Some new results in the mathematical theory of phage-reproduction

1969 ◽  
Vol 6 (3) ◽  
pp. 493-504 ◽  
Author(s):  
Prem S. Puri
Keyword(s):  
Mathematical Theory ◽  
Joint Distribution ◽  
Fixed Time ◽  
Fixed Number ◽  
Death Process ◽  
Birth And Death Process ◽  
Random Event ◽  
Realistic Assumption ◽  
Phage Multiplication ◽  
Burst Time

SummaryIn the theory of phage reproduction, the mathematical models considered thus far (see Gani [5]) assume that the bacterial burst occurs a fixed time after infection, after a fixed number of generations of phage multiplication, or when the number of mature bacteriophages has reached a fixed threshold. In the present paper, a more realistic assumption is considered: given that until timetthe bacterial burst has not taken place, its occurence betweentandt+ Δtis a random event with probabilityf(· |t)Δt+o(Δt), wherefis a non-negative and non-decreasing function of the numberX(t) of vegetative phages and ofZ(t), the number of mature bacteriophages at timet.More specifically it is assumed thatf=b(t)X(t) +c(t)Z(t) withb(t),c(t) ≦ 0. HereX(t) denotes the survivors in a linear birth and death process andZ(t) the number of deaths until timet.The joint distribution ofXTandZT, the respective numbers of vegetative and mature bacteriophages at the burst time is considered. The distribution ofZTis then fitted to some observed data of Delbrück [2].

Some new results in the mathematical theory of phage-reproduction

1969 ◽  
Vol 6 (03) ◽  
pp. 493-504 ◽  
Author(s):  
Prem S. Puri
Keyword(s):  
Mathematical Theory ◽  
Joint Distribution ◽  
Fixed Time ◽  
Fixed Number ◽  
Death Process ◽  
Birth And Death Process ◽  
Random Event ◽  
Realistic Assumption ◽  
Phage Multiplication ◽  
Burst Time

Summary In the theory of phage reproduction, the mathematical models considered thus far (see Gani [5]) assume that the bacterial burst occurs a fixed time after infection, after a fixed number of generations of phage multiplication, or when the number of mature bacteriophages has reached a fixed threshold. In the present paper, a more realistic assumption is considered: given that until time t the bacterial burst has not taken place, its occurence between tand t + Δt is a random event with probability f(· | t)Δt + o(Δt), where f is a non-negative and non-decreasing function of the number X(t) of vegetative phages and of Z(t), the number of mature bacteriophages at time t. More specifically it is assumed that f = b(t)X(t) + c(t)Z(t) with b(t), c(t) ≦ 0. Here X(t) denotes the survivors in a linear birth and death process and Z(t) the number of deaths until time t. The joint distribution of XT and ZT , the respective numbers of vegetative and mature bacteriophages at the burst time is considered. The distribution of ZT is then fitted to some observed data of Delbrück [2].

On the age of a randomly picked individual in a linear birth-and-death process

2018 ◽  
Vol 55 (1) ◽  
pp. 82-93 ◽  
Author(s):  
Fabian Kück ◽  
Dominic Schuhmacher
Keyword(s):  
Distribution Function ◽  
Exponential Distribution ◽  
Total Variation ◽  
Cumulative Distribution Function ◽  
Fixed Time ◽  
Cumulative Distribution ◽  
Death Process ◽  
Birth And Death Process

Abstract We consider the distribution of the age of an individual picked uniformly at random at some fixed time in a linear birth-and-death process. By exploiting a bijection between the birth-and-death tree and a contour process, we derive the cumulative distribution function for this distribution. In the critical and supercritical cases, we also give rates for the convergence in terms of the total variation and other metrics towards the appropriate exponential distribution.

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 extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

A stochastic process related to language change

1970 ◽  
Vol 7 (01) ◽  
pp. 69-78 ◽  
Author(s):  
Barron Brainerd
Keyword(s):  
Stochastic Process ◽  
Language Change ◽  
Death Process ◽  
Birth And Death Process ◽  
Standard Methods

The purpose of this note is two-fold. First, to introduce the mathematical reader to a group of problems in the study of language change which has received little attention from mathematicians and probabilists. Secondly, to introduce a birth and death process, arising naturally out of this group of problems, which has received little attention in the literature. This process can be solved using the standard methods and the solution is exhibited here.

Multi-Location Inventory Model Considering Bidirectional and Unidirectional Transshipments Simultaneously

2013 ◽  
Vol 694-697 ◽  
pp. 2742-2745
Author(s):  
Jin Hong Zhong ◽  
Yun Zhou
Keyword(s):  
Process Model ◽  
Approximate Calculation ◽  
Inventory Model ◽  
Inventory System ◽  
Death Process ◽  
Birth And Death Process ◽  
Inventory Models ◽  
Continuous Review ◽  
Poisson Demand ◽  
Queue Theory

Abstract. A cross-regional multi-site inventory system with independent Poisson demand and continuous review (S-1,S) policy, in which there is bidirectional transshipment between the locations at the same area, and unidirectional transshipment between the locations at the different area. According to the M/G/S/S queue theory, birth and death process model and approximate calculation policy, we established inventory models respectively for the loss sales case and backorder case, and designed corresponding procedures to solve them. Finally, we verify the effectiveness of proposed models and methods by means of a lot of contrast experiments.

scholarly journals Stochastic Models for a Chemostat and Long-Time Behavior

2013 ◽  
Vol 45 (03) ◽  
pp. 822-836 ◽  
Author(s):  
Pierre Collet ◽  
Servet Martínez ◽  
Sylvie Méléard ◽  
Jaime San Martín
Keyword(s):  
Population Size ◽  
Nutrient Concentration ◽  
Bacterial Population ◽  
Fixed Number ◽  
Death Process ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Nutrient Distribution ◽  
Long Time ◽  
Number Of Individuals

We introduce two stochastic chemostat models consisting of a coupled population-nutrient process reflecting the interaction between the nutrient and the bacteria in the chemostat with finite volume. The nutrient concentration evolves continuously but depends on the population size, while the population size is a birth-and-death process with coefficients depending on time through the nutrient concentration. The nutrient is shared by the bacteria and creates a regulation of the bacterial population size. The latter and the fluctuations due to the random births and deaths of individuals make the population go almost surely to extinction. Therefore, we are interested in the long-time behavior of the bacterial population conditioned to nonextinction. We prove the global existence of the process and its almost-sure extinction. The existence of quasistationary distributions is obtained based on a general fixed-point argument. Moreover, we prove the absolute continuity of the nutrient distribution when conditioned to a fixed number of individuals and the smoothness of the corresponding densities.

AN AGE-DEPENDENT BIRTH AND DEATH PROCESS

Biometrika ◽  
1955 ◽  
Vol 42 (3-4) ◽  
pp. 291-306 ◽  
Author(s):  
W. A. O'N WAUGH
Keyword(s):  
Death Process ◽  
Birth And Death Process ◽  
Age Dependent

On a property of finite-state birth and death processes

1986 ◽  
Vol 23 (04) ◽  
pp. 859-866
Author(s):  
A. J. Branford
Keyword(s):  
Simple Proof ◽  
Death Process ◽  
Birth And Death Process ◽  
Birth And Death Processes ◽  
Converse Result ◽  
Finite State ◽  
Event Times

A simple proof is given of the result that the ‘overflow' from a finite-state birth and death process is a renewal stream characterized by hyperexponential inter-event times. Our structure is utilized to give a converse result that any hyperexponential renewal stream can be so produced as the overflow from a finite-state birth and death process.

Sign in / Sign up

Export Citation Format

Share Document