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Symmetry
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Mathematical Modeling of Biological Systems: Geometry, Symmetry and Conservation Laws
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Mathematical Modeling of Biological Systems: Geometry, Symmetry and Conservation Laws

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

Deadline for manuscript submissions: closed (31 July 2021) | Viewed by 23521

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editors

Dr. Federico Papa

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Guest Editor
Institute for Systems Analysis and Computer Science “A. Ruberti”, National Research Council, 00185 Rome, Italy
Interests: mathematical modeling of biological systems; mathematical modeling and control of tumor growth and treatment; mathematical modeling of epidemics; systems biology; dynamical system identification and filtering; optimal control theory
Special Issues, Collections and Topics in MDPI journals
Dr. Carmela Sinisgalli

E-Mail Website
Guest Editor
Institute for Systems Analysis and Computer Science “A. Ruberti”, National Research Council, 00185 Rome, Italy
Interests: mathematical and computational modeling in biology and medicine; mathematical oncology; signal and image analysis; dynamical system identification

Special Issue Information

Dear Colleagues,

Mathematical modeling is a powerful approach supporting the investigation of open problems in natural sciences, in particular physics, biology and medicine. Applied mathematics allows to translate the available information about real-world phenomena into mathematical objects and concepts. Mathematical models are useful descriptive tools that allow to gather the salient aspects of complex biological systems along with their fundamental governing laws, by elucidating the system behavior in time and space, also evidencing symmetry, or symmetry breaking, in geometry and morphology. Additionally, mathematical models are useful predictive tools able to reliably forecast the future system evolution or its response to specific inputs. More importantly, concerning biomedical systems, such models can even become prescriptive tools, allowing effective, sometimes optimal, intervention strategies for the treatment and control of pathological states to be planned.

The application of mathematical physics, nonlinear analysis, systems and control theory to the study of biological and medical systems results in the formulation of new challenging problems for the scientific community. This Special Issue is looking for innovative contributions of experienced researchers in the field of mathematical modelling applied to biology and medicine. Original research papers and reviews are welcome.

Dr. Federico Papa
Dr. Carmela Sinisgalli
Guest Editors

“I know that in the study of material things number, order, and position are the threefold clue to exact knowledge: and that these three, in the mathematician’s hands furnish the first outlines for a sketch of the Universe.”

(D’Arcy Thompson, Growth and Form, 1917)

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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

  • Mathematical Modelling of Biological Systems
  • Mathematical Physics
  • Symmetry/Asymmetry in Biological Patterns and Structures
  • Computational Biology and Medicine
  • Systems Biology
  • Dynamical Systems
  • System Identification and Filtering
  • Optimal Control Theory
  • Mathematical Oncology
  • Epidemic Modelling
  • Structured Population Models
  • Biochemical Networks

Published Papers (10 papers)

