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Mathematics
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Stochastic Modeling in Biology
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Stochastic Modeling in Biology

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Computational and Applied Mathematics".

Deadline for manuscript submissions: closed (30 September 2020) | Viewed by 15561

Special Issue Editor

Dr. Mario Abundo

E-Mail Website
Guest Editor
Dipartimento di Matematica, Università Tor Vergata, via della Ricerca Scientifica, 00133 Roma, Italy
Interests: stochastic processes; stochastic modeling in biology; computer simulation
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The aim of this Special Issue is to publish original research articles covering advances in the theory of stochastic modeling in biology. In this framework, continuous and discrete time stochastic processes will be discussed, as well as stochastic differential equations, fractional differential equations, correlated processes, first-passage-time problems, stochastic optimal controls, parameter estimation, and simulation techniques. All the above topics are intended to be treated in the spirit of modeling the evolution of stochastic systems of interest in biology.

Potential topics include but are not limited to the following:

-Stochastic processes for neuronal activity;

-Jump-diffusion processes;

-Markov and semi-Markov processes;

-Time-changed processes;

-Markov chains;

-Fractional processes;

-Fractional Brownian motion.

Stochastic models to describe the evolution of biological systems are very important; in fact, often a mere deterministic description is insufficient to capture the essence of the qualitative behavior of such systems, when one varies the conditions of the environment and/or the physical parameters.

The purpose of this Special Issue is to gather a collection of articles reflecting the latest developments in stochastic modeling in biology, with the aim of studying the qualitative and quantitative behavior of phenomena in which the random component is essential.

Prof. Dr. Mario Abundo
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Neuronal models
  • Jump-diffusion processes
  • Markov and semi-Markov processes
  • Time-changed processes
  • Markov chains
  • Fractional processes
  • Fractional Brownian motion

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Published Papers (6 papers)

