Certain state-dependent processes for dichotomised parasite populations

1990 ◽  
Vol 27 (02) ◽  
pp. 251-258 ◽  
Author(s):  
A. W. Kemp ◽  
J. Newton
Keyword(s):  
Generating Functions ◽  
Death Process ◽  
Fixed Size ◽  
Probability Generating Functions ◽  
State Dependent ◽  
Parasite Populations ◽  
Dependent Processes ◽  
Birth Death

The paper re-examines Quinn and MacGillivray's (1986) stationary birth-death process for a population of fixed size N consisting of two types of parasite, active and passive, and sets up a more elaborate model for the dichotomy between parasites on hosts with and without open wounds resulting from previous parasite attacks. The probability generating functions for the stationary count distributions are obtained, allowing limiting forms of the distributions to be investigated.

Certain state-dependent processes for dichotomised parasite populations

1990 ◽  
Vol 27 (2) ◽  
pp. 251-258 ◽  
Author(s):  
A. W. Kemp ◽  
J. Newton
Keyword(s):  
Generating Functions ◽  
Death Process ◽  
Fixed Size ◽  
Probability Generating Functions ◽  
State Dependent ◽  
Parasite Populations ◽  
Dependent Processes ◽  
Birth Death

The paper re-examines Quinn and MacGillivray's (1986) stationary birth-death process for a population of fixed size N consisting of two types of parasite, active and passive, and sets up a more elaborate model for the dichotomy between parasites on hosts with and without open wounds resulting from previous parasite attacks. The probability generating functions for the stationary count distributions are obtained, allowing limiting forms of the distributions to be investigated.

scholarly journals Exact transient solution of a state-dependent birth-death process

2006 ◽  
Vol 2006 ◽  
pp. 1-16 ◽  
Author(s):  
P. R. Parthasarathy ◽  
R. Sudhesh
Keyword(s):  
Power Series ◽  
Closed Form ◽  
Death Process ◽  
Transient Solution ◽  
State Dependent ◽  
Transient Probabilities ◽  
Series Expression ◽  
Birth Death

A power series expression in closed form for the transient probabilities of a state-dependent birth-death process is presented with suitable illustrations.

Further approaches to computing fundamental characteristics of birth-death processes

2001 ◽  
Vol 38 (4) ◽  
pp. 995-1005 ◽  
Author(s):  
Frank Ball ◽  
Valeri T. Stefanov
Keyword(s):  
Discrete Time ◽  
Generating Functions ◽  
Continuous Time ◽  
Exponential Family ◽  
Death Process ◽  
Random Sum ◽  
First Passage ◽  
Reward Functions ◽  
Passage Times ◽  
Birth Death

General and unifying approaches are discussed for computing fundamental characteristics of both continuous-time and discrete-time birth-death processes. In particular, an exponential family framework is used to derive explicit expressions, in terms of continued fractions, for joint generating functions of first-passage times and a whole collection of associated random quantities, and a random sum representation is used to obtain formulae for means, variances and covariances of stopped reward functions defined on a birth-death process.

scholarly journals A note on three stochastic processes with immigration

2007 ◽  
Vol 48 (3) ◽  
pp. 409-418 ◽  
Author(s):  
J. Gani ◽  
L. Stals
Keyword(s):  
Stochastic Processes ◽  
Generating Functions ◽  
Probability Generating Functions ◽  
Birth Death

AbstractThree stochastic processes, the birth, death and birth-death processes, subject to immigration can be decomposed into the sum of each process in the absence of immigration and anindependent process. We examine these independent processes through their probability generating functions (pgfs) and derive their expectations.

Further approaches to computing fundamental characteristics of birth-death processes

2001 ◽  
Vol 38 (04) ◽  
pp. 995-1005 ◽  
Author(s):  
Frank Ball ◽  
Valeri T. Stefanov
Keyword(s):  
Discrete Time ◽  
Generating Functions ◽  
Continuous Time ◽  
Exponential Family ◽  
Death Process ◽  
Random Sum ◽  
First Passage ◽  
Reward Functions ◽  
Passage Times ◽  
Birth Death

General and unifying approaches are discussed for computing fundamental characteristics of both continuous-time and discrete-time birth-death processes. In particular, an exponential family framework is used to derive explicit expressions, in terms of continued fractions, for joint generating functions of first-passage times and a whole collection of associated random quantities, and a random sum representation is used to obtain formulae for means, variances and covariances of stopped reward functions defined on a birth-death process.

Normal approximations to discrete unimodal distributions

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.

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

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.

scholarly journals The Sampling Distribution of Disease-Associated Alleles

Genetics ◽  
1997 ◽  
Vol 147 (4) ◽  
pp. 1855-1861 ◽  
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.

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.

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