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Research

13 pages, 776 KiB  
Article
The Double Phospho/Dephosphorylation Cycle as a Benchmark to Validate an Effective Taylor Series Method to Integrate Ordinary Differential Equations
by Alessandro Borri, Francesco Carravetta and Pasquale Palumbo
Symmetry 2021, 13(9), 1684; https://doi.org/10.3390/sym13091684 - 13 Sep 2021
Cited by 1 | Viewed by 1295
Abstract
The double phosphorylation/dephosphorylation cycle consists of a symmetric network of biochemical reactions of paramount importance in many intracellular mechanisms. From a network perspective, they consist of four enzymatic reactions interconnected in a specular way. The general approach to model enzymatic reactions in a [...] Read more.
The double phosphorylation/dephosphorylation cycle consists of a symmetric network of biochemical reactions of paramount importance in many intracellular mechanisms. From a network perspective, they consist of four enzymatic reactions interconnected in a specular way. The general approach to model enzymatic reactions in a deterministic fashion is by means of stiff Ordinary Differential Equations (ODEs) that are usually hard to integrate according to biologically meaningful parameter settings. Indeed, the quest for model simplification started more than one century ago with the seminal works by Michaelis and Menten, and their Quasi Steady-State Approximation methods are still matter of investigation nowadays. This work proposes an effective algorithm based on Taylor series methods that manages to overcome the problems arising in the integration of stiff ODEs, without settling for model approximations. The double phosphorylation/dephosphorylation cycle is exploited as a benchmark to validate the methodology from a numerical viewpoint. Full article
(This article belongs to the Special Issue Mathematical Modeling of Biological Systems: Geometry, Symmetry and Conservation Laws)
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23 pages, 10908 KiB  
Article
On Systems of Active Particles Perturbed by Symmetric Bounded Noises: A Multiscale Kinetic Approach
by Bruno Felice Filippo Flora, Armando Ciancio and Alberto d’Onofrio
Symmetry 2021, 13(9), 1604; https://doi.org/10.3390/sym13091604 - 01 Sep 2021
Viewed by 2073
Abstract
We consider an ensemble of active particles, i.e., of agents endowed by internal variables u(t). Namely, we assume that the nonlinear dynamics of u is perturbed by realistic bounded symmetric stochastic perturbations acting nonlinearly or linearly. In the absence of [...] Read more.
We consider an ensemble of active particles, i.e., of agents endowed by internal variables u(t). Namely, we assume that the nonlinear dynamics of u is perturbed by realistic bounded symmetric stochastic perturbations acting nonlinearly or linearly. In the absence of birth, death and interactions of the agents (BDIA) the system evolution is ruled by a multidimensional Hypo-Elliptical Fokker–Plank Equation (HEFPE). In presence of nonlocal BDIA, the resulting family of models is thus a Partial Integro-differential Equation with hypo-elliptical terms. In the numerical simulations we focus on a simple case where the unperturbed dynamics of the agents is of logistic type and the bounded perturbations are of the Doering–Cai–Lin noise or the Arctan bounded noise. We then find the evolution and the steady state of the HEFPE. The steady state density is, in some cases, multimodal due to noise-induced transitions. Then we assume the steady state density as the initial condition for the full system evolution. Namely we modeled the vital dynamics of the agents as logistic nonlocal, as it depends on the whole size of the population. Our simulations suggest that both the steady states density and the total population size strongly depends on the type of bounded noise. Phenomena as transitions to bimodality and to asymmetry also occur. Full article
(This article belongs to the Special Issue Mathematical Modeling of Biological Systems: Geometry, Symmetry and Conservation Laws)
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16 pages, 1681 KiB  
Article
Primary Model for Biomass Growth Prediction in Batch Fermentation
by Blanca E. Garcia, Emmanuel Rodriguez, Yolocuauhtli Salazar, Paul A. Valle, Adriana C. Flores-Gallegos, O. Miriam Rutiaga-Quiñones and Raul Rodriguez-Herrera
Symmetry 2021, 13(8), 1468; https://doi.org/10.3390/sym13081468 - 11 Aug 2021
Cited by 6 | Viewed by 2753 | Correction
Abstract
Predictive models may be considered a tool to ensure food quality as they provide insights that support decision making on the design of processes, such as fermentation. Objective: To formulate a mathematical model that describes the growth of lactic acid bacteria (LAB) in [...] Read more.
