Formulas for average transition times between states of the Markov birth-death process

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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Time varying moments, regime switch, and crisis warning: The birth–death process with changing transition probability

Physica A Statistical Mechanics and its Applications ◽

10.1016/j.physa.2014.02.038 ◽

2014 ◽

Vol 404 ◽

pp. 56-64 ◽

Cited By ~ 4

Author(s):

Yinan Tang ◽

Ping Chen

Keyword(s):

Transition Probability ◽

Death Process ◽

Time Varying ◽

Regime Switch ◽

Birth Death

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

Methodology And Computing In Applied Probability ◽

10.1007/s11009-021-09882-6 ◽

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.

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Stochastic models for interference between searching insect parasites

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

10.1017/s0334270000006226 ◽

1989 ◽

Vol 30 (3) ◽

pp. 268-277 ◽

Cited By ~ 1

Author(s):

Phil Diamond

Keyword(s):

Stationary Distribution ◽

Intraspecific Competition ◽

Death Process ◽

Confluent Hypergeometric Functions ◽

Large Populations ◽

The Mean ◽

The One ◽

Globally Stable ◽

Insect Parasites ◽

Birth Death

AbstractCompetition between a finite number of searching insect parasites is modelled by differential equations and birth-death processes. In the one species case of intraspecific competition, the deterministic equilibrium is globally stable and, for large populations, approximates the mean of the stationary distribution of the process. For two species, both inter- and intraspecific competition occurs and the deterministic equilibrium is globally stable. When the birth-death process is reversible, it is shown that the mean of the stationary distribution is approximated by the equilibrium. Confluent hypergeometric functions of two variables are important to the theory.

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