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.

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.

Labelling experiments and phase-age distributions for multiphase branching processes

1973 ◽  
Vol 10 (4) ◽  
pp. 739-747 ◽  
Author(s):  
P. J. Brockwell ◽  
W. H. Kuo
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Joint Probability ◽  
Random Variable ◽  
Cell Labelling ◽  
Single Individual ◽  
Age Dependent ◽  
The Individual ◽  
Number Of Individuals ◽  
Joint Probability Density

A supercritical age-dependent branching process is considered in which the lifespan of each individual is composed of four phases whose durations have joint probability density f(x1, x2, x3, x4). Starting with a single individual of age zero at time zero we consider the asymptotic behaviour as t→ ∞ of the random variable Z(4) (a0,…, an, t) defined as the number of individuals in phase 4 at time t for which the elapsed phase durations Y01,…, Y04,…, Yi1,…, Yi4,…, Yn4 of the individual itself and its first n ancestors satisfy the inequalities Yij ≦ aij, i = 0,…, n, j = 1,…, 4. The application of the results to the analysis of cell-labelling experiments is described. Finally we state an analogous result which defines (conditional on eventual non-extinction of the population) the asymptotic joint distribution of the phase and elapsed phase durations of an individual drawn at random from the population and the phase durations of its ancestors.

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) < 1 and F′′ (1) < ∞; 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)}.

Taboo extinction, sojourn times, and asymptotic growth for the Markovian birth and death process

1972 ◽  
Vol 9 (3) ◽  
pp. 486-506 ◽  
Author(s):  
W. A. O'n. Waugh
Keyword(s):  
Growth Factor ◽  
Exponential Growth ◽  
Asymptotic Expression ◽  
Branching Processes ◽  
Random Variable ◽  
Death Process ◽  
Birth And Death Process ◽  
Asymptotic Growth ◽  
Size Dependent ◽  
Sojourn Times

A well-known result in the theory of branching processes provides an asymptotic expression for the population size (valid for large times) in terms of a single random variable, multiplied by a deterministic exponential growth factor. In the present paper this is generalized to a class of size-dependent population models. The work is based on the series of sojourn times. An essential tool is the use of probabilities conditional upon non-extinction (taboo probabilities).

Extinction times for a general birth, death and catastrophe process

2004 ◽  
Vol 41 (04) ◽  
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.

Inequalities for branching processes

1966 ◽  
Vol 3 (1) ◽  
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

SummaryIf 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 limn→∞m−n {1−Fn(s)}, 0 ≦ s ≦ 1 where m = F′(1) < 1 and F′′(1) < ∞; 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)}.

Labelling experiments and phase-age distributions for multiphase branching processes

1973 ◽  
Vol 10 (04) ◽  
pp. 739-747 ◽  
Author(s):  
P. J. Brockwell ◽  
W. H. Kuo
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Joint Probability ◽  
Random Variable ◽  
Cell Labelling ◽  
Single Individual ◽  
Age Dependent ◽  
The Individual ◽  
Number Of Individuals ◽  
Joint Probability Density

A supercritical age-dependent branching process is considered in which the lifespan of each individual is composed of four phases whose durations have joint probability density f(x 1, x 2, x 3, x 4). Starting with a single individual of age zero at time zero we consider the asymptotic behaviour as t→ ∞ of the random variable Z (4) (a 0,…, a n , t) defined as the number of individuals in phase 4 at time t for which the elapsed phase durations Y 01,…, Y 04,…, Yi 1,…, Yi 4,…, Yn 4 of the individual itself and its first n ancestors satisfy the inequalities Yij ≦ aij , i = 0,…, n, j = 1,…, 4. The application of the results to the analysis of cell-labelling experiments is described. Finally we state an analogous result which defines (conditional on eventual non-extinction of the population) the asymptotic joint distribution of the phase and elapsed phase durations of an individual drawn at random from the population and the phase durations of its ancestors.

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.

scholarly journals A Note on Extinction Times for the General Birth, Death and Catastrophe Process

2007 ◽  
Vol 44 (2) ◽  
pp. 566-569 ◽  
Author(s):  
Phil Pollett ◽  
Hanjun Zhang ◽  
Benjamin J. Cairns
Keyword(s):  
Population Size ◽  
Explicit Expression ◽  
Transition Rates ◽  
Expected Time ◽  
Time To Extinction ◽  
Birth Death

We consider a birth, death and catastrophe process where the transition rates are allowed to depend on the population size. We obtain an explicit expression for the expected time to extinction, which is valid in all cases where extinction occurs with probability 1.

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.

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