scholarly journals Continuous-time multitype branching processes conditioned on very late extinction

2011 ◽  
Vol 15 ◽  
pp. 417-442 ◽  
Author(s):  
Sophie Pénisson
Keyword(s):  
Continuous Time ◽  
Branching Processes ◽  
Multitype Branching Processes

scholarly journals Supercritical multitype branching processes: the ancestral types of typical individuals

2003 ◽  
Vol 35 (4) ◽  
pp. 1090-1110 ◽  
Author(s):  
Hans-Otto Georgii ◽  
Ellen Baake
Keyword(s):  
Continuous Time ◽  
Branching Processes ◽  
Almost Sure Convergence ◽  
Family Tree ◽  
Mutation Process ◽  
Convergence Theorems ◽  
Conceptual Proof ◽  
The Family ◽  
Multitype Branching Processes

For supercritical multitype Markov branching processes in continuous time, we investigate the evolution of types along those lineages that survive up to some time t. We establish almost-sure convergence theorems for both time and population averages of ancestral types (conditioned on nonextinction), and identify the mutation process describing the type evolution along typical lineages. An important tool is a representation of the family tree in terms of a suitable size-biased tree with trunk. As a by-product, this representation allows a ‘conceptual proof’ (in the sense of Kurtz et al.) of the continuous-time version of the Kesten-Stigum theorem.

scholarly journals Conditional Distributions and Waiting Times in Multitype Branching Processes

2013 ◽  
Vol 45 (03) ◽  
pp. 692-718 ◽  
Author(s):  
H. K. Alexander
Keyword(s):  
Continuous Time ◽  
Evolutionary Biology ◽  
Branching Processes ◽  
Waiting Times ◽  
Escape Mutant ◽  
Conditional Distributions ◽  
Specific Population ◽  
Potential Applications ◽  
Population Sizes ◽  
Multitype Branching Processes

In this paper we present novel results for discrete-time and Markovian continuous-time multitype branching processes. As a population develops, we are interested in the waiting time until a particular type of interest (such as an escape mutant) appears, and in how the distribution of individuals depends on whether this type has yet appeared. Specifically, both forward and backward equations for the distribution of type-specific population sizes over time, conditioned on the nonappearance of one or more particular types, are derived. In tandem, equations for the probability that one or more particular types have not yet appeared are also derived. Brief examples illustrate numerical methods and potential applications of these results in evolutionary biology and epidemiology.

scholarly journals Conditional Distributions and Waiting Times in Multitype Branching Processes

2013 ◽  
Vol 45 (3) ◽  
pp. 692-718 ◽  
Author(s):  
H. K. Alexander
Keyword(s):  
Continuous Time ◽  
Evolutionary Biology ◽  
Branching Processes ◽  
Waiting Times ◽  
Escape Mutant ◽  
Conditional Distributions ◽  
Specific Population ◽  
Potential Applications ◽  
Population Sizes ◽  
Multitype Branching Processes

In this paper we present novel results for discrete-time and Markovian continuous-time multitype branching processes. As a population develops, we are interested in the waiting time until a particular type of interest (such as an escape mutant) appears, and in how the distribution of individuals depends on whether this type has yet appeared. Specifically, both forward and backward equations for the distribution of type-specific population sizes over time, conditioned on the nonappearance of one or more particular types, are derived. In tandem, equations for the probability that one or more particular types have not yet appeared are also derived. Brief examples illustrate numerical methods and potential applications of these results in evolutionary biology and epidemiology.

scholarly journals Supercritical multitype branching processes: the ancestral types of typical individuals

2003 ◽  
Vol 35 (04) ◽  
pp. 1090-1110 ◽  
Author(s):  
Hans-Otto Georgii ◽  
Ellen Baake
Keyword(s):  
Continuous Time ◽  
Branching Processes ◽  
Almost Sure Convergence ◽  
Family Tree ◽  
Mutation Process ◽  
Convergence Theorems ◽  
Conceptual Proof ◽  
The Family ◽  
Multitype Branching Processes