Predictive models may be considered a tool to ensure food quality as they provide insights that support decision making on the design of processes, such as fermentation. Objective: To formulate a mathematical model that describes the growth of lactic acid bacteria (LAB) in batch fermentation. Methodology: Based on real-life experimental data from eight LAB strains, we formulated a primary model in the form of a third-degree polynomial function that successfully describes the four phases observed in LAB growth, i.e., lag, exponential, stationary, and death. Our cubic mathematical model allows us to understand the fundamental nonlinear dynamics of LAB as well as its time-variant dependencies. Parameters of the model are written in terms of initial biomass, maximum biomass, maximum growth rate, and lag phase duration. Further, a statistical analysis was performed to compare our cubic primary model with the ones proposed by Gompertz, Baranyi, and Vázquez-Murado by computing the coefficient of determination R2, the residual sum of squares RSS, and the Akaike Information Criterion AIC. Results: The average statistical results from the cubic model are as follows: R2=0.820 providing a better fit than the other three models, RSS=0.658 and AIC=−6.499, where both values are lower than the other models considered in this study. Conclusion: The cubic primary model formulated in this work describes the behavior of biomass as it accurately represents the four phases of biomass growth in batch fermentation process. Full article
(This article belongs to the Special Issue Mathematical Modeling of Biological Systems: Geometry, Symmetry and Conservation Laws)
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20 pages, 5548 KiB  
Article
Symmetric and Asymmetric Diffusions through Age-Varying Mixed-Species Stand Parameters
by Petras Rupšys and Edmundas Petrauskas
Symmetry 2021, 13(8), 1457; https://doi.org/10.3390/sym13081457 - 09 Aug 2021
Cited by 6 | Viewed by 1335
Abstract
(1) Background: This paper deals with unevenly aged, whole-stand models from mixed-effect parameters diffusion processes and Voronoi diagram points of view and concentrates on the mixed-species stands in Lithuania. We focus on the Voronoi diagram of potentially available areas to tree positions as [...] Read more.
(1) Background: This paper deals with unevenly aged, whole-stand models from mixed-effect parameters diffusion processes and Voronoi diagram points of view and concentrates on the mixed-species stands in Lithuania. We focus on the Voronoi diagram of potentially available areas to tree positions as the measure of the competition effect of individual trees and the tree diameter at breast height to relate their evolution through time. (2) Methods: We consider a bivariate hybrid mixed-effect parameters stochastic differential equation for the parameterization of the diameter and available polygon area at age to ensure a proper description of the link between them during the age (time) span of a forest stand. In this study, the Voronoi diagram was used as a mathematical tool for the quantitative characterization of inter-tree competition. (3) Results: The newly derived model considers bivariate correlated observations, tree diameter, and polygon area arising from a particular stand and enables defining equations for calculating diameter, polygon-area, and stand-density predictions and forecasts. (4) Conclusions: From a statistical point of view, the newly developed models produced acceptable statistical measures of predictions and forecasts. All the results were implemented in the Maple computer algebra system. Full article
(This article belongs to the Special Issue Mathematical Modeling of Biological Systems: Geometry, Symmetry and Conservation Laws)
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16 pages, 551 KiB  
Article
An Entropic Gradient Structure in the Network Dynamics of a Slime Mold
by Vincenzo Bonifaci
Symmetry 2021, 13(8), 1385; https://doi.org/10.3390/sym13081385 - 29 Jul 2021
Viewed by 1434
Abstract
The approach to equilibrium in certain dynamical systems can be usefully described in terms of information-theoretic functionals. Well-studied models of this kind are Markov processes, chemical reaction networks, and replicator dynamics, for all of which it can be proven, under suitable assumptions, that [...] Read more.
The approach to equilibrium in certain dynamical systems can be usefully described in terms of information-theoretic functionals. Well-studied models of this kind are Markov processes, chemical reaction networks, and replicator dynamics, for all of which it can be proven, under suitable assumptions, that the relative entropy (informational divergence) of the state of the system with respect to an equilibrium is nonincreasing over time. This work reviews another recent result of this type, which emerged in the study of the network optimization dynamics of an acellular slime mold, Physarum polycephalum. In this setting, not only the relative entropy of the state is nonincreasing, but its evolution over time is crucial to the stability of the entire system, and the equilibrium towards which the dynamics is attracted proves to be a global minimizer of the cost of the network. Full article
(This article belongs to the Special Issue Mathematical Modeling of Biological Systems: Geometry, Symmetry and Conservation Laws)
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15 pages, 1044 KiB  
Article
Nonlinear Analysis of the C-Peptide Variable Related to Type 1-Diabetes Mellitus
by Diana Gamboa, Carlos E. Vázquez-López, Rosana Gutierrez and Paul J. Campos
Symmetry 2021, 13(7), 1238; https://doi.org/10.3390/sym13071238 - 09 Jul 2021
Viewed by 1429
Abstract
Type-1 diabetes mellitus is a chronic disease that is constantly monitored worldwide by researchers who are strongly determined to establish mathematical and experimental strategies that lead to a breakthrough toward an immunological treatment or a mathematical model that would update the UVA/Padova algorithm. [...] Read more.
