scholarly journals Modeling Y-Linked Pedigrees through Branching Processes

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 256
Author(s):  
Miguel González ◽  
Cristina Gutiérrez ◽  
Rodrigo Martínez
Keyword(s):  
Hearing Loss ◽  
Asymptotic Behavior ◽  
Gibbs Sampler ◽  
Mutant Allele ◽  
Branching Process ◽  
Branching Processes ◽  
Dominant Allele ◽  
Linked Gene ◽  
Different Types ◽  
The Mean

A multidimensional two-sex branching process is introduced to model the evolution of a pedigree originating from the mutation of an allele of a Y-linked gene in a monogamous population. The study of the extinction of the mutant allele and the analysis of the dominant allele in the pedigree is addressed on the basis of the classical theory of multi-type branching processes. The asymptotic behavior of the number of couples of different types in the pedigree is also derived. Finally, using the estimates of the mean growth rates of the allele and its mutation provided by a Gibbs sampler, a real Y-linked pedigree associated with hearing loss is analyzed, concluding that this mutation will persist in the population although without dominating the pedigree.

scholarly journals Limiting Genotype Frequencies of Y-Linked Genes with a Mutant Allele in a Two-Sex Population

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 131
Author(s):  
Miguel González ◽  
Cristina Gutiérrez ◽  
Rodrigo Martínez
Keyword(s):  
Population Genetics ◽  
Growth Rates ◽  
Mutant Allele ◽  
Branching Process ◽  
Linked Gene ◽  
Genotype Frequencies ◽  
Different Types

A two-type two-sex branching process is considered to model the evolution of the number of carriers of an allele and its mutations of a Y-linked gene. The limiting growth rates of the different types of couples and males (depending on the allele, mutated or not, that they carry on) on the set of coexistence of both alleles and on the fixation set of the mutant allele are obtained. In addition, the limiting genotype of the Y-linked gene and the limiting sex frequencies on those sets are established. Finally, the main results have been illustrated with simulated studies contextualized in problems of population genetics.

Asymptotic behaviour of immigration-branching processes by general set of types. II: Supercritical branching part

1975 ◽  
Vol 7 (03) ◽  
pp. 468-494
Author(s):  
H. Hering
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Almost Sure Convergence ◽  
Transition Function ◽  
Mean Square ◽  
Second Moments ◽  
Mean Square Convergence ◽  
The Mean ◽  
Parameter Dependent ◽  
Dependent Probability

We construct an immigration-branching process from an inhomogeneous Poisson process, a parameter-dependent probability distribution of populations and a Markov branching process with homogeneous transition function. The set of types is arbitrary, and the parameter is allowed to be discrete or continuous. Assuming a supercritical branching part with primitive first moments and finite second moments, we prove propositions on the mean square convergence and the almost sure convergence of normalized averaging processes associated with the immigration-branching process.

Asymptotic behaviour of population-size-dependent branching processes in Markovian random environments

1999 ◽  
Vol 36 (2) ◽  
pp. 611-619 ◽  
Author(s):  
Han-Xing Wang ◽  
Dafan Fang
Keyword(s):  
Asymptotic Behaviour ◽  
Population Size ◽  
Branching Process ◽  
Branching Processes ◽  
Random Environments ◽  
Real Numbers ◽  
Size Dependent ◽  
Recurrent Markov Chain ◽  
The Mean ◽  
Positive Recurrent

A population-size-dependent branching process {Zn} is considered where the population's evolution is controlled by a Markovian environment process {ξn}. For this model, let mk,θ and be the mean and the variance respectively of the offspring distribution when the population size is k and a environment θ is given. Let B = {ω : Zn(ω) = 0 for some n} and q = P(B). The asymptotic behaviour of limnZn and is studied in the case where supθ|mk,θ − mθ| → 0 for some real numbers {mθ} such that infθmθ > 1. When the environmental sequence {ξn} is a irreducible positive recurrent Markov chain (particularly, when its state space is finite), certain extinction (q = 1) and non-certain extinction (q < 1) are studied.

scholarly journals Contagions in random networks with overlapping communities

2015 ◽  
Vol 47 (4) ◽  
pp. 973-988 ◽  
Author(s):  
Emilie Coupechoux ◽  
Marc Lelarge
Keyword(s):  
Phase Transition ◽  
Random Graph ◽  
Epidemic Model ◽  
Branching Process ◽  
Branching Processes ◽  
Graph Model ◽  
Single Individual ◽  
Overlapping Communities ◽  
Random Graph Model ◽  
The Mean

We consider a threshold epidemic model on a clustered random graph model obtained from local transformations in an alternating branching process that approximates a bipartite graph. In other words, our epidemic model is such that an individual becomes infected as soon as the proportion of his/her infected neighbors exceeds the threshold q of the epidemic. In our random graph model, each individual can belong to several communities. The distributions for the community sizes and the number of communities an individual belongs to are arbitrary. We consider the case where the epidemic starts from a single individual, and we prove a phase transition (when the parameter q of the model varies) for the appearance of a cascade, i.e. when the epidemic can be propagated to an infinite part of the population. More precisely, we show that our epidemic is entirely described by a multi-type (and alternating) branching process, and then we apply Sevastyanov's theorem about the phase transition of multi-type Galton-Watson branching processes. In addition, we compute the entries of the mean progeny matrix corresponding to the epidemic. The phase transition for the contagion is given in terms of the largest eigenvalue of this matrix.

