Logistic Growth Described by Birth-Death and Diffusion Processes

We consider the logistic growth model and analyze its relevant properties, such as the limits, the monotony, the concavity, the inflection point, the maximum specific growth rate, the lag time, and the threshold crossing time problem. We also perform a comparison with other growth models, such as the Gompertz, Korf, and modified Korf models. Moreover, we focus on some stochastic counterparts of the logistic model. First, we study a time-inhomogeneous linear birth-death process whose conditional mean satisfies an equation of the same form of the logistic one. We also find a sufficient and necessary condition in order to have a logistic mean even in the presence of an absorbing endpoint. Then, we obtain and analyze similar properties for a simple birth process, too. Then, we investigate useful strategies to obtain two time-homogeneous diffusion processes as the limit of discrete processes governed by stochastic difference equations that approximate the logistic one. We also discuss an interpretation of such processes as diffusion in a suitable potential. In addition, we study also a diffusion process whose conditional mean is a logistic curve. In more detail, for the considered processes we study the conditional moments, certain indices of dispersion, the first-passage-time problem, and some comparisons among the processes.

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Joint distributions of random variables and their integrals for certain birth-death and diffusion processes

Advances in Applied Probability ◽

10.1017/s0001867800037988 ◽

1971 ◽

Vol 3 (02) ◽

pp. 339-352 ◽

Cited By ~ 3

Author(s):

J. Gani ◽

D. R. Mcneil

Keyword(s):

Diffusion Processes ◽

First Passage Time ◽

Integral Functional ◽

Passage Time ◽

Death Process ◽

Traffic Jam ◽

Joint Distributions ◽

The Cost ◽

And Storage ◽

Birth Death

For the linear growth birth-death process with parameters λ n = nλ, μ n = nμ, Puri ((1966), (1968)) has investigated the joint distribution of the number X(t) of survivors in the process and the associated integral Y(t) = ∫0 t X(τ)dτ. In particular, he has obtained limiting results as t → ∞. Recently one of us (McNeil (1970)) has derived the distribution of the integral functional W x = ∫0 Tx g{X(τ)}dτ, where T x is the first passage time to the origin in a general birth-death process with X(0) = x and g(·) is an arbitrary function. Functionals of the form W x arise naturally in traffic and storage theory; for example W x may represent the total cost of a traffic jam, or the cost of storing a commodity until expiration of the stock. Moments of such functionals were found in the case of M/G/1 and GI/M/1 queues by Gaver (1969) and Daley (1969).

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Joint distributions of random variables and their integrals for certain birth-death and diffusion processes

Advances in Applied Probability ◽

10.2307/1426175 ◽

1971 ◽

Vol 3 (2) ◽

pp. 339-352 ◽

Cited By ~ 9

Author(s):

J. Gani ◽

D. R. Mcneil

Keyword(s):

Diffusion Processes ◽

First Passage Time ◽

Integral Functional ◽

Passage Time ◽

Death Process ◽

Traffic Jam ◽

Joint Distributions ◽

The Cost ◽

And Storage ◽

Birth Death

For the linear growth birth-death process with parameters λn = nλ, μn = nμ, Puri ((1966), (1968)) has investigated the joint distribution of the number X(t) of survivors in the process and the associated integral Y(t) = ∫0tX(τ)dτ. In particular, he has obtained limiting results as t → ∞. Recently one of us (McNeil (1970)) has derived the distribution of the integral functional Wx = ∫0Txg{X(τ)}dτ, where Tx is the first passage time to the origin in a general birth-death process with X(0) = x and g(·) is an arbitrary function. Functionals of the form Wx arise naturally in traffic and storage theory; for example Wx may represent the total cost of a traffic jam, or the cost of storing a commodity until expiration of the stock. Moments of such functionals were found in the case of M/G/1 and GI/M/1 queues by Gaver (1969) and Daley (1969).

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Bell Polynomial Approach for Time-Inhomogeneous Linear Birth–Death Process with Immigration

Mathematics ◽

10.3390/math8071123 ◽

2020 ◽

Vol 8 (7) ◽

pp. 1123

Author(s):

Virginia Giorno ◽

Amelia G. Nobile

Keyword(s):

Transition Probabilities ◽

Growth Models ◽

Death Process ◽

Bell Polynomials ◽

Bell Polynomial ◽

Conditional Mean ◽

Finite State ◽

Potential Applications ◽

Intensity Functions ◽

Birth Death

We considered the time-inhomogeneous linear birth–death processes with immigration. For these processes closed form expressions for the transition probabilities were obtained in terms of the complete Bell polynomials. The conditional mean and the conditional variance were explicitly evaluated. Several time-inhomogeneous processes were studied in detail in view of their potential applications in population growth models and in queuing systems. A time-inhomogeneous linear birth–death processes with finite state-space was also taken into account. Special attention was devoted to the cases of periodic immigration intensity functions that play an important role in the description of the evolution of dynamic systems influenced by seasonal immigration or other regular environmental cycles. Various numerical computations were performed for periodic immigration intensity functions.

