On eigenvalue bounds for the finite-state birth-death process intensity matrix

For a sequence of finite state space birth–death processes, each having a single absorbing state, we show that, under certain conditions, as the size of the state space tends to infinity, the quasistationary distributions converge to the stationary distribution of a limiting infinite state space birth–death process. This generalizes a result of Keilson and Ramaswamy by allowing birth and death rates to depend upon the size of the state space. We give sufficient conditions under which the convergence result of Keilson and Ramaswamy remains valid. The generalization allows us to apply our convergence result to examples from population biology: a Pearl–Verhulst logistic population growth model and the susceptible-infective-susceptible (SIS) model for infectious spread. The limit distributions obtained suggest new finite-population approximations to the quasistationary distributions of these models, obtained by the method of cumulant closure. The new approximations are found to be both simple in form and accurate.

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Approximating Quasistationary Distributions of Birth–Death Processes

Journal of Applied Probability ◽

10.1017/s0021900200012869 ◽

2012 ◽

Vol 49 (04) ◽

pp. 1036-1051

Author(s):

Damian Clancy

Keyword(s):

State Space ◽

Population Biology ◽

Sufficient Conditions ◽

Convergence Result ◽

The State ◽

Death Process ◽

Sis Model ◽

Finite State ◽

Quasistationary Distributions ◽

Birth Death

For a sequence of finite state space birth–death processes, each having a single absorbing state, we show that, under certain conditions, as the size of the state space tends to infinity, the quasistationary distributions converge to the stationary distribution of a limiting infinite state space birth–death process. This generalizes a result of Keilson and Ramaswamy by allowing birth and death rates to depend upon the size of the state space. We give sufficient conditions under which the convergence result of Keilson and Ramaswamy remains valid. The generalization allows us to apply our convergence result to examples from population biology: a Pearl–Verhulst logistic population growth model and the susceptible-infective-susceptible (SIS) model for infectious spread. The limit distributions obtained suggest new finite-population approximations to the quasistationary distributions of these models, obtained by the method of cumulant closure. The new approximations are found to be both simple in form and accurate.

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Bell Polynomial Approach for Time-Inhomogeneous Linear Birth–Death Process with Immigration

Mathematics ◽

10.3390/math8071123 ◽

2020 ◽

Vol 8 (7) ◽

pp. 1123

Author(s):

Virginia Giorno ◽

Amelia G. Nobile

Keyword(s):

Transition Probabilities ◽

Growth Models ◽

Death Process ◽

Bell Polynomials ◽

Bell Polynomial ◽

Conditional Mean ◽

Finite State ◽

Potential Applications ◽

Intensity Functions ◽

Birth Death

We considered the time-inhomogeneous linear birth–death processes with immigration. For these processes closed form expressions for the transition probabilities were obtained in terms of the complete Bell polynomials. The conditional mean and the conditional variance were explicitly evaluated. Several time-inhomogeneous processes were studied in detail in view of their potential applications in population growth models and in queuing systems. A time-inhomogeneous linear birth–death processes with finite state-space was also taken into account. Special attention was devoted to the cases of periodic immigration intensity functions that play an important role in the description of the evolution of dynamic systems influenced by seasonal immigration or other regular environmental cycles. Various numerical computations were performed for periodic immigration intensity functions.

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Large deviations and quasi-stationarity for density-dependent birth-death processes

The Journal of the Australian Mathematical Society Series B Applied Mathematics ◽

10.1017/s0334270000012492 ◽

1998 ◽

Vol 40 (2) ◽

pp. 238-256 ◽

Cited By ~ 5

Author(s):

Terence Chan

Keyword(s):

Large Deviations ◽

Stationary Distribution ◽

Death Process ◽

Large Deviations Principle ◽

Density Dependent ◽

Exponential Decay Rate ◽

Finite State ◽

Degenerate Distribution ◽

The One ◽

Birth Death

AbstractConsider a density-dependent birth-death process XN on a finite state space of size N. Let PN be the law (on D([0, T]) where T > 0 is arbitrary) of the density process XN/N and let πN be the unique stationary distribution (on[0,1]) of XN/N, if it exists. Typically, these distributions converge weakly to a degenerate distribution as N → ∞ so the probability of sets not containing the degenerate point will tend to 0; large deviations is concerned with obtaining the exponential decay rate of these probabilities. Friedlin-Wentzel theory is used to establish the large deviations behaviour (as N → ∞) of PN. In the one-dimensional case, a large deviations principle for the stationary distribution πN is obtained by elementary explicit computations. However, when the birth-death process has an absorbing state at 0 (so πN no longer exists), the same elementary computations are still applicable to the quasi-stationary distribution, and we show that the quasi-stationary distributions obey the same large deviations principle as in the recurrent case. In addition, we address some questions related to the estimated time to absorption and obtain a large deviations principle for the invariant distribution in higher dimensions by studying a quasi-potential.

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On Separation for Birth-Death Processes

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964800003193 ◽

1994 ◽

Vol 8 (1) ◽

pp. 51-68

Author(s):

Masaaki Kijima

Keyword(s):

Infinitesimal Generator ◽

Transition Probabilities ◽

Upper Bounds ◽

Death Process ◽

Exponential Functions ◽

Finite State ◽

Variation Distance ◽

Stationary Probabilities ◽

Finite State Space ◽

Birth Death

This article considers separation for a birth-death process on a finite state space S = [1,2,…, N]. Separation is defined by si(t) = 1 – minj∈sPij(t)/πj, as in Fill [5,6], where Pij(t) denotes the transition probabilities of the birth-death process and πj the stationary probabilities. Separation is a measure of nonstationarity of Markov chains and provides an upper bound of the variation distance. Easily computable upper bounds for si-(t) are given, which consist of simple exponential functions whose parameters are the eigenvalues of the infinitesimal generator or its submatrices of the birth-death process.