Geometric rate of growth in population-size-dependent branching processes

1984 ◽  
Vol 21 (01) ◽  
pp. 40-49 ◽  
Author(s):  
F. C. Klebaner
Keyword(s):  
Population Size ◽  
Process Model ◽  
Branching Process ◽  
Necessary Conditions ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Almost Sure Convergence ◽  
Size Dependent ◽  
Offspring Distribution ◽  
The Mean

We consider a branching-process model {Zn }, where the law of offspring distribution depends on the population size. We consider the case when the means mn (mn is the mean of offspring distribution when the population size is equal to n) tend to a limit m &gt; 1 as n →∞. For a certain class of processes {Zn } necessary conditions for convergence in L 1 and L 2 and sufficient conditions for almost sure convergence and convergence in L 2 of Wn = Zn/mn are given.

Homogeneous Branching Processes with Non-Homogeneous Immigration

2021 ◽  
Vol 36 (2) ◽  
pp. 165-183
Author(s):  
Ibrahim Rahimov
Keyword(s):  
Asymptotic Behavior ◽  
Branching Process ◽  
Branching Processes ◽  
Limited Effect ◽  
Immigration Rate ◽  
Current State ◽  
State Of Research ◽  
New Phenomena ◽  
Changes Over Time ◽  
Over Time

Abstract The stationary immigration has a limited effect over the asymptotic behavior of the underlying branching process. It affects mostly the limiting distribution and the life-period of the process. In contrast, if the immigration rate changes over time, then the asymptotic behavior of the process is significantly different and a variety of new phenomena are observed. In this review we discuss branching processes with time non-homogeneous immigration. Our goal is to help researchers interested in the topic to familiarize themselves with the current state of research.

Asymptotic behaviour of population-size-dependent branching processes in Markovian random environments

1999 ◽  
Vol 36 (02) ◽  
pp. 611-619 ◽  
Author(s):  
Han-Xing Wang ◽  
Dafan Fang
Keyword(s):  
Asymptotic Behaviour ◽  
Population Size ◽  
Branching Process ◽  
Branching Processes ◽  
Random Environments ◽  
Real Numbers ◽  
Size Dependent ◽  
Recurrent Markov Chain ◽  
The Mean ◽  
Positive Recurrent

A population-size-dependent branching process {Z n } is considered where the population's evolution is controlled by a Markovian environment process {ξ n }. For this model, let m k,θ and be the mean and the variance respectively of the offspring distribution when the population size is k and a environment θ is given. Let B = {ω : Z n (ω) = 0 for some n} and q = P(B). The asymptotic behaviour of lim n Z n and is studied in the case where supθ|m k,θ − m θ| → 0 for some real numbers {m θ} such that infθ m θ &gt; 1. When the environmental sequence {ξ n } is a irreducible positive recurrent Markov chain (particularly, when its state space is finite), certain extinction (q = 1) and non-certain extinction (q &lt; 1) are studied.

scholarly journals Asymptotic behavior of projections of supercritical multi-type continuous-state and continuous-time branching processes with immigration

2021 ◽  
Vol 53 (4) ◽  
pp. 1023-1060
Author(s):  
Mátyás Barczy ◽  
Sandra Palau ◽  
Gyula Pap
Keyword(s):  
Asymptotic Behavior ◽  
Continuous Time ◽  
Branching Process ◽  
Branching Processes ◽  
Order Moment ◽  
Moment Condition ◽  
First Order ◽  
Continuous State ◽  
Branching Process With Immigration ◽  
Continuous Time Branching Processes

AbstractUnder a fourth-order moment condition on the branching and a second-order moment condition on the immigration mechanisms, we show that an appropriately scaled projection of a supercritical and irreducible continuous-state and continuous-time branching process with immigration on certain left non-Perron eigenvectors of the branching mean matrix is asymptotically mixed normal. With an appropriate random scaling, under some conditional probability measure, we prove asymptotic normality as well. In the case of a non-trivial process, under a first-order moment condition on the immigration mechanism, we also prove the convergence of the relative frequencies of distinct types of individuals on a suitable event; for instance, if the immigration mechanism does not vanish, then this convergence holds almost surely.

scholarly journals Multitype linear fractional branching processes

2006 ◽  
Vol 43 (04) ◽  
pp. 1091-1106 ◽  
Author(s):  
A. Joffe ◽  
G. Letac
Keyword(s):  
Generating Functions ◽  
Branching Process ◽  
Branching Processes ◽  
Main Tool ◽  
Linear Functions ◽  
Multitype Branching Process ◽  
The Mean

We complete a paper written by Edward Pollak in 1974 on a multitype branching process the generating functions of whose birth law are fractional linear functions with the same denominator. The main tool is a parameterization of these functions adapted using the mean matrix M and an element w of the first quadrant. We use this opportunity to give a self-contained presentation of Pollak's theory.

Branching Process Modelling of COVID-19 Pandemic Including Immunity and Vaccination

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Dimitar Atanasov ◽  
Vessela Stoimenova ◽  
Nikolay M. Yanev
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Data Availability ◽  
Model Parameters ◽  
Reproduction Rate ◽  
Direct Computation ◽  
Infection Dynamics ◽  
Secondary Infections ◽  
Proposed Model ◽  
The Mean

Abstract We propose modeling COVID-19 infection dynamics using a class of two-type branching processes. These models require only observations on daily statistics to estimate the average number of secondary infections caused by a host and to predict the mean number of the non-observed infected individuals. The development of the epidemic process depends on the reproduction rate as well as on additional facets as immigration, adaptive immunity, and vaccination. Usually, in the existing deterministic and stochastic models, the officially reported and publicly available data are not sufficient for estimating model parameters. An important advantage of the proposed model, in addition to its simplicity, is the possibility of direct computation of its parameters estimates from the daily available data. We illustrate the proposed model and the corresponding data analysis with data from Bulgaria, however they are not limited to Bulgaria and can be applied to other countries subject to data availability.

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