Type-1 diabetes mellitus is a chronic disease that is constantly monitored worldwide by researchers who are strongly determined to establish mathematical and experimental strategies that lead to a breakthrough toward an immunological treatment or a mathematical model that would update the UVA/Padova algorithm. In this work, we aim at a nonlinear mathematical analysis related to a fifth-order ordinary differential equations model that describes the asymmetric relation between C-peptides, pancreatic cells, and the immunological response. The latter is based on both the Localization of Compact Invariant Set (LCIS) appliance and Lyapunov’s stability theory to discuss the viability of implementing a possible treatment that stabilizes a specific set of cell populations. Our main result is to establish conditions for the existence of a localizing compact invariant domain that contains all the dynamics of diabetes mellitus. These conditions become essential for the localizing domain and stabilize the cell populations within desired levels, i.e., a state where a patient with diabetes could consider a healthy stage. Moreover, these domains demonstrate the cell populations’ asymmetric behavior since both the dynamics and the localizing domain of each cell population are defined into the positive orthant. Furthermore, closed-loop analysis is discussed by proposing two regulatory inputs opening the possibility of nonlinear control. Additionally, numerical simulations show that all trajectories converge inside the positive domain once given an initial condition. Finally, there is a discussion about the biological implications derived from the analytical results. Full article
(This article belongs to the Special Issue Mathematical Modeling of Biological Systems: Geometry, Symmetry and Conservation Laws)
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37 pages, 7729 KiB  
Article
Mathematical Models for Some Aspects of Blood Microcirculation
by Angiolo Farina, Antonio Fasano and Fabio Rosso
Symmetry 2021, 13(6), 1020; https://doi.org/10.3390/sym13061020 - 06 Jun 2021
Cited by 6 | Viewed by 2605
Abstract
Blood rheology is a challenging subject owing to the fact that blood is a mixture of a fluid (plasma) and of cells, among which red blood cells make about 50% of the total volume. It is precisely this circumstance that originates the peculiar [...] Read more.
Blood rheology is a challenging subject owing to the fact that blood is a mixture of a fluid (plasma) and of cells, among which red blood cells make about 50% of the total volume. It is precisely this circumstance that originates the peculiar behavior of blood flow in small vessels (i.e., roughly speaking, vessel with a diameter less than half a millimeter). In this class we find arterioles, venules, and capillaries. The phenomena taking place in microcirculation are very important in supporting life. Everybody knows the importance of blood filtration in kidneys, but other phenomena, of not less importance, are known only to a small class of physicians. Overviewing such subjects reveals the fascinating complexity of microcirculation. Full article
(This article belongs to the Special Issue Mathematical Modeling of Biological Systems: Geometry, Symmetry and Conservation Laws)
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21 pages, 2641 KiB  
Article
A Control Based Mathematical Model for the Evaluation of Intervention Lines in COVID-19 Epidemic Spread: The Italian Case Study
by Paolo Di Giamberardino, Rita Caldarella and Daniela Iacoviello
Symmetry 2021, 13(5), 890; https://doi.org/10.3390/sym13050890 - 17 May 2021
Cited by 3 | Viewed by 1981
Abstract
This paper addresses the problem of describing the spread of COVID-19 by a mathematical model introducing all the possible control actions as prevention (informative campaign, use of masks, social distancing, vaccination) and medication. The model adopted is similar to SEIQR, with the infected [...] Read more.
This paper addresses the problem of describing the spread of COVID-19 by a mathematical model introducing all the possible control actions as prevention (informative campaign, use of masks, social distancing, vaccination) and medication. The model adopted is similar to SEIQR, with the infected patients split into groups of asymptomatic subjects and isolated ones. This distinction is particularly important in the current pandemic, due to the fundamental the role of asymptomatic subjects in the virus diffusion. The influence of the control actions is considered in analysing the model, from the calculus of the equilibrium points to the determination of the reproduction number. This choice is motivated by the fact that the available organised data have been collected since from the end of February 2020, and almost simultaneously containment measures, increasing in typology and effectiveness, have been applied. The characteristics of COVID-19, not fully understood yet, suggest an asymmetric diffusion among countries and among categories of subjects. Referring to the Italian situation, the containment measures, as applied by the population, have been identified, showing their relation with the government’s decisions; this allows the study of possible scenarios, comparing the impact of different possible choices. Full article
