Branching processes in generalized autoregressive conditional environments

2016 ◽  
Vol 48 (4) ◽  
pp. 1211-1234 ◽  
Author(s):  
Irene Hueter
Keyword(s):  
Ebola Virus ◽  
Negative Binomial ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Random Environments ◽  
Offspring Number ◽  
Generalized Poisson ◽  
Novel Approach ◽  
Necessary And Sufficient ◽  
Near Extinction

AbstractBranching processes in random environments have been widely studied and applied to population growth systems to model the spread of epidemics, infectious diseases, cancerous tumor growth, and social network traffic. However, Ebola virus, tuberculosis infections, and avian flu grow or change at rates that vary with time—at peak rates during pandemic time periods, while at low rates when near extinction. The branching processes in generalized autoregressive conditional environments we propose provide a novel approach to branching processes that allows for such time-varying random environments and instances of peak growth and near extinction-type rates. Offspring distributions we consider to illustrate the model include the generalized Poisson, binomial, and negative binomial integer-valued GARCH models. We establish conditions on the environmental process that guarantee stationarity and ergodicity of the mean offspring number and environmental processes and provide equations from which their variances, autocorrelation, and cross-correlation functions can be deduced. Furthermore, we present results on fundamental questions of importance to these processes—the survival-extinction dichotomy, growth behavior, necessary and sufficient conditions for noncertain extinction, characterization of the phase transition between the subcritical and supercritical regimes, and survival behavior in each phase and at criticality.

scholarly journals Multitype branching processes in random environments

1971 ◽  
Vol 8 (1) ◽  
pp. 17-31 ◽  
Author(s):  
Edward W. Weissner
Keyword(s):  
Random Environment ◽  
Branching Process ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Random Environments ◽  
Restrictive Assumption ◽  
Multitype Branching Processes ◽  
Watson Process ◽  
Necessary And Sufficient

Consider the classical Galton-Watson process described by Harris ((1963), Chapter 1). Recently this model has been generalized in Smith (1968), Smith and Wilkinson (1969), and Wilkinson (1967). They removed the restrictive assumption that the particles always divide in accordance with the same p.g.f. Instead, they assumed that at each unit of time, Nature be allowed to choose a p.g.f. from a class of p.g.f.'s, independently of the population, past and present, and the previously selected p.g.f.'s, which would then be assigned to the present population. Each particle of the present population would then split, independently of the others, in accordance with the selected p.g.f. This process, called a branching process in a random environment (BPRE), is clearly more applicable than the Galton-Watson process. Moreover, Smith and Wilkinson have found necessary and sufficient conditions for almost certain extinction of the BPRE which are almost as easy to verify as those for the Galton-Watson process.

scholarly journals Multitype branching processes in random environments

1971 ◽  
Vol 8 (01) ◽  
pp. 17-31 ◽  
Author(s):  
Edward W. Weissner
Keyword(s):  
Random Environment ◽  
Branching Process ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Random Environments ◽  
Restrictive Assumption ◽  
Multitype Branching Processes ◽  
Watson Process ◽  
Necessary And Sufficient

Consider the classical Galton-Watson process described by Harris ((1963), Chapter 1). Recently this model has been generalized in Smith (1968), Smith and Wilkinson (1969), and Wilkinson (1967). They removed the restrictive assumption that the particles always divide in accordance with the same p.g.f. Instead, they assumed that at each unit of time, Nature be allowed to choose a p.g.f. from a class of p.g.f.'s, independently of the population, past and present, and the previously selected p.g.f.'s, which would then be assigned to the present population. Each particle of the present population would then split, independently of the others, in accordance with the selected p.g.f. This process, called a branching process in a random environment (BPRE), is clearly more applicable than the Galton-Watson process. Moreover, Smith and Wilkinson have found necessary and sufficient conditions for almost certain extinction of the BPRE which are almost as easy to verify as those for the Galton-Watson process.

scholarly journals Impulsive controllability of linear dynamical systems with applications to maneuvers of spacecraft

1996 ◽  
Vol 2 (4) ◽  
pp. 277-299 ◽  
Author(s):  
Xinzhi Liu ◽  
Allan R. Willms
Keyword(s):  
Dynamical Systems ◽  
Discrete Time ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear Dynamical Systems ◽  
Novel Approach ◽  
Necessary And Sufficient

Necessary and sufficient conditions for impulsive controllability of linear dynamical systems are obtained, which provide a novel approach to problems that are basically defined by continuous dynamical systems, but on which only discrete-time actions are exercised. As an application, impulsive maneuvering of a spacecraft is discussed.

On the class of controlled branching processes with random control functions

2002 ◽  
Vol 39 (4) ◽  
pp. 804-815 ◽  
Author(s):  
M. González ◽  
M. Molina ◽  
I. Del Puerto
Keyword(s):  
Population Size ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Control Functions ◽  
Controlled Branching Processes ◽  
Necessary And Sufficient ◽  
Random Control

In this paper, the class of controlled branching processes with random control functions introduced by Yanev (1976) is considered. For this class, necessary and sufficient conditions are established for the process to become extinct with probability 1 and the limit probabilistic behaviour of the population size, suitably normed, is investigated.

