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Symmetry
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Mathematical Models: Methods and Applications
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Mathematical Models: Methods and Applications

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (30 April 2023) | Viewed by 19228

Special Issue Editors

Dr. Ophir Nave

E-Mail Website
Guest Editor
Department of Mathematics and Computer Science, Jerusalem College of Technology (JCT), Jerusalem, Israel
Interests: mathematical models of cancer; asymptotic analysis; method of integral invariant manifold; fixed point; numerical analysis; singular perturbe; gradient algorithms
Dr. Svetlana Bunimovich-Mendrazitsky

E-Mail Website
Co-Guest Editor
Department of Mathematics, Ariel University, Ariel, Israel
Interests: mathematical modelling

Special Issue Information

Dear Colleagues,

The Special Issue “Mathematical Modeling: Methods and Applications” aims to publish papers that provide new concepts, ideas, insight, or new understanding of biological systems using a mathematical model. The mathematical models can be of a theoretical or practical nature.

In order to study models in the natural sciences, various methods of mathematics can be applied, such as: moving beyond a dimensionless model and using symmetry, various asymptotic methods, numerical methods, system decomposition methods providing a fast system and a slow subsystem, approximation by columns, use of beads and more. We are interested in mathematical models implemented to analyze the phenomenon under investigation in order to advance science and discover new phenomena and knowledge.

Papers submitted to the journal should provide a deep, biological insight as a result of mathematical analysis using numerical analysis by new algorithms. The papers should identify and open up challenging new types of mathematical problems that are derived from biological knowledge.

Please note that all submitted papers must be within the general scope of the Symmetry journal.

Dr. Ophir Nave
Guest Editor
Dr. Svetlana Bunimovich-Mendrazitsky
Co-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. Symmetry is an international peer-reviewed open access monthly 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 2400 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

  • Computer algorithms
  • Symmetry
  • Mathematical modeling
  • Numerical analysis
  • Asymptotic analysis
  • ODE and PDE
  • Biology
  • Multiscale and multiphysics modeling.

Published Papers (12 papers)