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10 pages, 928 KiB  
Article
On the Entropy of Fractionally Integrated Gauss–Markov Processes
by Mario Abundo and Enrica Pirozzi
Mathematics 2020, 8(11), 2031; https://doi.org/10.3390/math8112031 - 14 Nov 2020
Cited by 1 | Viewed by 1435
Abstract
This paper is devoted to the estimation of the entropy of the dynamical system {Xα(t),t≥0}, where the stochastic process Xα(t) consists of the fractional Riemann–Liouville integral of order [...] Read more.
This paper is devoted to the estimation of the entropy of the dynamical system {Xα(t),t≥0}, where the stochastic process Xα(t) consists of the fractional Riemann–Liouville integral of order α∈(0,1) of a Gauss–Markov process. The study is based on a specific algorithm suitably devised in order to perform the simulation of sample paths of such processes and to evaluate the numerical approximation of the entropy. We focus on fractionally integrated Brownian motion and Ornstein–Uhlenbeck process due their main rule in the theory and application fields. Their entropy is specifically estimated by computing its approximation (ApEn). We investigate the relation between the value of α and the complexity degree; we show that the entropy of Xα(t) is a decreasing function of α∈(0,1). Full article
(This article belongs to the Special Issue Stochastic Modeling in Biology)
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29 pages, 1691 KiB  
Article
Bell Polynomial Approach for Time-Inhomogeneous Linear Birth–Death Process with Immigration
by Virginia Giorno and Amelia G. Nobile
Mathematics 2020, 8(7), 1123; https://doi.org/10.3390/math8071123 - 9 Jul 2020
Cited by 7 | Viewed by 2398
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Stochastic Modeling in Biology)
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27 pages, 424 KiB  
Article
Connections between Weighted Generalized Cumulative Residual Entropy and Variance
by Abdolsaeed Toomaj and Antonio Di Crescenzo
Mathematics 2020, 8(7), 1072; https://doi.org/10.3390/math8071072 - 2 Jul 2020
Cited by 22 | Viewed by 2586
Abstract
A shift-dependent information measure is favorable to handle in some specific applied contexts such as mathematical neurobiology and survival analysis. For this reason, the weighted differential entropy has been introduced in the literature. In accordance with this measure, we propose the weighted generalized [...] Read more.
A shift-dependent information measure is favorable to handle in some specific applied contexts such as mathematical neurobiology and survival analysis. For this reason, the weighted differential entropy has been introduced in the literature. In accordance with this measure, we propose the weighted generalized cumulative residual entropy as well. Despite existing apparent similarities between these measures, however, there are quite substantial and subtle differences between them because of their different metrics. In this paper, particularly, we show that the proposed measure is equivalent to the generalized cumulative residual entropy of the cumulative weighted random variable. Thus, we first provide expressions for the variance and the new measure in terms of the weighted mean residual life function and then elaborate on some characteristics of such measures, including equivalent expressions, stochastic comparisons, bounds, and connection with the excess wealth transform. Finally, we also illustrate some applications of interest in system reliability with reference to shock models and random minima. Full article
(This article belongs to the Special Issue Stochastic Modeling in Biology)
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31 pages, 15397 KiB  
Article
On Exact and Approximate Approaches for Stochastic Receptor-Ligand Competition Dynamics—An Ecological Perspective
by Polly-Anne Jeffrey, Martín López-García, Mario Castro, Grant Lythe and Carmen Molina-París
Mathematics 2020, 8(6), 1014; https://doi.org/10.3390/math8061014 - 20 Jun 2020
Viewed by 3317
Abstract
Cellular receptors on the cell membrane can bind ligand molecules in the extra-cellular medium to form ligand-bound monomers. These interactions ultimately determine the fate of a cell through the resulting intra-cellular signalling cascades. Often, several receptor types can bind a shared ligand leading [...] Read more.
Cellular receptors on the cell membrane can bind ligand molecules in the extra-cellular medium to form ligand-bound monomers. These interactions ultimately determine the fate of a cell through the resulting intra-cellular signalling cascades. Often, several receptor types can bind a shared ligand leading to the formation of different monomeric complexes, and in turn to competition for the common ligand. Here, we describe competition between two receptors which bind a common ligand in terms of a bi-variate stochastic process. The stochastic description is important to account for fluctuations in the number of molecules. Our interest is in computing two summary statistics—the steady-state distribution of the number of bound monomers and the time to reach a threshold number of monomers of a given kind. The matrix-analytic approach developed in this manuscript is exact, but becomes impractical as the number of molecules in the system increases. Thus, we present novel approximations which can work under low-to-moderate competition scenarios. Our results apply to systems with a larger number of population species (i.e., receptors) competing for a common resource (i.e., ligands), and to competition systems outside the area of molecular dynamics, such as Mathematical Ecology. Full article
(This article belongs to the Special Issue Stochastic Modeling in Biology)
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20 pages, 340 KiB  
Article
Least-Squares Estimators of Drift Parameter for Discretely Observed Fractional Ornstein–Uhlenbeck Processes
by Pavel Kříž and Leszek Szała
Mathematics 2020, 8(5), 716; https://doi.org/10.3390/math8050716 - 3 May 2020
Cited by 3 | Viewed by 2150
Abstract
We introduce three new estimators of the drift parameter of a fractional Ornstein–Uhlenbeck process. These estimators are based on modifications of the least-squares procedure utilizing the explicit formula for the process and covariance structure of a fractional Brownian motion. We demonstrate their advantageous [...] Read more.
We introduce three new estimators of the drift parameter of a fractional Ornstein–Uhlenbeck process. These estimators are based on modifications of the least-squares procedure utilizing the explicit formula for the process and covariance structure of a fractional Brownian motion. We demonstrate their advantageous properties in the setting of discrete-time observations with fixed mesh size, where they outperform the existing estimators. Numerical experiments by Monte Carlo simulations are conducted to confirm and illustrate theoretical findings. New estimation techniques can improve calibration of models in the form of linear stochastic differential equations driven by a fractional Brownian motion, which are used in diverse fields such as biology, neuroscience, finance and many others. Full article
(This article belongs to the Special Issue Stochastic Modeling in Biology)
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17 pages, 345 KiB  
Article
Modeling Y-Linked Pedigrees through Branching Processes
by Miguel González, Cristina Gutiérrez and Rodrigo Martínez
Mathematics 2020, 8(2), 256; https://doi.org/10.3390/math8020256 - 15 Feb 2020
Viewed by 2555
Abstract
A multidimensional two-sex branching process is introduced to model the evolution of a pedigree originating from the mutation of an allele of a Y-linked gene in a monogamous population. The study of the extinction of the mutant allele and the analysis of the [...] Read more.
A multidimensional two-sex branching process is introduced to model the evolution of a pedigree originating from the mutation of an allele of a Y-linked gene in a monogamous population. The study of the extinction of the mutant allele and the analysis of the dominant allele in the pedigree is addressed on the basis of the classical theory of multi-type branching processes. The asymptotic behavior of the number of couples of different types in the pedigree is also derived. Finally, using the estimates of the mean growth rates of the allele and its mutation provided by a Gibbs sampler, a real Y-linked pedigree associated with hearing loss is analyzed, concluding that this mutation will persist in the population although without dominating the pedigree. Full article
(This article belongs to the Special Issue Stochastic Modeling in Biology)
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