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A Time-Inhomogeneous Prendiville Model with Failures and Repairs

Mathematics ◽

10.3390/math10020251 ◽

2022 ◽

Vol 10 (2) ◽

pp. 251

Author(s):

Virginia Giorno ◽

Amelia G. Nobile

Keyword(s):

Markov Chain ◽

State Space ◽

The State ◽

Random Time ◽

Death Process ◽

Inhomogeneous Markov Chain ◽

Finite State ◽

Intensity Functions ◽

Finite State Space ◽

Birth Death

We consider a time-inhomogeneous Markov chain with a finite state-space which models a system in which failures and repairs can occur at random time instants. The system starts from any state j (operating, F, R). Due to a failure, a transition from an operating state to F occurs after which a repair is required, so that a transition leads to the state R. Subsequently, there is a restore phase, after which the system restarts from one of the operating states. In particular, we assume that the intensity functions of failures, repairs and restores are proportional and that the birth-death process that models the system is a time-inhomogeneous Prendiville process.

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HITTING TIME DISTRIBUTIONS FOR BIRTH–DEATH PROCESSES WITH BILATERAL ABSORBING BOUNDARIES

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964816000280 ◽

2016 ◽

Vol 31 (3) ◽

pp. 345-356

Author(s):

Yong-Hua Mao ◽

Chi Zhang

Keyword(s):

State Space ◽

Hitting Time ◽

Death Process ◽

Absorbing Boundaries ◽

Finite State ◽

Finite State Space ◽

Birth Death

For the birth–death process on a finite state space with bilateral boundaries, we give a simpler derivation of the hitting time distributions by h-transform and φ-transform. These transforms can then be used to construct a quick derivation of the hitting time distributions of the minimal birth–death process on a denumerable state space with exit/regular boundaries.

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Normal approximations to discrete unimodal distributions

Journal of Applied Probability ◽

10.1017/s0021900200115931 ◽

1986 ◽

Vol 23 (04) ◽

pp. 1013-1018

Author(s):

B. G. Quinn ◽

H. L. MacGillivray

Keyword(s):

Stationary Distribution ◽

Sufficient Conditions ◽

Random Variables ◽

Hypergeometric Distribution ◽

Death Process ◽

Normal Approximations ◽

Discrete Random Variables ◽

Birth Death

Sufficient conditions are presented for the limiting normality of sequences of discrete random variables possessing unimodal distributions. The conditions are applied to obtain normal approximations directly for the hypergeometric distribution and the stationary distribution of a special birth-death process.

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A generalized Gompertz growth model with applications and related birth-death processes

Ricerche di Matematica ◽

10.1007/s11587-020-00548-y ◽

2020 ◽

Author(s):

Majid Asadi ◽

Antonio Di Crescenzo ◽

Farkhondeh A. Sajadi ◽

Serena Spina

Keyword(s):

Growth Model ◽

Growth Curve ◽

Goodness Of Fit ◽

Gompertz Model ◽

Death Process ◽

Birth Process ◽

Failure Data ◽

First Case ◽

Gompertz Growth Model ◽

Birth Death

AbstractIn this paper, we propose a flexible growth model that constitutes a suitable generalization of the well-known Gompertz model. We perform an analysis of various features of interest, including a sensitivity analysis of the initial value and the three parameters of the model. We show that the considered model provides a good fit to some real datasets concerning the growth of the number of individuals infected during the COVID-19 outbreak, and software failure data. The goodness of fit is established on the ground of the ISRP metric and the $$d_2$$ d 2 -distance. We also analyze two time-inhomogeneous stochastic processes, namely a birth-death process and a birth process, whose means are equal to the proposed growth curve. In the first case we obtain the probability of ultimate extinction, being 0 an absorbing endpoint. We also deal with a threshold crossing problem both for the proposed growth curve and the corresponding birth process. A simulation procedure for the latter process is also exploited.

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The Sampling Distribution of Disease-Associated Alleles

Genetics ◽

10.1093/genetics/147.4.1855 ◽

1997 ◽

Vol 147 (4) ◽

pp. 1855-1861 ◽

Cited By ~ 1

Author(s):

Montgomery Slatkin ◽

Bruce Rannala

Keyword(s):

Low Frequency ◽

Null Model ◽

Sampling Distribution ◽

Death Process ◽

Published Data ◽

Data Sets ◽

Likelihood Functions ◽

Alternative Hypotheses ◽

Size Standard ◽

Birth Death

Abstract A theory is developed that provides the sampling distribution of low frequency alleles at a single locus under the assumption that each allele is the result of a unique mutation. The numbers of copies of each allele is assumed to follow a linear birth-death process with sampling. If the population is of constant size, standard results from theory of birth-death processes show that the distribution of numbers of copies of each allele is logarithmic and that the joint distribution of numbers of copies of k alleles found in a sample of size n follows the Ewens sampling distribution. If the population from which the sample was obtained was increasing in size, if there are different selective classes of alleles, or if there are differences in penetrance among alleles, the Ewens distribution no longer applies. Likelihood functions for a given set of observations are obtained under different alternative hypotheses. These results are applied to published data from the BRCA1 locus (associated with early onset breast cancer) and the factor VIII locus (associated with hemophilia A) in humans. In both cases, the sampling distribution of alleles allows rejection of the null hypothesis, but relatively small deviations from the null model can account for the data. In particular, roughly the same population growth rate appears consistent with both data sets.

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