Extinction of population-size-dependent branching processes in random environments

1999 ◽  
Vol 36 (1) ◽  
pp. 146-154 ◽  
Author(s):  
Han-xing Wang
Keyword(s):  
Population Size ◽  
Branching Process ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Random Environments ◽  
Size Dependent ◽  
Branching Model

We generalize a population-size-dependent branching process to a more general branching model called the population-size-dependent branching process in random environments. For the model where {Zn}n≥0 is associated with the stationary environment ξ− = {ξn}n≥0, let B = {ω : Zn(ω) = for some n}, and q(ξ−) = P(B | ξ−, Z0 = 1). The result is that P(q(̅ξ) = 1) is either 1 or 0, and sufficient conditions for certain extinction (i.e. P(q(ξ−) = 1) = 1) and for non-certain extinction (i.e. P(q(ξ−) < 1) = 1) are obtained for the model.

Extinction of population-size-dependent branching processes in random environments

1999 ◽  
Vol 36 (01) ◽  
pp. 146-154 ◽  
Author(s):  
Han-xing Wang
Keyword(s):  
Population Size ◽  
Branching Process ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Random Environments ◽  
Size Dependent ◽  
Branching Model

We generalize a population-size-dependent branching process to a more general branching model called the population-size-dependent branching process in random environments. For the model where {Z n } n≥0 is associated with the stationary environment ξ− = {ξ n } n≥0, let B = {ω : Z n (ω) = for some n}, and q(ξ−) = P(B | ξ−, Z 0 = 1). The result is that P(q(̅ξ) = 1) is either 1 or 0, and sufficient conditions for certain extinction (i.e. P(q(ξ−) = 1) = 1) and for non-certain extinction (i.e. P(q(ξ−) &lt; 1) = 1) are obtained for the model.

A continuous time Markov branching model with random environments

1973 ◽  
Vol 5 (1) ◽  
pp. 37-54 ◽  
Author(s):  
Norman Kaplan
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Population Model ◽  
Limit Theorems ◽  
Branching Process ◽  
Sufficient Conditions ◽  
Random Environments ◽  
Markov Branching Process ◽  
Branching Model ◽  
Necessary And Sufficient

A population model is constructed which combines the ideas of a discrete time branching process with random environments and a continuous time non-homogeneous Markov branching process. The extinction problem is considered and necessary and sufficient conditions for extinction are determined. Also discussed are limit theorems for what corresponds to the supercritical case.

scholarly journals Extinction and explosion of nonlinear Markov branching processes

2007 ◽  
Vol 82 (3) ◽  
pp. 403-428 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Branching Processes ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Integral Condition ◽  
Expected Time ◽  
Time To Extinction ◽  
Positive Jump ◽  
Unique Invariant Measure ◽  
Necessary And Sufficient ◽  
Forward Equations

AbstractThis paper concerns a generalization of the Markov branching process that preserves the random walk jump chain, but admits arbitrary positive jump rates. Necessary and sufficient conditions are found for regularity, including a generalization of the Harris-Dynkin integral condition when the jump rates are reciprocals of a Hausdorff moment sequence. Behaviour of the expected time to extinction is found, and some asymptotic properties of the explosion time are given for the case where extinction cannot occur. Existence of a unique invariant measure is shown, and conditions found for unique solution of the Forward equations. The ergodicity of a resurrected version is investigated.

A continuous time Markov branching model with random environments

1973 ◽  
Vol 5 (01) ◽  
pp. 37-54 ◽  
Author(s):  
Norman Kaplan
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Population Model ◽  
Limit Theorems ◽  
Branching Process ◽  
Sufficient Conditions ◽  
Random Environments ◽  
Markov Branching Process ◽  
Branching Model ◽  
Necessary And Sufficient

A population model is constructed which combines the ideas of a discrete time branching process with random environments and a continuous time non-homogeneous Markov branching process. The extinction problem is considered and necessary and sufficient conditions for extinction are determined. Also discussed are limit theorems for what corresponds to the supercritical case.

scholarly journals The Population Dynamics of File-Sharing Peer-to-Peer Networks

2016 ◽  
Author(s):  
Dang Dinh Trang ◽  
Roland Pereczes ◽  
Sándor Molnár
Keyword(s):  
Population Dynamics ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
File Sharing ◽  
Peer To Peer ◽  
Modeling Framework ◽  
Population Extinction ◽  
P2p Systems ◽  
Necessary And Sufficient ◽  
And Control

Recently  peer-to-peer  file-sharing applications  have  shown  an  extreme  popularity  and the  workload  generated  to  the  Internet  has  been dominated  by  the  traffic  coming  from  these applications.  In  this  paper  we  develop  a  simple  but effective  mathematical  model  to  capture  the  file population  dynamics  of  such  systems.  Our  modeling framework  is  based  on  the  theory  of  branching processes. We describe analytically the behavior of the proposed  model.  The  precise  characterization  of  the necessary  and  sufficient  conditions  of  population extinction  or  explosion  is  given  based  on  the  system parameters.  We  also  present  the  expected  ratio  of active,  passive  and  dead  peers  for  the  long-term regime.  We  validate  and  demonstrate  our  results  in several  simulation  studies.  Based  on  our  results  we propose  a  number  of  engineering  guidelines  to  the design and control of file-sharing P2P systems.

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