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A generalized Gompertz growth model with applications and related birth-death processes

Ricerche di Matematica ◽

10.1007/s11587-020-00548-y ◽

2020 ◽

Author(s):

Majid Asadi ◽

Antonio Di Crescenzo ◽

Farkhondeh A. Sajadi ◽

Serena Spina

Keyword(s):

Growth Model ◽

Growth Curve ◽

Goodness Of Fit ◽

Gompertz Model ◽

Death Process ◽

Birth Process ◽

Failure Data ◽

First Case ◽

Gompertz Growth Model ◽

Birth Death

AbstractIn this paper, we propose a flexible growth model that constitutes a suitable generalization of the well-known Gompertz model. We perform an analysis of various features of interest, including a sensitivity analysis of the initial value and the three parameters of the model. We show that the considered model provides a good fit to some real datasets concerning the growth of the number of individuals infected during the COVID-19 outbreak, and software failure data. The goodness of fit is established on the ground of the ISRP metric and the $$d_2$$ d 2 -distance. We also analyze two time-inhomogeneous stochastic processes, namely a birth-death process and a birth process, whose means are equal to the proposed growth curve. In the first case we obtain the probability of ultimate extinction, being 0 an absorbing endpoint. We also deal with a threshold crossing problem both for the proposed growth curve and the corresponding birth process. A simulation procedure for the latter process is also exploited.

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Population processes allowing emigration of families

Journal of Applied Probability ◽

10.2307/3213635 ◽

1984 ◽

Vol 21 (2) ◽

pp. 225-232 ◽

Cited By ~ 4

Author(s):

Abebe Tessera

Keyword(s):

Population Growth ◽

Growth Models ◽

Death Process ◽

Population Processes ◽

The Individual ◽

Population Growth Models ◽

Birth Death

In the familiar immigration–birth–death process the events of immigration, birth and death relate to the individual. There are processes in which the whole family and not just an individual migrates. Such population growth models are studied in some detail.

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The First Passage Time Problem for Gauss-Diffusion Processes: Algorithmic Approaches and Applications to LIF Neuronal Model

Methodology And Computing In Applied Probability ◽

10.1007/s11009-009-9132-8 ◽

2009 ◽

Vol 13 (1) ◽

pp. 29-57 ◽

Cited By ~ 27

Author(s):

Aniello Buonocore ◽

Luigia Caputo ◽

Enrica Pirozzi ◽

Luigi M. Ricciardi

Keyword(s):

Diffusion Processes ◽

First Passage Time ◽

Passage Time ◽

Neuronal Model ◽

First Passage ◽

Time Problem ◽

Algorithmic Approaches ◽

First Passage Time Problem

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Hazard rate and reversed hazard rate monotonicities in continuous-time Markov chains

Journal of Applied Probability ◽

10.1239/jap/1032265203 ◽

1998 ◽

Vol 35 (3) ◽

pp. 545-556 ◽

Cited By ~ 11

Author(s):

Masaaki Kijima

Keyword(s):

Markov Chain ◽

Markov Chains ◽

Continuous Time ◽

Hazard Rate ◽

First Passage Time ◽

Death Process ◽

Continuous Time Markov Chain ◽

Reversed Hazard Rate ◽

The Right ◽

Birth Death

A continuous-time Markov chain on the non-negative integers is called skip-free to the right (left) if only unit increments to the right (left) are permitted. If a Markov chain is skip-free both to the right and to the left, it is called a birth–death process. Karlin and McGregor (1959) showed that if a continuous-time Markov chain is monotone in the sense of likelihood ratio ordering then it must be an (extended) birth–death process. This paper proves that if an irreducible Markov chain in continuous time is monotone in the sense of hazard rate (reversed hazard rate) ordering then it must be skip-free to the right (left). A birth–death process is then characterized as a continuous-time Markov chain that is monotone in the sense of both hazard rate and reversed hazard rate orderings. As an application, the first-passage-time distributions of such Markov chains are also studied.

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Spectral structure of the first-passage-time densities for classes of Markov chains

Journal of Applied Probability ◽

10.1017/s0021900200031363 ◽

1987 ◽

Vol 24 (03) ◽

pp. 631-643 ◽

Cited By ~ 4

Author(s):

Masaaki Kijima

Keyword(s):

Markov Chains ◽

First Passage Time ◽

Passage Time ◽

Death Process ◽

Spectral Structure ◽

First Passage ◽

Completely Monotone ◽

Spectral Representations ◽

Absorbing States ◽

Birth Death

Keilson [7] showed that for a birth-death process defined on non-negative integers with reflecting barrier at 0 the first-passage-time density from 0 to N (N to N + 1) has Pólya frequency of order infinity (is completely monotone). Brown and Chaganty [3] and Assaf et al. [1] studied the first-passage-time distribution for classes of discrete-time Markov chains and then produced the essentially same results as these through a uniformization. This paper addresses itself to an extension of Keilson's results to classes of Markov chains such as time-reversible Markov chains, skip-free Markov chains and birth-death processes with absorbing states. The extensions are due to the spectral representations of the infinitesimal generators governing these Markov chains. Explicit densities for those first-passage times are also given.

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