The accuracy of the diffusion approximation to the expected time to extinction for some discrete stochastic processes

1983 ◽  
Vol 20 (2) ◽  
pp. 305-321 ◽  
Author(s):  
J. Grasman ◽  
D. Ludwig
Keyword(s):  
Stochastic Processes ◽  
Diffusion Approximation ◽  
Transition Matrix ◽  
Branching Process ◽  
Substantial Improvement ◽  
Death Process ◽  
Asymptotic Approximations ◽  
Birth And Death Process ◽  
Expected Time ◽  
Time To Extinction

Asymptotic approximations and numerical computations are used to estimate the accuracy of the diffusion approximation for the expected time to extinction for some stochastic processes. The results differ for processes with a continuant transition matrix (e.g. a birth and death process), and those with a noncontinuant transition matrix (e.g. a non-linear branching process). In the latter case, the diffusion equation does not hold near the point of exit. Consequently, high-order corrections do not result in substantial improvement over the diffusion approximation.

The accuracy of the diffusion approximation to the expected time to extinction for some discrete stochastic processes

1983 ◽  
Vol 20 (02) ◽  
pp. 305-321
Author(s):  
J. Grasman ◽  
D. Ludwig
Keyword(s):  
Stochastic Processes ◽  
Diffusion Approximation ◽  
Transition Matrix ◽  
Branching Process ◽  
Substantial Improvement ◽  
Death Process ◽  
Asymptotic Approximations ◽  
Birth And Death Process ◽  
Expected Time ◽  
Time To Extinction

Asymptotic approximations and numerical computations are used to estimate the accuracy of the diffusion approximation for the expected time to extinction for some stochastic processes. The results differ for processes with a continuant transition matrix (e.g. a birth and death process), and those with a noncontinuant transition matrix (e.g. a non-linear branching process). In the latter case, the diffusion equation does not hold near the point of exit. Consequently, high-order corrections do not result in substantial improvement over the diffusion approximation.

The extinction time of a birth, death and catastrophe process and of a related diffusion model

1985 ◽  
Vol 17 (01) ◽  
pp. 42-52 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Population Size ◽  
Diffusion Processes ◽  
Death Process ◽  
Arbitrary Distribution ◽  
Initial Population ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Expected Time ◽  
Time To Extinction ◽  
Initial Population Size

The distribution of the extinction time for a linear birth and death process subject to catastrophes is determined. The catastrophes occur at a rate proportional to the population size and their magnitudes are random variables having an arbitrary distribution with generating function d(·). The asymptotic behaviour (for large initial population size) of the expected time to extinction is found under the assumption that d(.) has radius of convergence greater than 1. Corresponding results are derived for a related class of diffusion processes interrupted by catastrophes with sizes having an arbitrary distribution function.

The extinction time of a birth, death and catastrophe process and of a related diffusion model

1985 ◽  
Vol 17 (1) ◽  
pp. 42-52 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Population Size ◽  
Diffusion Processes ◽  
Death Process ◽  
Arbitrary Distribution ◽  
Initial Population ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Expected Time ◽  
Time To Extinction ◽  
Initial Population Size

The distribution of the extinction time for a linear birth and death process subject to catastrophes is determined. The catastrophes occur at a rate proportional to the population size and their magnitudes are random variables having an arbitrary distribution with generating function d(·). The asymptotic behaviour (for large initial population size) of the expected time to extinction is found under the assumption that d(.) has radius of convergence greater than 1. Corresponding results are derived for a related class of diffusion processes interrupted by catastrophes with sizes having an arbitrary distribution function.

Multivariate birth-and-death processes as approximations to epidemic processes

1973 ◽  
Vol 10 (1) ◽  
pp. 15-26 ◽  
Author(s):  
D. A. Griffiths
Keyword(s):  
Transient Process ◽  
Branching Process ◽  
Large Population ◽  
Disease Spread ◽  
Death Process ◽  
Birth And Death Process ◽  
Birth And Death Processes ◽  
Epidemic Size

This paper presents the theory of a multivariate birth-and-death process and its representation as a branching process. The bivariate linear birth-and-death process may be used as a model for various epidemic situations involving two types of infective. Various properties of the transient process are discussed and the distribution of epidemic size is investigated. For the case of a disease spread solely by carriers when the two types of infective are carriers and clinical infectives the large population version of a model proposed by Downton (1968) is further developed and shown under appropriate circumstances to closely approximate Downton's model.

Extinction times for a general birth, death and catastrophe process

2004 ◽  
Vol 41 (4) ◽  
pp. 1211-1218 ◽  
Author(s):  
Ben Cairns ◽  
P. K. Pollett
Keyword(s):  
Population Size ◽  
Rate Coefficients ◽  
Death Process ◽  
Transition Rates ◽  
Linear Rate ◽  
Expected Time ◽  
Current Population ◽  
Time To Extinction ◽  
Limited Class ◽  
Birth Death

The birth, death and catastrophe process is an extension of the birth–death process that incorporates the possibility of reductions in population of arbitrary size. We will consider a general form of this model in which the transition rates are allowed to depend on the current population size in an arbitrary manner. The linear case, where the transition rates are proportional to current population size, has been studied extensively. In particular, extinction probabilities, the expected time to extinction, and the distribution of the population size conditional on nonextinction (the quasi-stationary distribution) have all been evaluated explicitly. However, whilst these characteristics are of interest in the modelling and management of populations, processes with linear rate coefficients represent only a very limited class of models. We address this limitation by allowing for a wider range of catastrophic events. Despite this generalisation, explicit expressions can still be found for the expected extinction times.

