Advances in Differential Dynamical Systems with Applications to Economics and Biology

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Dynamical Systems".

Deadline for manuscript submissions: closed (31 March 2022) | Viewed by 25414

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1. Department of Economics and Business Administration, West University of Timişoara, 300115 Timişoara, Romania
2. Institute for Advanced Environmental Research, West University of Timişoara, 300223 Timişoara, Romania
Interests: nonlinear dynamics; economic modeling; differential equations; stability analysis; biomathematics; numerical simulation; mathematical modeling
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Guest Editor
1. Department of Mathematics and Computer Science, West University of Timişoara, 300223 Timişoara, Romania
2. Institute for Advanced Environmental Research, West University of Timişoara, 300223 Timişoara, Romania
Interests: dynamical systems; fractional-order differential equation; delay differential equations; mathematical modeling
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Department of Mathematics, State University of New York at New Paltz, New Paltz, NY 12561, USA
Interests: dynamical systems; mathematical biology; computational neuroscience
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Nowadays, in order to study economic and biological processes, mathematical modeling is a very useful tool. In economics and biology, a delay between cause and effect is very often taken into consideration. Sometimes, it is more practical to add a distributed time delay because it illustrates the situation where delays arise in certain ranges of values for certain related probability distributions, taking into account the variables’ entire historical behavior. Moreover, fractional derivatives instead of integer order derivatives may reflect the memory and the inherited properties of different systems. In terms of realistic conditions, stochastic perturbation framed by a stochastic differential delay system is used to explain the ambiguity about the context in which the system operates.

This Special Issue focuses on the dynamical analysis of mathematical models arising from economy and biology and innovative developments of mathematical techniques for their applications. Submissions that involve interdisciplinary collaborations are welcome, as are recent advances in both discrete and continuous techniques and significant applications. Numerical simulations can be used to emphasize the theoretical findings. Finally, an economic or biological interpretation of the obtained results is desired.

Prof. Dr. Mihaela Neamțu
Dr. Eva Kaslik
Dr. Anca Rădulescu
Guest Editors

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Keywords

  • dynamical systems
  • time delay
  • stability
  • bifurcation analysis in economic and biological systems
  • chaotic behavior
  • fractional order systems
  • numerical methods
  • numerical simulations

Published Papers (17 papers)

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Editorial

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3 pages, 174 KiB  
Editorial
Preface to the Special Issue on “Advances in Differential Dynamical Systems with Applications to Economics and Biology”
by Eva Kaslik, Mihaela Neamţu and Anca Rădulescu
Mathematics 2022, 10(19), 3561; https://doi.org/10.3390/math10193561 - 29 Sep 2022
Viewed by 727
Abstract
In recent research on natural processes, mathematical modeling has become a very useful tool [...] Full article

