Analysis of a growth model inspired by Gompertz and Korf laws, and an analogous birth-death process

The aim of this paper is to provide a fractional generalization of the Gompertz law via a Caputo-like definition of fractional derivative of a function with respect to another function. In particular, we observe that the model presented appears to be substantially different from the other attempt of fractional modifications of this model, since the fractional nature is carried along by the general solution even in its asymptotic behavior for long times. We then validate the presented model by employing it as a reference frame to model three biological systems of peculiar interest for biophysics and environmental engineering, namely: dark fermentation, photofermentation and microalgae biomass growth.

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“…It is also worth remarking that the Gompertz law is also recovered as a simple Malthusian model (see [3]), namely…”

Section: Fractional Gompertzian-type Models: Previous and New Resultsmentioning

confidence: 96%

“…The Gompertz law, from a mathematical perspective, is a Malthusian-type growth model with a time-dependent exponentially decreasing rate. The stochastic roots of this model have been carefully addressed by De Lauro et al in [2] and an interesting generalization of this model has been recently presented by Di Crescenzo and Spina in [3].…”

Section: Introductionmentioning

confidence: 99%

Modeling biological systems with an improved fractional Gompertz law

Frunzo

Garra

Giusti

et al. 2019

Communications in Nonlinear Science and Numerical Simulation

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“…So, when the conditions (15) and (5) hold, the logistic curve N(t) and the mean of the birth-death process X(t) are governed by the same differential equation. The transition probabilities, the conditional mean and the conditional variance of the process can be expressed in a simpler form, as shown in [4]. In particular, the transition probabilities of the process X(t) with rates (10), are given by…”

Section: Analysis Of a Special Time-inhomogeneous Linear Birth-death mentioning

confidence: 99%

“…respectively. Note that, if ξ(t) is a positive function for all t > 0, we have that both the conditional mean E y (t) and the conditional variance Var y (t) are strictly increasing (see [4], as a reference). Since the state 0 is an absorbing endpoint, P y,0 (t) represents the probability that the population dies out prior to time t conditional by the initial size X(0) = y.…”

Section: Analysis Of a Special Time-inhomogeneous Linear Birth-death mentioning

confidence: 99%

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Logistic Growth Described by Birth-Death and Diffusion Processes

Crescenzo

Paraggio

2019

Mathematics

Self Cite

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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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Modeling the random spreading of fake news through a two‐dimensional time‐inhomogeneous birth‐death process

Di Crescenzo,

Paraggio

2024

Math Methods in App Sciences

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We consider a two‐dimensional time‐inhomogeneous birth‐death process to model the time‐evolution of fake news in a population. The two components of the process represent, respectively: (i) the number of individuals (say spreaders) who know the rumor and intend to spread it and (ii) the number of individuals (say inactives) who have forgotten the rumor previously received. We employ the probability generating function‐based approach to obtain the moments and the covariance of the two‐dimensional process. We also analyze a new adimensional index to study the correlation between the two components. Some special cases are considered in which both the expected numbers of spreaders and inactives are equal to suitable sigmoidal curves, which are often adopted in modeling growth phenomena. Finally, we provide an application based on real data related to the diffusion of fake news, in which the optimal choice of the sigmoidal curve that fit the datasets is based on the minimization of the mean square error and of the relative absolute error.

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Analysis of a growth model inspired by Gompertz and Korf laws, and an analogous birth-death process

Cited by 33 publications

References 35 publications

Modeling biological systems with an improved fractional Gompertz law

Modeling biological systems with an improved fractional Gompertz law

Logistic Growth Described by Birth-Death and Diffusion Processes

Modeling the random spreading of fake news through a two‐dimensional time‐inhomogeneous birth‐death process

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