scholarly journals Exact representations for tumour incidence for some density-dependent models

2005 ◽  
Vol 2005 (16) ◽  
pp. 2655-2667
Author(s):  
P. R. Parthasarathy ◽  
Klaus Dietz
Keyword(s):  
Continued Fraction ◽  
Death Process ◽  
Transition Rates ◽  
Birth And Death Process ◽  
System Of Difference Equations ◽  
Normal Cells ◽  
Death Rates ◽  
Density Dependent ◽  
Proliferation And Differentiation ◽  
Complete Solutions

Carcinogenesis is a multistage random process involving generic changes and stochastic proliferation and differentiation of normal cells and genetically altered stem cells. In this paper, we present the probability of time to tumour onset for a carcinogenesis model wherein the cells grow according to a birth and death process with density-dependent birth and death rates. This is achieved by transforming the underlying system of difference equations which results in a continued fraction. This continued fraction approach helps us to find the complete solutions. The popular Moolgavkar-Venzon-Knudson (MVK) model assumes constant birth, death, and transition rates.

A linear birth and death process under the influence of another process

1975 ◽  
Vol 12 (1) ◽  
pp. 1-17 ◽  
Author(s):  
Prem S. Puri
Keyword(s):  
Markov Process ◽  
Closed Form ◽  
Limit Distribution ◽  
Explicit Solution ◽  
Negative Integer ◽  
Death Process ◽  
Birth And Death Process ◽  
Death Rates ◽  
Time To Extinction

Let {X1 (t), X2 (t), t ≧ 0} be a bivariate birth and death (Markov) process taking non-negative integer values, such that the process {X2(t), t ≧ 0} may influence the growth of the process {X1(t), t ≧ 0}, while the process X2 (·) itself grows without any influence whatsoever of the first process. The process X2 (·) is taken to be a simple linear birth and death process with λ2 and µ2 as its birth and death rates respectively. The process X1 (·) is also assumed to be a linear birth and death process but with its birth and death rates depending on X2 (·) in the following manner: λ (t) = λ1 (θ + X2 (t)); µ(t) = µ1 (θ + X2 (t)). Here λ i, µi and θ are all non-negative constants. By studying the process X1 (·), first conditionally given a realization of the process {X2 (t), t ≧ 0} and then by unconditioning it later on by taking expectation over the process {X2 (t), t ≧ 0} we obtain explicit solution for G in closed form. Again, it is shown that a proper limit distribution of X1 (t) always exists as t→∞, except only when both λ1 > µ1 and λ2 > µ2. Also, certain problems concerning moments of the process, regression of X1 (t) on X2 (t); time to extinction, and the duration of the interaction between the two processes, etc., are studied in some detail.

Excursions of birth and death processes, orthogonal polynomials, and continued fractions

1999 ◽  
Vol 36 (3) ◽  
pp. 752-770 ◽  
Author(s):  
Fabrice Guillemin ◽  
Didier Pinchon
Keyword(s):  
Orthogonal Polynomial ◽  
Continued Fractions ◽  
Spectral Measure ◽  
Continued Fraction ◽  
Transition Probability ◽  
Polynomial System ◽  
Death Process ◽  
Probabilistic Interpretation ◽  
Birth And Death Process ◽  
Birth And Death Processes

On the basis of the Karlin and McGregor result, which states that the transition probability functions of a birth and death process can be expressed via the introduction of an orthogonal polynomial system and a spectral measure, we investigate in this paper how the Laplace transforms and the distributions of different transient characteristics related to excursions of a birth and death process can be expressed by means of the basic orthogonal polynomial system and the spectral measure. This allows us in particular to give a probabilistic interpretation of the series introduced by Stieltjes to study the convergence of the fundamental continued fraction associated with the system. Throughout the paper, we pay special attention to the case when the birth and death process is ergodic. Under the assumption that the spectrum of the spectral measure is discrete, we show how the distributions of different random variables associated with excursions depend on the fundamental continued fraction, the orthogonal polynomial system and the spectral measure.

Application of birth-and-death processes to the cascade theory

1968 ◽  
Vol 46 (10) ◽  
pp. S212-S215 ◽  
Author(s):  
K. Kobayakawa ◽  
S. Miono
Keyword(s):  
Complete Solution ◽  
Initial Energy ◽  
Death Process ◽  
Recent Experimental Data ◽  
Birth And Death Process ◽  
Birth And Death Processes ◽  
Death Rates ◽  
Electromagnetic Showers ◽  
Better Than ◽  
Number Of Particles

A birth-and-death process taking immigration into account is considered. The complete solution of the equation governing the generalized birth-and-death process when the birth and death rates λ(t)and μ(t) and also the immigration factor ν(t) may be any specified functions of the time t is given. This solution can be applied to the fluctuation problem in electromagnetic showers. The distribution function of the number of particles at given depth t with given initial energy is derived. The results obtained are compared with recent experimental data, and the agreement is much better than in past theoretical works.