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Research

18 pages, 747 KiB  
Article
Analysis of a Heterogeneous Queuing Model with Intermittently Obtainable Servers under a Hybrid Vacation Schedule
by Divya Kothandaraman and Indhira Kandaiyan
Symmetry 2023, 15(7), 1304; https://0-doi-org.brum.beds.ac.uk/10.3390/sym15071304 - 24 Jun 2023
Cited by 2 | Viewed by 1374
Abstract
This paper investigates the concept of a Markovian queueing model with heterogeneous, intermittently available servers with feedback under a hybrid vacation policy. Both the asymmetric transition representation and the hybrid vacation policy are addressed in this article. The necessary and sufficient conditions for [...] Read more.
This paper investigates the concept of a Markovian queueing model with heterogeneous, intermittently available servers with feedback under a hybrid vacation policy. Both the asymmetric transition representation and the hybrid vacation policy are addressed in this article. The necessary and sufficient conditions for system stability are presented. In addition, the steady-state probability distribution of the queueing model was derived by employing the matrix geometric method. Furthermore, a few formulae were constructed to determine the model’s performance indicators. Finally, the influence of system parameters was also investigated using some numerical examples. Full article
(This article belongs to the Special Issue Mathematical Models: Methods and Applications)
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17 pages, 1383 KiB  
Article
Quantifying the Effects of Global Warming on the Plankton Population: An Asymmetric Multifactor Mathematical Model-Based Approach
by Junbin Zhong, Jianji Li, Jingtian Deng and Jinwei Fang
Symmetry 2023, 15(5), 1047; https://0-doi-org.brum.beds.ac.uk/10.3390/sym15051047 - 09 May 2023
Cited by 1 | Viewed by 1358
Abstract
A nonlinear dynamical model for the plankton population in a fixed sea area under the influence of asymmetric multiple factors, including atmospheric CO2 concentration, atmospheric temperature, nutrient concentration, seawater temperature, light intensity, and predator density is proposed to address the survival of [...] Read more.
A nonlinear dynamical model for the plankton population in a fixed sea area under the influence of asymmetric multiple factors, including atmospheric CO2 concentration, atmospheric temperature, nutrient concentration, seawater temperature, light intensity, and predator density is proposed to address the survival of the plankton population due to global warming. The model’s accuracy is confirmed by comparison with actual data, and numerical simulations are carried out to justify the relevant findings. The results suggest that increasing plankton’s ability to absorb atmospheric CO2 or regulate atmospheric temperature can help to mitigate global warming. Furthermore, if the population density of fish, the primary predator of plankton, falls within a certain range, the increase in atmospheric temperature will be mitigated. Additionally, the stability conditions for the suggested model are obtained, along with the equilibrium point of the system. Overall, this paper considers the effects of asymmetric multifactor interaction on plankton population density and establishes a mathematical connection between environmental ecosystems and plankton that might aid in addressing the challenges posed by global warming and preserving the plankton population. Full article
(This article belongs to the Special Issue Mathematical Models: Methods and Applications)
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11 pages, 414 KiB  
Article
Numerical Modeling of Peridynamic Richards’ Equation with Piecewise Smooth Initial Conditions Using Spectral Methods
by Fabio V. Difonzo and Francesco Di Lena
Symmetry 2023, 15(5), 960; https://0-doi-org.brum.beds.ac.uk/10.3390/sym15050960 - 23 Apr 2023
Cited by 2 | Viewed by 1034
Abstract
In this paper, we introduce peridynamic theory and its application to Richards’ equation with a piecewise smooth initial condition. Peridynamic theory is a non-local continuum theory that models the deformation and failure of materials. Richards’ equation describes the unsaturated flow of water through [...] Read more.
In this paper, we introduce peridynamic theory and its application to Richards’ equation with a piecewise smooth initial condition. Peridynamic theory is a non-local continuum theory that models the deformation and failure of materials. Richards’ equation describes the unsaturated flow of water through porous media, and it plays an essential role in many applications, such as groundwater management, soil science, and environmental engineering. We develop a peridynamic formulation of Richards’ equation that includes the effect of peridynamic forces and a piecewise smooth initial condition, further introducing a non-standard symmetric influence function to describe such peridynamic interactions, which turns out to provide beneficial effects from a numerical point of view. Moreover, we implement a numerical scheme based on Chebyshev polynomials and symmetric Gauss–Lobatto nodes, providing a powerful spectral method able to capture singularities and critical issues of Richards’ equation with piecewise smooth initial conditions. We also present numerical simulations that illustrate the performance of the proposed approach. In particular, we perform a computational investigation into the spatial order of convergence, showing that, despite the discontinuity in the initial condition, the order of convergence is retained. Full article