A linear birth and death process under the influence of another process

1975 ◽  
Vol 12 (1) ◽  
pp. 1-17 ◽  
Author(s):  
Prem S. Puri
Keyword(s):  
Markov Process ◽  
Closed Form ◽  
Limit Distribution ◽  
Explicit Solution ◽  
Negative Integer ◽  
Death Process ◽  
Birth And Death Process ◽  
Death Rates ◽  
Time To Extinction

Let {X1 (t), X2 (t), t ≧ 0} be a bivariate birth and death (Markov) process taking non-negative integer values, such that the process {X2(t), t ≧ 0} may influence the growth of the process {X1(t), t ≧ 0}, while the process X2 (·) itself grows without any influence whatsoever of the first process. The process X2 (·) is taken to be a simple linear birth and death process with λ2 and µ2 as its birth and death rates respectively. The process X1 (·) is also assumed to be a linear birth and death process but with its birth and death rates depending on X2 (·) in the following manner: λ (t) = λ1 (θ + X2 (t)); µ(t) = µ1 (θ + X2 (t)). Here λ i, µi and θ are all non-negative constants. By studying the process X1 (·), first conditionally given a realization of the process {X2 (t), t ≧ 0} and then by unconditioning it later on by taking expectation over the process {X2 (t), t ≧ 0} we obtain explicit solution for G in closed form. Again, it is shown that a proper limit distribution of X1 (t) always exists as t→∞, except only when both λ1 > µ1 and λ2 > µ2. Also, certain problems concerning moments of the process, regression of X1 (t) on X2 (t); time to extinction, and the duration of the interaction between the two processes, etc., are studied in some detail.

Bounds on the extinction time distribution of a branching process

1974 ◽  
Vol 6 (2) ◽  
pp. 322-335 ◽  
Author(s):  
Alan Agresti
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Time Distribution ◽  
Branching Process ◽  
Probability Generating Function ◽  
Extinction Time ◽  
Expected Time ◽  
Time To Extinction ◽  
Special Case ◽  
Better Than

The class of fractional linear generating functions, one of the few known classes of probability generating functions whose iterates can be explicitly stated, is examined. The method of bounding a probability generating function g (satisfying g″(1) < ∞) by two fractional linear generating functions is used to derive bounds for the extinction time distribution of the Galton-Watson branching process with offspring probability distribution represented by g. For the special case of the Poisson probability generating function, the best possible bounding fractional linear generating functions are obtained, and the bounds for the expected time to extinction of the corresponding Poisson branching process are better than any previously published.

Inequalities for branching processes

1966 ◽  
Vol 3 (01) ◽  
pp. 261-267 ◽  
Author(s):  
C. R. Heathcote ◽  
E. Seneta
Keyword(s):  
Generating Function ◽  
Conditional Distribution ◽  
Branching Process ◽  
Branching Processes ◽  
Probability Generating Function ◽  
Random Variable ◽  
Expected Time ◽  
Time To Extinction ◽  
Explicit Formulae

Summary If F(s) is the probability generating function of a non-negative random variable, the nth functional iterate Fn (s) = Fn– 1 (F(s)) generates the distribution of the size of the nth generation of a simple branching process. In general it is not possible to obtain explicit formulae for many quantities involving Fn (s), and this paper considers certain bounds and approximations. Bounds are found for the Koenigs-type iterates lim n→∞ m −n {1−Fn (s)}, 0 ≦ s ≦ 1 where m = F′ (1) &lt; 1 and F′′ (1) &lt; ∞; for the expected time to extinction and for the limiting conditional-distribution generating function limn→∞{Fn (s) − Fn (0)} [1 – Fn (0)]–1. Particular attention is paid to the case F(s) = exp {m(s − 1)}.

scholarly journals On expected durations of birth–death processes, with applications to branching processes and SIS epidemics

2016 ◽  
Vol 53 (1) ◽  
pp. 203-215 ◽  
Author(s):  
Frank Ball ◽  
Tom Britton ◽  
Peter Neal
Keyword(s):  
Asymptotic Expression ◽  
Branching Processes ◽  
Random Variable ◽  
Death Process ◽  
Single Individual ◽  
Expected Time ◽  
Time To Extinction ◽  
Number Of Individuals ◽  
Sis Epidemic ◽  
Birth Death

Abstract We study continuous-time birth–death type processes, where individuals have independent and identically distributed lifetimes, according to a random variable Q, with E[Q] = 1, and where the birth rate if the population is currently in state (has size) n is α(n). We focus on two important examples, namely α(n) = λ n being a branching process, and α(n) = λn(N - n) / N which corresponds to an SIS (susceptible → infective → susceptible) epidemic model in a homogeneously mixing community of fixed size N. The processes are assumed to start with a single individual, i.e. in state 1. Let T, An, C, and S denote the (random) time to extinction, the total time spent in state n, the total number of individuals ever alive, and the sum of the lifetimes of all individuals in the birth–death process, respectively. We give expressions for the expectation of all these quantities and show that these expectations are insensitive to the distribution of Q. We also derive an asymptotic expression for the expected time to extinction of the SIS epidemic, but now starting at the endemic state, which is not independent of the distribution of Q. The results are also applied to the household SIS epidemic, showing that, in contrast to the household SIR (susceptible → infective → recovered) epidemic, its threshold parameter R* is insensitive to the distribution of Q.

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