Multitype infinite-allele branching processes in continuous time

2017 ◽  
Vol 54 (2) ◽  
pp. 550-568
Author(s):  
Thomas O. McDonald ◽  
Marek Kimmel
Keyword(s):  
Tumor Growth ◽  
Continuous Time ◽  
Branching Process ◽  
Branching Processes ◽  
Clonal Evolution ◽  
Asymptotic Results ◽  
General Process ◽  
Binary Splitting ◽  
Passenger Mutations ◽  
Viral Reproduction

Abstract We introduce extensions to an infinite-allele branching process that allows for multiple types to exist alongside labels. We consider a Markov branching process and general branching process under different assumptions, and show asymptotic results about the growth of the labels as well as the frequency spectrum. These results are motivated by two separate models. The Markov binary splitting results are motivated by a model of clonal evolution in cancer that considers the effect of both driver and passenger mutations on tumor growth. The general process has applications in viral reproduction and dynamics.

scholarly journals Extinction Times in Multitype Markov Branching Processes

2009 ◽  
Vol 46 (01) ◽  
pp. 296-307 ◽  
Author(s):  
Dominik Heinzmann
Keyword(s):  
Limit Theorem ◽  
Continuous Time ◽  
Branching Process ◽  
Branching Processes ◽  
Markov Branching Process ◽  
Time To Extinction

In this paper, a distributional approximation to the time to extinction in a subcritical continuous-time Markov branching process is derived. A limit theorem for this distribution is established and the error in the approximation is quantified. The accuracy of the approximation is illustrated in an epidemiological example. Since Markov branching processes serve as approximations to nonlinear epidemic processes in the initial and final stages, our results can also be used to describe the time to extinction for such processes.

scholarly journals Asymptotic behavior of critical, irreducible multi-type continuous state and continuous time branching processes with immigration

2016 ◽  
Vol 16 (04) ◽  
pp. 1650008 ◽  
Author(s):  
Mátyás Barczy ◽  
Gyula Pap
Keyword(s):  
Continuous Time ◽  
Branching Process ◽  
Branching Processes ◽  
Bessel Process ◽  
Step Functions ◽  
Squared Bessel Process ◽  
Continuous State ◽  
Random Step ◽  
Branching Process With Immigration ◽  
Continuous Time Branching Processes

Under natural assumptions, a Feller type diffusion approximation is derived for critical, irreducible multi-type continuous state and continuous time branching processes with immigration. Namely, it is proved that a sequence of appropriately scaled random step functions formed from a critical, irreducible multi-type continuous state and continuous time branching process with immigration converges weakly towards a squared Bessel process supported by a ray determined by the Perron vector of a matrix related to the branching mechanism of the branching process in question.

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 Extinction Times in Multitype Markov Branching Processes

2009 ◽  
Vol 46 (1) ◽  
pp. 296-307 ◽  
Author(s):  
Dominik Heinzmann
Keyword(s):  
Limit Theorem ◽  
Continuous Time ◽  
Branching Process ◽  
Branching Processes ◽  
Markov Branching Process ◽  
Time To Extinction

In this paper, a distributional approximation to the time to extinction in a subcritical continuous-time Markov branching process is derived. A limit theorem for this distribution is established and the error in the approximation is quantified. The accuracy of the approximation is illustrated in an epidemiological example. Since Markov branching processes serve as approximations to nonlinear epidemic processes in the initial and final stages, our results can also be used to describe the time to extinction for such processes.

scholarly journals Upper Deviations for Split Times of Branching Processes

2012 ◽  
Vol 49 (4) ◽  
pp. 1134-1143
Author(s):  
Hamed Amini ◽  
Marc Lelarge
Keyword(s):  
Continuous Time ◽  
Branching Process ◽  
Branching Processes ◽  
Deviation Probability ◽  
Markov Branching Process ◽  
Split Time

Upper deviation results are obtained for the split time of a supercritical continuous-time Markov branching process. More precisely, we establish the existence of logarithmic limits for the likelihood that the split times of the process are greater than an identified value and determine an expression for the limiting quantity. We also give an estimation for the lower deviation probability of the split times, which shows that the scaling is completely different from the upper deviations.