(This article belongs to the Special Issue Mathematical Models: Methods and Applications)
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13 pages, 316 KiB  
Article
Empirical Bayes Decision for a Generalized Exponential Distribution with Contaminated Data
by Han Qiu, Jiaqing Chen, Zihao Yuan and Yangxin Huang
Symmetry 2023, 15(2), 511; https://0-doi-org.brum.beds.ac.uk/10.3390/sym15020511 - 14 Feb 2023
Viewed by 966
Abstract
The two-sided and one-sided empirical Bayes test (EBT) rules for the parameter of a generalized exponential distribution with contaminated data (errors in variables) are constructed by a deconvolution kernel method, respectively. Under the type of the supersmooth error distributions and the supersmooth errors [...] Read more.
The two-sided and one-sided empirical Bayes test (EBT) rules for the parameter of a generalized exponential distribution with contaminated data (errors in variables) are constructed by a deconvolution kernel method, respectively. Under the type of the supersmooth error distributions and the supersmooth errors with the error level can be controlled situations, the asymptotically optimal uniformly over a class of prior distributions and uniform rates of convergence of the corresponding regret for the proposed EBT rules are obtained with suitable conditions. The example study shows that the assumptions and conditions of the main results of this paper are satisfied easily by calculating. Full article
(This article belongs to the Special Issue Mathematical Models: Methods and Applications)
18 pages, 450 KiB  
Article
Analysis of an M/G/1 Retrial Queue with Delayed Repair and Feedback under Working Vacation policy with Impatient Customers
by Micheal Mathavavisakan Nicholas GnanaSekar and Indhira Kandaiyan
Symmetry 2022, 14(10), 2024; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14102024 - 27 Sep 2022
Cited by 12 | Viewed by 1643
Abstract
The concept of a single server retrial queueing system with delayed repair and feedback under a working vacation policy, along with the asymmetric transition representation, is discussed in this article. In addition, consumers are entitled to balk and renege in some situations. The [...] Read more.
The concept of a single server retrial queueing system with delayed repair and feedback under a working vacation policy, along with the asymmetric transition representation, is discussed in this article. In addition, consumers are entitled to balk and renege in some situations. The steady-state probability generating function for system size and orbit size is derived by using the approach of supplementary variables. Discussions include key metrics of the system and a few significant special conditions. Moreover, the impact of system parameters is examined through the analysis of some numerical examples. Full article
(This article belongs to the Special Issue Mathematical Models: Methods and Applications)
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15 pages, 369 KiB  
Article
Threshold of Stochastic SIRS Epidemic Model from Infectious to Susceptible Class with Saturated Incidence Rate Using Spectral Method
by Ishtiaq Ali and Sami Ullah Khan
Symmetry 2022, 14(9), 1838; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14091838 - 05 Sep 2022
Cited by 19 | Viewed by 1614
Abstract
Stochastic SIRS models play a key role in formulating and analyzing the transmission of infectious diseases. These models reflect the environmental changes of the diseases and their biological mechanisms. Therefore, it is very important to study the uniqueness and existence of the global [...] Read more.
Stochastic SIRS models play a key role in formulating and analyzing the transmission of infectious diseases. These models reflect the environmental changes of the diseases and their biological mechanisms. Therefore, it is very important to study the uniqueness and existence of the global positive solution to investigate the asymptotic properties of the model. In this article, we investigate the dynamics of the stochastic SIRS epidemic model with a saturated incidence rate. The effects of both deterministic and stochastic distribution from infectious to susceptible are analyzed. Our findings show that the occurrence of symmetry breaking as a function of the stochastic noise has a significant advantage over the deterministic one to prevent the spread of the infectious diseases. The larger stochastic noise will guarantee the control of epidemic diseases with symmetric Brownian motion. Periodic outbreaks and re-infection may occur due to the existence of feedback memory. It is shown that the endemic equilibrium is stable under some suitable initial conditions, taking advantage of the symmetry of the large amount of contact structure. A numerical method based on Legendre polynomials that converts the given stochastic SIRS model into a nonlinear algebraic system is used for the approximate solution. Finally, some numerical experiments are performed to verify the theoretical results and clearly show the sharpness of the obtained conditions and thresholds. Full article
(This article belongs to the Special Issue Mathematical Models: Methods and Applications)
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28 pages, 1757 KiB  
Article
Double Penalized Expectile Regression for Linear Mixed Effects Model
by Sihan Gao, Jiaqing Chen, Zihao Yuan, Jie Liu and Yangxin Huang
Symmetry 2022, 14(8), 1538; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14081538 - 27 Jul 2022
Viewed by 1262
Abstract
