On age dependent branching processes

1966 ◽  
Vol 3 (2) ◽  
pp. 383-402 ◽  
Author(s):  
Howard Weiner
Keyword(s):  
Branching Processes ◽  
Asymptotic Properties ◽  
Age Dependent

This paper deals with asymptotic properties of various models of age dependent branching processes, relying heavily on Harris [3].

Applications of the age distribution in age dependent branching processes

1966 ◽  
Vol 3 (1) ◽  
pp. 179-201 ◽  
Author(s):  
Howard J. Weiner
Keyword(s):  
Markov Process ◽  
Markov Models ◽  
Branching Processes ◽  
Asymptotic Properties ◽  
Age Distribution ◽  
Lifetime Distributions ◽  
Age Dependent ◽  
Semi Markov Process ◽  
Model Section ◽  
Number Of Cells

This paper considers asymptotic properties of increasing population age dependent branching processes which have a limiting age distribution. In Section I, the Bellman-Harris model [2] is altered in accord with a suggestion by Kendall [3]. The effect of this modification on the applicable results of [3] is indicated. An extension to the case where each cell passes through a sequence of states to mitosis or proceeds from state to state in accord with a semi-Markov process to mitosis is considered. Section 1.1 considers asymptotic moments for the total number of cells. Section 1.2 treats moments in a sequence-of-states model [4]. Section 1.3 extends the results of 1.2 to a semi-Markov model. Section 1.4 treats various asymptotic lifetime distributions and fraction of cells in a given state for both the sequence-of-states and semi-Markov models.

On age dependent branching processes

1966 ◽  
Vol 3 (02) ◽  
pp. 383-402 ◽  
Author(s):  
Howard Weiner
Keyword(s):  
Branching Processes ◽  
Asymptotic Properties ◽  
Age Dependent

This paper deals with asymptotic properties of various models of age dependent branching processes, relying heavily on Harris [3].

Applications of the age distribution in age dependent branching processes

1966 ◽  
Vol 3 (01) ◽  
pp. 179-201 ◽  
Author(s):  
Howard J. Weiner
Keyword(s):  
Markov Process ◽  
Markov Models ◽  
Branching Processes ◽  
Asymptotic Properties ◽  
Age Distribution ◽  
Lifetime Distributions ◽  
Age Dependent ◽  
Semi Markov Process ◽  
Model Section ◽  
Number Of Cells

This paper considers asymptotic properties of increasing population age dependent branching processes which have a limiting age distribution. In Section I, the Bellman-Harris model [2] is altered in accord with a suggestion by Kendall [3]. The effect of this modification on the applicable results of [3] is indicated. An extension to the case where each cell passes through a sequence of states to mitosis or proceeds from state to state in accord with a semi-Markov process to mitosis is considered. Section 1.1 considers asymptotic moments for the total number of cells. Section 1.2 treats moments in a sequence-of-states model [4]. Section 1.3 extends the results of 1.2 to a semi-Markov model. Section 1.4 treats various asymptotic lifetime distributions and fraction of cells in a given state for both the sequence-of-states and semi-Markov models.

Multi-type age-dependent branching processes as models of metastasis evolution

2019 ◽  
Vol 35 (3) ◽  
pp. 284-299
Author(s):  
Maroussia Slavtchova-Bojkova ◽  
Kaloyan Vitanov
Keyword(s):  
Branching Processes ◽  
Age Dependent

Stochastic models in cell kinetics

1988 ◽  
Vol 25 (A) ◽  
pp. 91-111
Author(s):  
Peter J. Brockwell
Keyword(s):  
Life Cycle ◽  
Stochastic Models ◽  
Death Rate ◽  
Cell Kinetics ◽  
Branching Processes ◽  
Experimental Studies ◽  
Renewal Theory ◽  
Biological Cell ◽  
Age Dependent

We discuss the role of stochastic processes in modelling the life-cycle of a biological cell and the growth of cell populations. Results for multiphase age-dependent branching processes have proved invaluable for the interpretation of many of the basic experimental studies of the life-cycle. Moreover problems from cell kinetics, in particular those related to diurnal rhythm in cell-growth, have stimulated research into ‘periodic' renewal theory, and the asymptotic behaviour of populations of cells with periodic death rate.

A renewal density theorem in the multi-dimensional case

1967 ◽  
Vol 4 (1) ◽  
pp. 62-76 ◽  
Author(s):  
Charles J. Mode
Keyword(s):  
Density Function ◽  
Covariance Matrix ◽  
Branching Processes ◽  
Random Variable ◽  
Density Theorem ◽  
Positive Definite ◽  
Dimensional Case ◽  
Age Dependent ◽  
Image Position ◽  
Mean Vector

SummaryIn this note a renewal density theorem in the multi-dimensional case is formulated and proved. Let f(x) be the density function of a p-dimensional random variable with positive mean vector μ and positive-definite covariance matrix Σ, let hn(x) be the n-fold convolution of f(x) with itself, and set Then for arbitrary choice of integers k1, …, kp–1 distinct or not in the set (1, 2, …, p), it is shown that under certain conditions as all elements in the vector x = (x1, …, xp) become large. In the above expression μ‵ is interpreted as a row vector and μ a column vector. An application to the theory of a class of age-dependent branching processes is also presented.

scholarly journals Subcritical Sevastyanov branching processes with nonhomogeneous Poisson immigration

2017 ◽  
Vol 54 (2) ◽  
pp. 569-587 ◽  
Author(s):  
Ollivier Hyrien ◽  
Kosto V. Mitov ◽  
Nikolay M. Yanev
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Branching Processes ◽  
Asymptotic Properties ◽  
Cell Populations ◽  
Subcritical Case ◽  
Immigration Process ◽  
Large Numbers

Abstract We consider a class of Sevastyanov branching processes with nonhomogeneous Poisson immigration. These processes relax the assumption required by the Bellman–Harris process which imposes the lifespan and offspring of each individual to be independent. They find applications in studies of the dynamics of cell populations. In this paper we focus on the subcritical case and examine asymptotic properties of the process. We establish limit theorems, which generalize classical results due to Sevastyanov and others. Our key findings include a novel law of large numbers and a central limit theorem which emerge from the nonhomogeneity of the immigration process.

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