SIMPLE MODEL OF ASYMMETRICAL BUSINESS CYCLES: INTERACTIVE DYNAMICS OF A LARGE NUMBER OF AGENTS WITH DISCRETE CHOICES

1998 ◽  
Vol 2 (4) ◽  
pp. 427-442 ◽  
Author(s):  
Masanao Aoki
Keyword(s):  
Business Cycles ◽  
Basins Of Attraction ◽  
Death Process ◽  
Transition Rates ◽  
Birth And Death Process ◽  
Nonlinear Functions ◽  
Stable Equilibria ◽  
Probability Of Transition ◽  
Interactive Dynamics ◽  
Passage Times

A (jump) Markov process (generalized birth-and-death process) is used to model interactions of a large number of agents subject to field-type externalities. Transition rates are (nonlinear) functions of the composition of the population of agents classified by the choices they make. The model state randomly moves from one equilibrium to another, and exhibits asymmetrical oscillations (business cycles). It is shown that the processes can have several locally stable equilibria, depending on the degree of uncertainty associated with consequences of alternative choices. There is a positive probability of transition between any pair of such basins of attraction, and mean first-passage times between equilibria can be different when different pairs of basins are calculated.

A linear birth and death process under the influence of another process

1975 ◽  
Vol 12 (01) ◽  
pp. 1-17 ◽  
Author(s):  
Prem S. Puri
Keyword(s):  
Markov Process ◽  
Closed Form ◽  
Limit Distribution ◽  
Explicit Solution ◽  
Negative Integer ◽  
Death Process ◽  
Birth And Death Process ◽  
Death Rates ◽  
Time To Extinction

Let {X 1 (t), X 2 (t), t ≧ 0} be a bivariate birth and death (Markov) process taking non-negative integer values, such that the process {X 2(t), t ≧ 0} may influence the growth of the process {X 1(t), t ≧ 0}, while the process X 2 (·) itself grows without any influence whatsoever of the first process. The process X 2 (·) is taken to be a simple linear birth and death process with λ 2 and µ 2 as its birth and death rates respectively. The process X 1 (·) is also assumed to be a linear birth and death process but with its birth and death rates depending on X 2 (·) in the following manner: λ (t) = λ 1 (θ + X 2 (t)); µ(t) = µ 1 (θ + X 2 (t)). Here λ i, µi and θ are all non-negative constants. By studying the process X 1 (·), first conditionally given a realization of the process {X 2 (t), t ≧ 0} and then by unconditioning it later on by taking expectation over the process {X 2 (t), t ≧ 0} we obtain explicit solution for G in closed form. Again, it is shown that a proper limit distribution of X 1 (t) always exists as t→∞, except only when both λ 1 > µ 1 and λ 2 > µ 2. Also, certain problems concerning moments of the process, regression of X 1 (t) on X 2 (t); time to extinction, and the duration of the interaction between the two processes, etc., are studied in some detail.

scholarly journals A Birth and Death Process Related to the Rogers–Ramanujan Continued Fraction

1998 ◽  
Vol 224 (2) ◽  
pp. 297-315 ◽  
Author(s):  
P.R Parthasarathy ◽  
R.B Lenin ◽  
W Schoutens ◽  
W Van Assche
Keyword(s):  
Continued Fraction ◽  
Death Process ◽  
Birth And Death Process

Excursions of birth and death processes, orthogonal polynomials, and continued fractions

1999 ◽  
Vol 36 (03) ◽  
pp. 752-770 ◽  
Author(s):  
Fabrice Guillemin ◽  
Didier Pinchon
Keyword(s):  
Orthogonal Polynomial ◽  
Continued Fractions ◽  
Spectral Measure ◽  
Continued Fraction ◽  
Transition Probability ◽  
Polynomial System ◽  
Death Process ◽  
Probabilistic Interpretation ◽  
Birth And Death Process ◽  
Birth And Death Processes

On the basis of the Karlin and McGregor result, which states that the transition probability functions of a birth and death process can be expressed via the introduction of an orthogonal polynomial system and a spectral measure, we investigate in this paper how the Laplace transforms and the distributions of different transient characteristics related to excursions of a birth and death process can be expressed by means of the basic orthogonal polynomial system and the spectral measure. This allows us in particular to give a probabilistic interpretation of the series introduced by Stieltjes to study the convergence of the fundamental continued fraction associated with the system. Throughout the paper, we pay special attention to the case when the birth and death process is ergodic. Under the assumption that the spectrum of the spectral measure is discrete, we show how the distributions of different random variables associated with excursions depend on the fundamental continued fraction, the orthogonal polynomial system and the spectral measure.