Research

Jump to: Editorial

17 pages, 3264 KiB  
Article
Evolutionary Optimization of Control Strategies for Non-Stationary Immersion Environments
by Alexander Musaev, Andrey Makshanov and Dmitry Grigoriev
Mathematics 2022, 10(11), 1797; https://doi.org/10.3390/math10111797 - 24 May 2022
Cited by 6 | Viewed by 1372
Abstract
We consider the problem of evolutionary self-organization of control strategies using the example of speculative trading in a non-stationary immersion market environment. The main issue that obstructs obtaining real profit is the extremely high instability of the system component of observation series which [...] Read more.
We consider the problem of evolutionary self-organization of control strategies using the example of speculative trading in a non-stationary immersion market environment. The main issue that obstructs obtaining real profit is the extremely high instability of the system component of observation series which implement stochastic chaos. In these conditions, traditional techniques for increasing the stability of control strategies are ineffective. In particular, the use of adaptive computational schemes is difficult due to the high volatility and non-stationarity of observation series. That leads to significant statistical errors of both kinds in the generated control decisions. An alternative approach based on the use of dynamic robustification technologies significantly reduces the effectiveness of the decisions. In the current work, we propose a method based on evolutionary modeling, which supplies structural and parametric self-organization of the control model. Full article
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20 pages, 8139 KiB  
Article
Numerical Studies of Channel Management Strategies for Nonstationary Immersion Environments: EURUSD Case Study
by Alexander Musaev, Andrey Makshanov and Dmitry Grigoriev
Mathematics 2022, 10(9), 1408; https://doi.org/10.3390/math10091408 - 22 Apr 2022
Cited by 7 | Viewed by 1207
Abstract
This article considers a short-term forecasting of a process that is an output signal of a nonlinear system observed on the background of additive noise. Forecasting is made possible thanks to the technique of nonparametric estimation of local trends. The main problem in [...] Read more.
This article considers a short-term forecasting of a process that is an output signal of a nonlinear system observed on the background of additive noise. Forecasting is made possible thanks to the technique of nonparametric estimation of local trends. The main problem in this case is the instability of the time of the existence of these local trends. The average duration of relatively stable intervals can be estimated from earlier observation history. Such approaches are called channel strategies. The task of constructing such strategies for EURUSD asset management in the conditions of market chaos is considered, as well as the potential capabilities of these management strategies via computational experiments. We demonstrated the fundamental possibility of achieving profit even for areas with complex dynamics with abrupt changes in the considered process. We propose improved channel strategies and also denote the main directions of increasing their effectiveness. Full article
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32 pages, 2659 KiB  
Article
A Study on Dynamics of CD4+ T-Cells under the Effect of HIV-1 Infection Based on a Mathematical Fractal-Fractional Model via the Adams-Bashforth Scheme and Newton Polynomials
by Hashem Najafi, Sina Etemad, Nichaphat Patanarapeelert, Joshua Kiddy K. Asamoah, Shahram Rezapour and Thanin Sitthiwirattham
Mathematics 2022, 10(9), 1366; https://doi.org/10.3390/math10091366 - 19 Apr 2022
Cited by 32 | Viewed by 1488
Abstract
In recent decades, AIDS has been one of the main challenges facing the medical community around the world. Due to the large human deaths of this disease, researchers have tried to study the dynamic behaviors of the infectious factor of this disease in [...] Read more.
In recent decades, AIDS has been one of the main challenges facing the medical community around the world. Due to the large human deaths of this disease, researchers have tried to study the dynamic behaviors of the infectious factor of this disease in the form of mathematical models in addition to clinical trials. In this paper, we study a new mathematical model in which the dynamics of CD4+ T-cells under the effect of HIV-1 infection are investigated in the context of a generalized fractal-fractional structure for the first time. The kernel of these new fractal-fractional operators is of the generalized Mittag-Leffler type. From an analytical point of view, we first derive some results on the existence theory and then the uniqueness criterion. After that, the stability of the given fractal-fractional system is reviewed under four different cases. Next, from a numerical point of view, we obtain two numerical algorithms for approximating the solutions of the system via the Adams-Bashforth method and Newton polynomials method. We simulate our results via these two algorithms and compare both of them. The numerical results reveal some stability and a situation of lacking a visible order in the early days of the disease dynamics when one uses the Newton polynomial. Full article
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10 pages, 305 KiB  
Article
Finite-Time Stability Analysis of Linear Differential Systems with Pure Delay
by Ahmed M. Elshenhab, Xingtao Wang, Omar Bazighifan and Jan Awrejcewicz
Mathematics 2022, 10(9), 1359; https://doi.org/10.3390/math10091359 - 19 Apr 2022
Cited by 6 | Viewed by 1221
Abstract
Nonhomogeneous systems governed by second-order linear differential equations with pure delay are considered. As an application, the exact solutions of these systems and their delayed matrix functions are used to obtain the finite-time stability results. Our results extend and improve some previous results [...] Read more.