Supercritical branching processes with density independent catastrophes

1988 ◽  
Vol 104 (2) ◽  
pp. 413-416 ◽  
Author(s):  
D. R. Grey
Keyword(s):  
Continuous Time ◽  
Branching Process ◽  
Transition Probabilities ◽  
Branching Processes ◽  
Constant Rate ◽  
Watson Process ◽  
Stationary Transition ◽  
Supercritical Branching Processes ◽  
Independent Identically Distributed ◽  
Independent Poisson Process

A Markov branching process in either discrete time (the Galton–Watson process) or continuous time is modified by the introduction of a process of catastrophes which remove some individuals (and, by implication, their descendants) from the population. The catastrophe process is independent of the reproduction mechanism and takes the form of a sequence of independent identically distributed non-negative integer-valued random variables. In the continuous time case, these catastrophes occur at the points of an independent Poisson process with constant rate. If at any time the size of a catastrophe is at least the current population size, then the population becomes extinct. Thus in both discrete and continuous time we still have a Markov chain with stationary transition probabilities and an absorbing state at zero. Some authors use the term ‘emigration’ as an alternative to ‘catastrophe’.

scholarly journals Limit Properties of Transition Functions of Continuous-Time Markov Branching Processes

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Azam A. Imomov
Keyword(s):  
Continuous Time ◽  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
Local Limit ◽  
Markov Branching Process ◽  
Transition Functions ◽  
Basic Lemma ◽  
Invariant Properties ◽  
Local Limit Theorems

Consider the Markov Branching Process with continuous time. Our focus is on the limit properties of transition functions of this process. Using differential analogue of the Basic Lemma we prove local limit theorems for all cases and observe invariant properties of considering process.

Asymptotic results for the extinction time of Markov branching processes allowing emigration, I. Random walk decrements

1989 ◽  
Vol 21 (02) ◽  
pp. 243-269 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Population Size ◽  
Large Scale ◽  
Branching Process ◽  
Branching Processes ◽  
Initial Population ◽  
Extinction Time ◽  
Asymptotic Results ◽  
Time To Extinction ◽  
Initial Population Size ◽  
The Mathematical Model

The mathematical model is a Markov branching process which is subjected to catastrophes or large-scale emigration. Catastrophes reduce the population size by independent and identically distributed decrements, and two mechanisms for generating catastrophe epochs are given separate consideration. These are that catastrophes occur at a rate proportional to population size, and as an independent Poisson process. The paper studies some properties of the time to extinction of the modified process in those cases where extinction occurs almost surely. Particular attention is given to limit theorems and the behaviour of the expected extinction time as the initial population size grows. These properties are contrasted with known properties for the case when there is no catastrophe component.

Skeletal stochastic differential equations for continuous-state branching processes

2019 ◽  
Vol 56 (4) ◽  
pp. 1122-1150 ◽  
Author(s):  
D. Fekete ◽  
J. Fontbona ◽  
A. E. Kyprianou
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Continuous Time ◽  
Branching Process ◽  
Branching Processes ◽  
Continuous State ◽  
Continuous State Branching Process ◽  
Common Framework ◽  
Watson Process ◽  
Skeletal Representation

AbstractIt is well understood that a supercritical continuous-state branching process (CSBP) is equal in law to a discrete continuous-time Galton–Watson process (the skeleton of prolific individuals) whose edges are dressed in a Poissonian way with immigration which initiates subcritical CSBPs (non-prolific mass). Equally well understood in the setting of CSBPs and superprocesses is the notion of a spine or immortal particle dressed in a Poissonian way with immigration which initiates copies of the original CSBP, which emerges when conditioning the process to survive eternally. In this article we revisit these notions for CSBPs and put them in a common framework using the well-established language of (coupled) stochastic differential equations (SDEs). In this way we are able to deal simultaneously with all types of CSBPs (supercritical, critical, and subcritical) as well as understanding how the skeletal representation becomes, in the sense of weak convergence, a spinal decomposition when conditioning on survival. We have two principal motivations. The first is to prepare the way to expand the SDE approach to the spatial setting of superprocesses, where recent results have increasingly sought the use of skeletal decompositions to transfer results from the branching particle setting to the setting of measure valued processes. The second is to provide a pathwise decomposition of CSBPs in the spirit of genealogical coding of CSBPs via Lévy excursions, albeit precisely where the aforesaid coding fails to work because the underlying CSBP is supercritical.

The extinction time of Markov branching processes

1976 ◽  
Vol 13 (2) ◽  
pp. 345-347 ◽  
Author(s):  
Robert C. Wang
Keyword(s):  
Density Function ◽  
Continuous Time ◽  
Branching Process ◽  
Branching Processes ◽  
Extinction Time ◽  
Markov Branching Process

This paper considers the expression of expectations and density function of the extinction time for a continuous-time Markov branching process.

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