scholarly journals The Kolmogorov equation in the stochastic fragmentation theory and branching processes with infinite collection of particle types

2006 ◽  
Vol 2006 ◽  
pp. 1-10
Author(s):  
R. Ye. Brodskii ◽  
Yu. P. Virchenko
Keyword(s):  
Stochastic Model ◽  
Basic Equation ◽  
Branching Process ◽  
Branching Processes ◽  
Fragmentation Process ◽  
Kolmogorov Equation ◽  
Infinite Collection ◽  
Kolmogorov Theory

The stochastic model for the description of the so-called fragmentation process in frameworks of Kolmogorov approach is proposed. This model is represented as the branching process with continuum set(0,∞)of particle types. Each typer∈(0,∞)corresponds to the set of fragments having the sizer. It is proved that the branching condition of this process represents the basic equation of the Kolmogorov theory.

scholarly journals Reduced Branching Processes with Very Heavy Tails

2008 ◽  
Vol 45 (1) ◽  
pp. 190-200 ◽  
Author(s):  
Andreas N. Lagerås ◽  
Serik Sagitov
Keyword(s):  
Stochastic Model ◽  
Branching Process ◽  
Critical Case ◽  
Heavy Tails ◽  
Regularity Condition ◽  
Branching Processes ◽  
Time Interval ◽  
Biological Population ◽  
Long Time ◽  
Heavy Tailed

The reduced Markov branching process is a stochastic model for the genealogy of an unstructured biological population. Its limit behavior in the critical case is well studied for the Zolotarev-Slack regularity parameter α ∈ (0, 1]. We turn to the case of very heavy-tailed reproduction distribution α = 0 assuming that Zubkov's regularity condition holds with parameter β ∈ (0, ∞). Our main result gives a new asymptotic pattern for the reduced branching process conditioned on nonextinction during a long time interval.

scholarly journals Reduced Branching Processes with Very Heavy Tails

2008 ◽  
Vol 45 (01) ◽  
pp. 190-200
Author(s):  
Andreas N. Lagerås ◽  
Serik Sagitov
Keyword(s):  
Stochastic Model ◽  
Branching Process ◽  
Critical Case ◽  
Heavy Tails ◽  
Regularity Condition ◽  
Branching Processes ◽  
Time Interval ◽  
Biological Population ◽  
Long Time ◽  
Heavy Tailed

The reduced Markov branching process is a stochastic model for the genealogy of an unstructured biological population. Its limit behavior in the critical case is well studied for the Zolotarev-Slack regularity parameter α ∈ (0, 1]. We turn to the case of very heavy-tailed reproduction distribution α = 0 assuming that Zubkov's regularity condition holds with parameter β ∈ (0, ∞). Our main result gives a new asymptotic pattern for the reduced branching process conditioned on nonextinction during a long time interval.

Some Contributions to the Theory of Near-Critical Bisexual Branching Processes

2007 ◽  
Vol 44 (02) ◽  
pp. 492-505
Author(s):  
M. Molina ◽  
M. Mota ◽  
A. Ramos
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Positive Probability ◽  
Limit Laws ◽  
Bisexual Branching Processes ◽  
Probabilistic Evolution

We investigate the probabilistic evolution of a near-critical bisexual branching process with mating depending on the number of couples in the population. We determine sufficient conditions which guarantee either the almost sure extinction of such a process or its survival with positive probability. We also establish some limiting results concerning the sequences of couples, females, and males, suitably normalized. In particular, gamma, normal, and degenerate distributions are proved to be limit laws. The results also hold for bisexual Bienaymé–Galton–Watson processes, and can be adapted to other classes of near-critical bisexual branching processes.