This paper constructs the double penalized expectile regression for linear mixed effects model, which can estimate coefficient and choose variable for random and fixed effects simultaneously. The method based on the linear mixed effects model by cojoining double penalized expectile regression. For this [...] Read more.
This paper constructs the double penalized expectile regression for linear mixed effects model, which can estimate coefficient and choose variable for random and fixed effects simultaneously. The method based on the linear mixed effects model by cojoining double penalized expectile regression. For this model, this paper proposes the iterative Lasso expectile regression algorithm to solve the parameter for this mode, and the Schwarz Information Criterion (SIC) and Generalized Approximate Cross-Validation Criterion (GACV) are used to choose the penalty parameters. Additionally, it establishes the asymptotic normality of the expectile regression coefficient estimators that are suggested. Though simulation studies, we examine the effects of coefficient estimation and the variable selection at varying expectile levels under various conditions, including different signal-to-noise ratios, random effects, and the sparsity of the model. In this work, founding that the proposed method is robust to various error distributions at every expectile levels, and is superior to the double penalized quantile regression method in the robustness of excluding inactive variables. The suggested method may still accurately exclude inactive variables and select important variables with a high probability for high-dimensional data. The usefulness of doubly penalized expectile regression in applications is illustrated through a case study using CD4 cell real data. Full article
(This article belongs to the Special Issue Mathematical Models: Methods and Applications)
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9 pages, 2220 KiB  
Article
On Filters of Bitonic Algebras
by Şule Ayar Özbal
Symmetry 2022, 14(8), 1509; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14081509 - 23 Jul 2022
Viewed by 1088
Abstract
With the deep study in this work, we introduce the concept of filters of a bitonic algebra A. We study some fundamental structures of such determined filters. We also focus on features of filters with respect to homomorphisms. With the help of [...] Read more.
With the deep study in this work, we introduce the concept of filters of a bitonic algebra A. We study some fundamental structures of such determined filters. We also focus on features of filters with respect to homomorphisms. With the help of the idea of upper sets, we investigate basic ideas of filters in a bitonic algebra, and we also state some important theorems related to them. We obtain some relations between filters of bitonic algebras and upper sets. We obtain an equivalent condition of the filters with the help of the notion of upper sets. Full article
(This article belongs to the Special Issue Mathematical Models: Methods and Applications)
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21 pages, 1695 KiB  
Article
Improve Stock Price Model-Based Stochastic Pantograph Differential Equation
by Mahmoud A. Eissa and M. Elsayed
Symmetry 2022, 14(7), 1358; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14071358 - 01 Jul 2022
Cited by 1 | Viewed by 1682
Abstract
Although the concept of symmetry is widely used in many fields, it is almost not discussed in finance. This concept appears to be relevant in relation, for example, to mathematical models that can predict stock prices to contribute to the decision-making process. This [...] Read more.
Although the concept of symmetry is widely used in many fields, it is almost not discussed in finance. This concept appears to be relevant in relation, for example, to mathematical models that can predict stock prices to contribute to the decision-making process. This work considers the stock price of European options with a new class of the non-constant delay model. The stochastic pantograph differential equation (SPDE) with a variable delay is provided in order to overcome the weaknesses of using stochastic models with constant delay. The proposed model is constructed to improve the evaluation process and prediction accuracy for stock prices. The feasibility of the proposed model is introduced under relatively weak conditions imposed on its volatility function. Furthermore, the sensitivity of time lag is discussed. The robust stochastic theta Milstein (STM) method is combined with the Monte Carlo simulation to compute asset prices within the proposed model. In addition, we prove that the numerical solution can preserve the non-negativity of the solution of the model. Numerical experiments using real financial data indicate that there is an increasing possibility of prediction accuracy for the proposed model with a variable delay compared to non-linear models with constant delay and the classical Black and Scholes model. Full article
(This article belongs to the Special Issue Mathematical Models: Methods and Applications)
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12 pages, 926 KiB  
Article
Differential Response to Cytotoxic Drugs Explains the Dynamics of Leukemic Cell Death: Insights from Experiments and Mathematical Modeling
by Ekaterina Guzev, Svetlana Bunimovich-Mendrazitsky and Michael A. Firer
Symmetry 2022, 14(6), 1269; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14061269 - 20 Jun 2022
Cited by 2 | Viewed by 1839
Abstract
This study presents a framework whereby cancer chemotherapy could be improved through collaboration between mathematicians and experimentalists. Following on from our recently published model, we use A20 murine leukemic cells transfected with monomeric red fluorescent proteins cells (mCherry) to compare the simulated and [...] Read more.