For supercritical multitype Markov branching processes in continuous time, we investigate the evolution of types along those lineages that survive up to some time t. We establish almost-sure convergence theorems for both time and population averages of ancestral types (conditioned on nonextinction), and identify the mutation process describing the type evolution along typical lineages. An important tool is a representation of the family tree in terms of a suitable size-biased tree with trunk. As a by-product, this representation allows a ‘conceptual proof’ (in the sense of Kurtz et al.) of the continuous-time version of the Kesten-Stigum theorem.

scholarly journals ESTIpop: a computational tool to simulate and estimate parameters for continuous-time Markov branching processes

2020 ◽  
Vol 36 (15) ◽  
pp. 4372-4373
Author(s):  
James P Roney ◽  
Jeremy Ferlic ◽  
Franziska Michor ◽  
Thomas O McDonald
Keyword(s):  
Parameter Estimation ◽  
Continuous Time ◽  
Likelihood Function ◽  
Branching Processes ◽  
R Package ◽  
Supplementary Information ◽  
Cell Counts ◽  
The Central Limit Theorem ◽  
Multitype Branching Processes ◽  
Analytical Approaches

Abstract Summary ESTIpop is an R package designed to simulate and estimate parameters for continuous-time Markov branching processes with constant or time-dependent rates, a common model for asexually reproducing cell populations. Analytical approaches to parameter estimation quickly become intractable in complex branching processes. In ESTIpop, parameter estimation is based on a likelihood function with respect to a time series of cell counts, approximated by the Central Limit Theorem for multitype branching processes. Additionally, simulation in ESTIpop via approximation can be performed many times faster than exact simulation methods with similar results. Availability and implementation ESTIpop is available as an R package on Github (https://github.com/michorlab/estipop). Supplementary information Supplementary data are available at Bioinformatics online.

On multitype branching processes in a random environment

1981 ◽  
Vol 13 (3) ◽  
pp. 464-497 ◽  
Author(s):  
David Tanny
Keyword(s):  
Random Environment ◽  
Stability Condition ◽  
Branching Processes ◽  
Classification Theorem ◽  
Regularity Conditions ◽  
Exponential Rate ◽  
Probabilistic Computation ◽  
Multitype Branching Processes ◽  
The Stability ◽  
Functional Iteration

This paper is concerned with the growth of multitype branching processes in a random environment (mbpre). It is shown that, under suitable regularity conditions, the process either explodes of becomes extinct. A classification theorem is given delineating the cases of explosion or extinction. Furthermore, it is shown that the process grows at an exponential rate on its set of non-extinction provided the process is stable. Criteria is given for non-certain extinction of the mbpre to occur, and an example shows that the stability condition cannot be removed. The method of proof used, in general, is direct probabilistic computation rather than the classical functional iteration techniques. Growth theorems are first proved for increasing mbpre and subsequently transferred to general mbpre using the associated mbpre and the reduced mbpre.

scholarly journals Extinction Probabilities of Supercritical Decomposable Branching Processes

2012 ◽  
Vol 49 (03) ◽  
pp. 639-651 ◽  
Author(s):  
Sophie Hautphenne
Keyword(s):  
Nonnegative Solution ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Equivalence Classes ◽  
Specific Class ◽  
Initial Particle ◽  
Whole Process ◽  
Extinction Probabilities ◽  
The Minimal Nonnegative Solution ◽  
Multitype Branching Processes

We focus on supercritical decomposable (reducible) multitype branching processes. Types are partitioned into irreducible equivalence classes. In this context, extinction of some classes is possible without the whole process becoming extinct. We derive criteria for the almost-sure extinction of the whole process, as well as of a specific class, conditionally given the class of the initial particle. We give sufficient conditions under which the extinction of a class implies the extinction of another class or of the whole process. Finally, we show that the extinction probability of a specific class is the minimal nonnegative solution of the usual extinction equation but with added constraints.

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