Stochastic Modelling of Particle Segregation in a Horizontal Drum Mixer

1992 ◽  
Vol 57 (10) ◽  
pp. 2100-2112 ◽  
Author(s):  
Vladimír Kudrna ◽  
Pavel Hasal ◽  
Andrzej Rochowiecki
Keyword(s):  
Differential Equation ◽  
Diffusion Coefficient ◽  
Stochastic Model ◽  
Diffusion Process ◽  
Stochastic Modelling ◽  
Solid Particles ◽  
Kolmogorov Equation ◽  
Particle Segregation ◽  
Drum Mixer ◽  
Horizontal Drum

A process of segregation of two distinct fractions of solid particles in a rotating horizontal drum mixer was described by stochastic model assuming the segregation to be a diffusion process with varying diffusion coefficient. The model is based on description of motion of particles inside the mixer by means of a stochastic differential equation. Results of stochastic modelling were compared to the solution of the corresponding Kolmogorov equation and to results of earlier carried out experiments.

scholarly journals Skeletal stochastic differential equations for superprocesses

2020 ◽  
Vol 57 (4) ◽  
pp. 1111-1134
Author(s):  
Dorottya Fekete ◽  
Joaquin Fontbona ◽  
Andreas E. Kyprianou
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Subcritical Case ◽  
Markov Branching Process ◽  
Continuous State ◽  
Common Framework ◽  
Infinite Line ◽  
Current Article ◽  
Lines Of Descent ◽  
Skeletal Representation

AbstractIt is well understood that a supercritical superprocess is equal in law to a discrete Markov branching process whose genealogy is dressed in a Poissonian way with immigration which initiates subcritical superprocesses. The Markov branching process corresponds to the genealogical description of prolific individuals, that is, individuals who produce eternal genealogical lines of descent, and is often referred to as the skeleton or backbone of the original superprocess. The Poissonian dressing along the skeleton may be considered to be the remaining non-prolific genealogical mass in the superprocess. Such skeletal decompositions are equally well understood for continuous-state branching processes (CSBP).In a previous article [16] we developed an SDE approach to study the skeletal representation of CSBPs, which provided a common framework for the skeletal decompositions of supercritical and (sub)critical CSBPs. It also helped us to understand how the skeleton thins down onto one infinite line of descent when conditioning on survival until larger and larger times, and eventually forever.Here our main motivation is to show the robustness of the SDE approach by expanding it to the spatial setting of superprocesses. The current article only considers supercritical superprocesses, leaving the subcritical case open.

Branching processes and functional differential equations determining steady-size distributions in cell populations

1995 ◽  
Vol 32 (01) ◽  
pp. 1-10
Author(s):  
Ziad Taib
Keyword(s):  
Differential Equation ◽  
Branching Process ◽  
Branching Processes ◽  
Functional Differential Equations ◽  
Cell Populations ◽  
Size Distributions ◽  
Multitype Branching Process ◽  
Functional Differential ◽  
Intuitive Interpretation ◽  
Infinite Sum

The functional differential equation y′(x) = ay(λx) + by(x) arises in many different situations. The purpose of this note is to show how it arises in some multitype branching process cell population models. We also show how its solution can be given an intuitive interpretation as the probability density function of an infinite sum of independent but not identically distributed random variables.

Continuity for multi-type branching processes with varying environments

1999 ◽  
Vol 36 (01) ◽  
pp. 139-145 ◽  
Author(s):  
Owen Dafydd Jones
Keyword(s):  
Linear Combination ◽  
Branching Process ◽  
Branching Processes ◽  
Random Variables ◽  
Main Tool ◽  
Concentration Function ◽  
Independent Random Variables ◽  
Varying Environments

Conditions are derived for the components of the normed limit of a multi-type branching process with varying environments, to be continuous on (0, ∞). The main tool is an inequality for the concentration function of sums of independent random variables, due originally to Petrov. Using this, we show that if there is a discontinuity present, then a particular linear combination of the population types must converge to a non-random constant (Equation (1)). Ensuring this can not happen provides the desired continuity conditions.

scholarly journals Interacting branching processes and linear file-sharing networks