Some further results on the birth-and-death process and its integral

1968 ◽  
Vol 64 (1) ◽  
pp. 141-154 ◽  
Author(s):  
Prem S. Puri
Keyword(s):  
Population Size ◽  
Joint Distribution ◽  
Probability Generating Function ◽  
Weighted Average ◽  
Limiting Distribution ◽  
Death Process ◽  
Birth And Death Process ◽  
Chi Square ◽  
Death Rates ◽  
Image Position

AbstractIn a simple homogeneous birth-and-death process with λ and μ as the constant birth and death rates respectively, let X(t) denote the population size at time t, Z(t) the number of deaths and N(t) the number of events (births and deaths combined) occurring during (0, t). Also let . The results obtained include the following:(a) An explicit formula for the characteristic quasi-probability generating function of the joint distribution of X(t), Y(t) and Z(t).(b) Let X(0) = 1. It is shown that, if t → ∞ while λ ≤ μ, N(t) ↑ N a.s., where N takes only positive odd integral values. If λ > μ, then P[N(t) ↑ ∞] = 1 − μ/λ. Given that N(t)∞, the limiting distribution of N(t) is similar to that of N. It was reported earlier (Puri (11)), that the limiting distribution of Y(t) is a weighted average of certain chi-square distributions. It is now found that these weights are nothing but the probabilities P[N = 2k + 1] (k = 0, 1,…).(c) Let λ = μ, and MXω), MYω and MZω be defined as in (36), then aswhere the c.f. of (X*; Y*; Z*) is given by (38).(d) Exact expressions for the p.d.f. of Y(t) are derived for the cases (i) λ = 0, μ > 0, (ii) λ > 0, μ = 0. For the case (iii) λ gt; 0, μ > 0, since the complete expression is complicated, only the procedure of derivation is indicated.(e) Finally, it is shown that the regressions of Y(t) and of Z(t) on X(t) are linear for X(t) ≥ 1.

The asymptotic behaviour of birth and death and some related processes

1975 ◽  
Vol 7 (01) ◽  
pp. 28-43 ◽  
Author(s):  
Andrew D. Barbour
Keyword(s):  
Asymptotic Behaviour ◽  
Central Limit ◽  
Markov Processes ◽  
Continuous Time ◽  
Death Process ◽  
Transition Rates ◽  
Birth And Death Process ◽  
Iterated Logarithm ◽  
The Matrix ◽  
Positive Integers

The paper examines those continuous time Markov processes Z(·) on the positive integers which have the ‘skip free upwards’ property, with regard to their asymptotic behaviour in the event of Z(t) tending to infinity. The behaviour is characterised in terms of the convergence or divergence of an appropriate function of Z(t), and the description is improved by central limit and iterated logarithm theorems. The conditions of the theorems are expressed entirely in terms of the matrix Q of instantaneous transition rates for Z(·). The method is applied, by way of example, to the super-critical linear birth and death process.

The asymptotic behaviour of birth and death and some related processes

1975 ◽  
Vol 7 (1) ◽  
pp. 28-43 ◽  
Author(s):  
Andrew D. Barbour
Keyword(s):  
Asymptotic Behaviour ◽  
Central Limit ◽  
Markov Processes ◽  
Continuous Time ◽  
Death Process ◽  
Transition Rates ◽  
Birth And Death Process ◽  
Iterated Logarithm ◽  
The Matrix ◽  
Positive Integers

The paper examines those continuous time Markov processes Z(·) on the positive integers which have the ‘skip free upwards’ property, with regard to their asymptotic behaviour in the event of Z(t) tending to infinity. The behaviour is characterised in terms of the convergence or divergence of an appropriate function of Z(t), and the description is improved by central limit and iterated logarithm theorems. The conditions of the theorems are expressed entirely in terms of the matrix Q of instantaneous transition rates for Z(·). The method is applied, by way of example, to the super-critical linear birth and death process.

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