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 (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 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.

scholarly journals Generalized fractional Poisson process and related stochastic dynamics

2020 ◽  
Vol 23 (3) ◽  
pp. 656-693 ◽  
Author(s):  
Thomas M. Michelitsch ◽  
Alejandro P. Riascos
Keyword(s):  
Poisson Process ◽  
Continuous Time ◽  
Counting Process ◽  
Stochastic Dynamics ◽  
Waiting Times ◽  
Fractional Diffusion ◽  
Diffusion Limit ◽  
Fractional Operators ◽  
Fractional Poisson Process ◽  
Erlang Process

AbstractWe survey the ‘generalized fractional Poisson process’ (GFPP). The GFPP is a renewal process generalizing Laskin’s fractional Poisson counting process and was first introduced by Cahoy and Polito. The GFPP contains two index parameters with admissible ranges 0 < β ≤ 1, α > 0 and a parameter characterizing the time scale. The GFPP involves Prabhakar generalized Mittag-Leffler functions and contains for special choices of the parameters the Laskin fractional Poisson process, the Erlang process and the standard Poisson process. We demonstrate this by means of explicit formulas. We develop the Montroll-Weiss continuous-time random walk (CTRW) for the GFPP on undirected networks which has Prabhakar distributed waiting times between the jumps of the walker. For this walk, we derive a generalized fractional Kolmogorov-Feller equation which involves Prabhakar generalized fractional operators governing the stochastic motions on the network. We analyze in d dimensions the ‘well-scaled’ diffusion limit and obtain a fractional diffusion equation which is of the same type as for a walk with Mittag-Leffler distributed waiting times. The GFPP has the potential to capture various aspects in the dynamics of certain complex systems.

scholarly journals Clustered continuous-time random walks: diffusion and relaxation consequences

2012 ◽  
Vol 468 (2142) ◽  
pp. 1615-1628 ◽  
Author(s):  
Karina Weron ◽  
Aleksander Stanislavsky ◽  
Agnieszka Jurlewicz ◽  
Mark M. Meerschaert ◽  
Hans-Peter Scheffler
Keyword(s):  
Random Walks ◽  
Power Law ◽  
Continuous Time ◽  
Waiting Times ◽  
Scaling Limit ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
The Novel ◽  
Fractional Diffusion Equations ◽  
Continuous Time Random Walks

We present a class of continuous-time random walks (CTRWs), in which random jumps are separated by random waiting times. The novel feature of these CTRWs is that the jumps are clustered. This introduces a coupled effect, with longer waiting times separating larger jump clusters. We show that the CTRW scaling limits are time-changed processes. Their densities solve two different fractional diffusion equations, depending on whether the waiting time is coupled to the preceding jump, or the following one. These fractional diffusion equations can be used to model all types of experimentally observed two power-law relaxation patterns. The parameters of the scaling limit process determine the power-law exponents and loss peak frequencies.

scholarly journals Computer simulation for the growing probability of additional offspring with an advantageous reversal allele in the decoupled continuous-time mutation–selection model

2016 ◽  
Vol 27 (06) ◽  
pp. 1650070
Author(s):  
Wonpyong Gill
Keyword(s):  
Computer Simulation ◽  
Continuous Time ◽  
Selective Advantage ◽  
Selection Model ◽  
Sequence Length ◽  
Theoretical Formula ◽  
Fitness Parameters ◽  
Population Sizes ◽  
Fitness Parameter ◽  
Allele Model

This study calculated the growing probability of additional offspring with the advantageous reversal allele in an asymmetric sharply-peaked landscape using the decoupled continuous-time mutation–selection model. The growing probability was calculated for various population sizes, N, sequence lengths, L, selective advantages, s, fitness parameters, k and measuring parameters, C. The saturated growing probability in the stochastic region was approximately the effective selective advantage, [Formula: see text], when [Formula: see text] and [Formula: see text]. The present study suggests that the growing probability in the stochastic region in the decoupled continuous-time mutation–selection model can be described using the theoretical formula for the growing probability in the Moran two-allele model. The selective advantage ratio, which represents the ratio of the effective selective advantage to the selective advantage, does not depend on the population size, selective advantage, measuring parameter and fitness parameter; instead the selective advantage ratio decreases with the increasing sequence length.

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.

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