the context of the evaluation of the likelihood pertaining to a time series, which can then be optimized over its parameters to obtain the maximum likelihood estimator. More specifically, through a series of examples with simulated and real-life data, we show how it can be deployed in model selection problems that involve the choice between a qbd and its non-modulated counterpart.

Geometric Markov chains

1995 ◽  
Vol 32 (02) ◽  
pp. 349-374
Author(s):  
William Rising
Keyword(s):  
Markov Chains ◽  
Stationary Distribution ◽  
Nearest Neighbor ◽  
First Passage Times ◽  
First Passage ◽  
Death Rates ◽  
Passage Times ◽  
Birth Death

A generalization of the familiar birth–death chain, called the geometric chain, is introduced and explored. By the introduction of two families of parameters in addition to the infinitesimal birth and death rates, the geometric chain allows transitions beyond the nearest neighbor, but is shown to retain the simple computational formulas of the birth–death chain for the stationary distribution and the expected first-passage times between states. It is also demonstrated that even when not reversible, a reversed geometric chain is again a geometric chain.

scholarly journals On mutations in the branching model for multitype populations

2018 ◽  
Vol 50 (2) ◽  
pp. 543-564 ◽  
Author(s):  
Loïc Chaumont ◽  
Thi Ngoc Anh Nguyen
Keyword(s):  
Infectious Diseases ◽  
Asymptotic Behaviour ◽  
Continuous Time ◽  
Mutation Rates ◽  
Genetic Evolution ◽  
Connected Component ◽  
Branching Model ◽  
Limiting Behaviour ◽  
Number Of Individuals

AbstractThe forest of mutations associated to a multitype branching forest is obtained by merging together all vertices in each of its clusters and by preserving connections between them. (Here, by cluster, we mean a maximal connected component of the forest in which all vertices have the same type.) We first show that the forest of mutations of any multitype branching forest is itself a branching forest. Then we give its progeny distribution and we describe some of its crucial properties in terms of the initial progeny distribution. We also obtain the limiting behaviour of the number of mutations both when the total number of individuals tends to ∞ and when the number of roots tends to ∞. The continuous-time case is then investigated by considering multitype branching forests with edge lengths. When mutations are nonreversible, we give a representation of their emergence times which allows us to describe the asymptotic behaviour of the latter, under certain conditions on the mutation rates. These results have potential relevance for emergence of mutations in population cells, particularly for genetic evolution of cancer or development of infectious diseases.

The Relaxation time for truncated birth-death processes

1987 ◽  
Vol 1 (4) ◽  
pp. 367-381 ◽  
Author(s):  
Julian Keilson ◽  
Ravi Ramaswamy
Keyword(s):  
Relaxation Time ◽  
Exit Time ◽  
Relaxation Times ◽  
Dual Process ◽  
Death Process ◽  
Exit Times ◽  
Initial State ◽  
Death Rates ◽  
Absorbing States ◽  
Birth Death

The relaxation time for an ergodic Markov process is a measure of the time until ergodicity is reached from its initial state. In this paper the relaxation time for an ergodic truncated birth-death process is studied. It is shown that the relaxation time for such a process on states {0,1, …, N} is the quasi-stationary exit time from the set {,2, …, N{0,1,…, N, N + 1} with two-sided absorption at states 0 and N + 1. The existence of such a dual process has been observed by Siegmund [15] for stochastically monotone Markov processes on the real line. Exit times for birth- death processes with two absorbing states are studied and an efficient algorithm for the numerical evaluation of mean exit times is presented. Simple analytical lower bounds for the relaxation times are obtained. These bounds are numerically accessible. Finally, the sensitivity of the relaxation time to variations in birth and death rates is studied.

scholarly journals Differential Galois Theory and Lie Symmetries

2015 ◽  
Author(s):  
David Blázquez-Sanz ◽  
◽  
Juan J. Morales-Ruiz ◽  
Jacques-Arthur Weil ◽  
◽  
...  
Keyword(s):  
Galois Theory ◽  
Lie Symmetries ◽  
Differential Galois Theory
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