This study presents a framework whereby cancer chemotherapy could be improved through collaboration between mathematicians and experimentalists. Following on from our recently published model, we use A20 murine leukemic cells transfected with monomeric red fluorescent proteins cells (mCherry) to compare the simulated and experimental cytotoxicity of two Federal Drug Administration (FDA)-approved anticancer drugs, Cytarabine (Cyt) and Ibrutinib (Ibr) in an in vitro model system of Chronic Lymphocytic Leukemia (CLL). Maximum growth inhibition with Cyt (95%) was reached at an 8-fold lower drug concentration (6.25 μM) than for Ibr (97%, 50 μM). For the proposed ordinary differential equations (ODE) model, a multistep strategy was used to estimate the parameters relevant to the analysis of in vitro experiments testing the effects of different drug concentrations. The simulation results demonstrate that our model correctly predicts the effects of drugs on leukemic cells. To assess the closeness of the fit between the simulations and experimental data, RMSEs for both drugs were calculated (both RMSEs < 0.1). The numerical solutions of the model show a symmetrical dynamical evolution for two drugs with different modes of action. Simulations of the combinatorial effect of Cyt and Ibr showed that their synergism enhanced the cytotoxic effect by 40%. We suggest that this model could predict a more personalized drug dose based on the growth rate of an individual’s cancer cells. Full article
(This article belongs to the Special Issue Mathematical Models: Methods and Applications)
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17 pages, 1781 KiB  
Article
Experimental Validation of a Mathematical Model to Describe the Drug Cytotoxicity of Leukemic Cells
by Ekaterina Guzev, Galia Luboshits, Svetlana Bunimovich-Mendrazitsky and Michael A. Firer
Symmetry 2021, 13(10), 1760; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13101760 - 22 Sep 2021
Cited by 5 | Viewed by 1980
Abstract
Chlorambucil (Chl), Melphalan (Mel), and Cytarabine (Cyt) are recognized drugs used in the chemotherapy of patients with advanced Chronic Lymphocytic Leukemia (CLL). The optimal treatment schedule and timing of Chl, Mel, and Cyt administration remains unknown and has traditionally been decided empirically and [...] Read more.
Chlorambucil (Chl), Melphalan (Mel), and Cytarabine (Cyt) are recognized drugs used in the chemotherapy of patients with advanced Chronic Lymphocytic Leukemia (CLL). The optimal treatment schedule and timing of Chl, Mel, and Cyt administration remains unknown and has traditionally been decided empirically and independently of preclinical in vitro efficacy studies. As a first step toward mathematical prediction of in vivo drug efficacy from in vitro cytotoxicity studies, we used murine A20 leukemic cells as a test case of CLL. We first found that logistic growth best described the proliferation of the cells in vitro. Then, we tested in vitro the cytotoxic efficacy of Chl, Mel, and Cyt against A20 cells. On the basis of these experimental data, we found the parameters for cancer cell death rates that were dependent on the concentration of the respective drugs and developed a mathematical model involving nonlinear ordinary differential equations. For the proposed mathematical model, three equilibrium states were analyzed using the general method of Lyapunov, with only one equilibrium being stable. We obtained a very good symmetry between the experimental results and numerical simulations of the model. Our novel model can be used as a general tool to study the cytotoxic activity of various drugs with different doses and modes of action by appropriate adjustment of the values for the selected parameters. Full article
(This article belongs to the Special Issue Mathematical Models: Methods and Applications)
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15 pages, 328 KiB  
Article
Novel Method to Analytically Obtain the Asymptotic Stable Equilibria States of Extended SIR-Type Epidemiological Models
by Teddy Lazebnik, Svetlana Bunimovich-Mendrazitsky and Leonid Shaikhet
Symmetry 2021, 13(7), 1120; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13071120 - 23 Jun 2021
Cited by 9 | Viewed by 1788
Abstract
We present a new analytical method to find the asymptotic stable equilibria states based on the Markov chain technique. We reveal this method on the Susceptible-Infectious-Recovered (SIR)-type epidemiological model that we developed for viral diseases with long-term immunity memory. This is a large-scale [...] Read more.
We present a new analytical method to find the asymptotic stable equilibria states based on the Markov chain technique. We reveal this method on the Susceptible-Infectious-Recovered (SIR)-type epidemiological model that we developed for viral diseases with long-term immunity memory. This is a large-scale model containing 15 nonlinear ordinary differential equations (ODEs), and classical methods have failed to analytically obtain its equilibria. The proposed method is used to conduct a comprehensive analysis by a stochastic representation of the dynamics of the model, followed by finding all asymptotic stable equilibrium states of the model for any values of parameters and initial conditions thanks to the symmetry of the population size over time. Full article
(This article belongs to the Special Issue Mathematical Models: Methods and Applications)
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