2010 ◽  
Vol 42 (03) ◽  
pp. 834-854 ◽  
Author(s):  
Lasse Leskelä ◽  
Philippe Robert ◽  
Florian Simatos
Keyword(s):  
Communication Network ◽  
Asymptotic Behavior ◽  
Distributed Systems ◽  
Stochastic Model ◽  
Branching Processes ◽  
File Sharing ◽  
Independent Interest ◽  
Constant Flow

File-sharing networks are distributed systems used to disseminate files among nodes of a communication network. The general simple principle of these systems is that once a node has retrieved a file, it may become a server for this file. In this paper, the capacity of these networks is analyzed with a stochastic model when there is a constant flow of incoming requests for a given file. It is shown that the problem can be solved by analyzing the asymptotic behavior of a class of interacting branching processes. Several results of independent interest concerning these branching processes are derived and then used to study the file-sharing systems.

scholarly journals A Stochastic Model for Virus Growth in a Cell Population

2014 ◽  
Vol 51 (3) ◽  
pp. 599-612 ◽  
Author(s):  
J. E. Björnberg ◽  
T. Britton ◽  
E. I. Broman ◽  
E. Natan
Keyword(s):  
Stochastic Model ◽  
Host Cell ◽  
Cell Population ◽  
Branching Process ◽  
Virus Population ◽  
Virus Growth ◽  
Optimal Balance ◽  
Large Numbers ◽  
A Cell

In this work we introduce a stochastic model for the spread of a virus in a cell population where the virus has two ways of spreading: either by allowing its host cell to live and duplicate, or by multiplying in large numbers within the host cell, causing the host cell to burst and thereby let the virus enter new uninfected cells. The model is a kind of interacting Markov branching process. We focus in particular on the probability that the virus population survives and how this depends on a certain parameter λ which quantifies the ‘aggressiveness’ of the virus. Our main goal is to determine the optimal balance between aggressive growth and long-term success. Our analysis shows that the optimal strategy of the virus (in terms of survival) is obtained when the virus has no effect on the host cell's life cycle, corresponding to λ = 0. This is in agreement with experimental data about real viruses.

scholarly journals Simulating the impact of vaccination rates on the initial stages of a COVID-19 outbreak in New Zealand (Aotearoa) with a stochastic model

2021 ◽  
Author(s):  
Leighton M Watson
Keyword(s):  
New Zealand ◽  
Stochastic Model ◽  
Process Model ◽  
Branching Process ◽  
Total Population ◽  
Median Number ◽  
Vaccination Rate ◽  
Vaccination Rates ◽  
The Impact ◽  
Different Levels

Aim: The August 2021 COVID-19 outbreak in Auckland has caused the New Zealand government to transition from an elimination strategy to suppression, which relies heavily on high vaccination rates in the population. As restrictions are eased and as COVID-19 leaks through the Auckland boundary, there is a need to understand how different levels of vaccination will impact the initial stages of COVID-19 outbreaks that are seeded around the country. Method: A stochastic branching process model is used to simulate the initial spread of a COVID-19 outbreak for different vaccination rates. Results: High vaccination rates are effective at minimizing the number of infections and hospitalizations. Increasing vaccination rates from 20% (approximate value at the start of the August 2021 outbreak) to 80% (approximate proposed target) of the total population can reduce the median number of infections that occur within the first four weeks of an outbreak from 1011 to 14 (25th and 75th quantiles of 545-1602 and 2-32 for V=20% and V=80%, respectively). As the vaccination rate increases, the number of breakthrough infections (infections in fully vaccinated individuals) and hospitalizations of vaccinated individuals increases. Unvaccinated individuals, however, are 3.3x more likely to be infected with COVID-19 and 25x more likely to be hospitalized. Conclusion: This work demonstrates the importance of vaccination in protecting individuals from COVID-19, preventing high caseloads, and minimizing the number of hospitalizations and hence limiting the pressure on the healthcare system.

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