(This article belongs to the Special Issue Mathematical Modeling of Biological Systems: Geometry, Symmetry and Conservation Laws)
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29 pages, 1646 KiB  
Article
Dynamics of an Eco-Epidemic Predator–Prey Model Involving Fractional Derivatives with Power-Law and Mittag–Leffler Kernel
by Hasan S. Panigoro, Agus Suryanto, Wuryansari Muharini Kusumawinahyu and Isnani Darti
Symmetry 2021, 13(5), 785; https://doi.org/10.3390/sym13050785 - 02 May 2021
Cited by 27 | Viewed by 2472
Abstract
In this paper, we consider a fractional-order eco-epidemic model based on the Rosenzweig–MacArthur predator–prey model. The model is derived by assuming that the prey may be infected by a disease. In order to take the memory effect into account, we apply two fractional [...] Read more.
In this paper, we consider a fractional-order eco-epidemic model based on the Rosenzweig–MacArthur predator–prey model. The model is derived by assuming that the prey may be infected by a disease. In order to take the memory effect into account, we apply two fractional differential operators, namely the Caputo fractional derivative (operator with power-law kernel) and the Atangana–Baleanu fractional derivative in the Caputo (ABC) sense (operator with Mittag–Leffler kernel). We take the same order of the fractional derivative in all equations for both senses to maintain the symmetry aspect. The existence and uniqueness of solutions of both eco-epidemic models (i.e., in the Caputo sense and in ABC sense) are established. Both models have the same equilibrium points, namely the trivial (origin) equilibrium point, the extinction of infected prey and predator point, the infected prey free point, the predator-free point and the co-existence point. For a model in the Caputo sense, we also show the non-negativity and boundedness of solution, perform the local and global stability analysis and establish the conditions for the existence of Hopf bifurcation. It is found that the trivial equilibrium point is a saddle point while other equilibrium points are conditionally asymptotically stable. The numerical simulations show that the solutions of the model in the Caputo sense strongly agree with analytical results. Furthermore, it is indicated numerically that the model in the ABC sense has quite similar dynamics as the model in the Caputo sense. The essential difference between the two models is the convergence rate to reach the stable equilibrium point. When a Hopf bifurcation occurs, the bifurcation points and the diameter of the limit cycles of both models are different. Moreover, we also observe a bistability phenomenon which disappears via Hopf bifurcation. Full article
(This article belongs to the Special Issue Mathematical Modeling of Biological Systems: Geometry, Symmetry and Conservation Laws)
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14 pages, 2126 KiB  
Article
SARS-COV-2: SIR Model Limitations and Predictive Constraints
by Charles Roberto Telles, Henrique Lopes and Diogo Franco
Symmetry 2021, 13(4), 676; https://doi.org/10.3390/sym13040676 - 14 Apr 2021
Cited by 12 | Viewed by 4241
Abstract
Background: The main purpose of this research is to describe the mathematical asymmetric patterns of susceptible, infectious, or recovered (SIR) model equation application in the light of coronavirus disease 2019 (COVID-19) skewness patterns worldwide. Methods: The research modeled severe acute respiratory syndrome coronavirus [...] Read more.
Background: The main purpose of this research is to describe the mathematical asymmetric patterns of susceptible, infectious, or recovered (SIR) model equation application in the light of coronavirus disease 2019 (COVID-19) skewness patterns worldwide. Methods: The research modeled severe acute respiratory syndrome coronavirus 2 (SARS-COV-2) spreading and dissemination patterns sensitivity by redesigning time series data extraction of daily new cases in terms of deviation consistency concerning variables that sustain COVID-19 transmission. The approach opened a new scenario where seasonality forcing behavior was introduced to understand SARS-COV-2 non-linear dynamics due to heterogeneity and confounding epidemics scenarios. Results: The main research results are the elucidation of three birth- and death-forced seasonality persistence phases that can explain COVID-19 skew patterns worldwide. They are presented in the following order: (1) the environmental variables (Earth seasons and atmospheric conditions); (2) health policies and adult learning education (HPALE) interventions; (3) urban spaces (local indoor and outdoor spaces for transit and social-cultural interactions, public or private, with natural physical features (river, lake, terrain). Conclusions: Three forced seasonality phases (positive to negative skew) phases were pointed out as a theoretical framework to explain uncertainty found in the predictive SIR model equations that might diverge in outcomes expected to express the disease’s behaviour. Full article
(This article belongs to the Special Issue Mathematical Modeling of Biological Systems: Geometry, Symmetry and Conservation Laws)
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