Nonhomogeneous systems governed by second-order linear differential equations with pure delay are considered. As an application, the exact solutions of these systems and their delayed matrix functions are used to obtain the finite-time stability results. Our results extend and improve some previous results by removing some restrictive conditions. Finally, an example is provided to illustrate our theoretical results. Full article
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16 pages, 5048 KiB  
Article
Statistical Analysis of Current Financial Instrument Quotes in the Conditions of Market Chaos
by Alexander Musaev, Andrey Makshanov and Dmitry Grigoriev
Mathematics 2022, 10(4), 587; https://doi.org/10.3390/math10040587 - 14 Feb 2022
Cited by 6 | Viewed by 1479
Abstract
In this paper, the problem of estimating the current value of financial instruments using multidimensional statistical analysis is considered. The research considers various approaches to constructing regression computational schemes using quotes of financial instruments correlated to the data as regressors. An essential feature [...] Read more.
In this paper, the problem of estimating the current value of financial instruments using multidimensional statistical analysis is considered. The research considers various approaches to constructing regression computational schemes using quotes of financial instruments correlated to the data as regressors. An essential feature of the problem is the chaotic nature of its observation series, which is due to the instability of the probabilistic structure of the initial data. These conditions invalidate the constraints under which traditional statistical estimates remain non-biased and effective. Violation of experiment repeatability requirements obstructs the use of the conventional data averaging approach. In this case, numeric experiments become the main method for investigating the efficiency of forecasting and analysis algorithms of observation series. The empirical approach does not provide guaranteed results. However, it can be used to build sufficiently effective rational strategies for managing trading operations. Full article
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14 pages, 836 KiB  
Article
The Least Squares Homotopy Perturbation Method for Systems of Differential Equations with Application to a Blood Flow Model
by Mădălina Sofia Paşca, Olivia Bundău, Adina Juratoni and Bogdan Căruntu
Mathematics 2022, 10(4), 546; https://doi.org/10.3390/math10040546 - 10 Feb 2022
Cited by 5 | Viewed by 1405
Abstract
In this paper, least squares homotopy perturbation is presented as a straightforward and accurate method to compute approximate analytical solutions for systems of ordinary differential equations. The method is employed to solve a problem related to a laminar flow of a viscous fluid [...] Read more.
In this paper, least squares homotopy perturbation is presented as a straightforward and accurate method to compute approximate analytical solutions for systems of ordinary differential equations. The method is employed to solve a problem related to a laminar flow of a viscous fluid in a semi-porous channel, which may be used to model the blood flow through a blood vessel, taking into account the effects of a magnetic field. The numerical computations show that the method is both easy to use and very accurate compared to the other methods previously used to solve the given problem. Full article
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19 pages, 320 KiB  
Article
Lie Geometric Methods in the Study of Driftless Control Affine Systems with Holonomic Distribution and Economic Applications
by Liviu Popescu, Daniel Militaru and Gabriel Tică
Mathematics 2022, 10(4), 545; https://doi.org/10.3390/math10040545 - 10 Feb 2022
Cited by 2 | Viewed by 848
Abstract
In the present paper, two optimal control problems are studied using Lie geometric methods and applying the Pontryagin Maximum Principle at the level of a new working space, called Lie algebroid. It is proved that the framework of a Lie algebroid is more [...] Read more.
In the present paper, two optimal control problems are studied using Lie geometric methods and applying the Pontryagin Maximum Principle at the level of a new working space, called Lie algebroid. It is proved that the framework of a Lie algebroid is more suitable than the cotangent bundle in order to find the optimal solutions of some driftless control affine systems with holonomic distributions. Finally, an economic application is given. Full article
19 pages, 1803 KiB  
Article
Stability Analysis of Equilibria for a Model of Maintenance Therapy in Acute Lymphoblastic Leukemia
by Irina Badralexi, Andrei-Dan Halanay and Ragheb Mghames
Mathematics 2022, 10(3), 313; https://doi.org/10.3390/math10030313 - 20 Jan 2022
Cited by 2 | Viewed by 972
Abstract
In this paper, we study two mathematical models, involving delay differential equations, which describe the processes of erythropoiesis and leukopoiesis in the case of maintenance therapy for acute lymphoblastic leukemia. All types of possible equilibrium points were determined, and their stability was analyzed. [...] Read more.
In this paper, we study two mathematical models, involving delay differential equations, which describe the processes of erythropoiesis and leukopoiesis in the case of maintenance therapy for acute lymphoblastic leukemia. All types of possible equilibrium points were determined, and their stability was analyzed. For some of the equilibrium points, conditions for parameters that imply stability were obtained. When this was not feasible, due to the complexity of the characteristic equation, we discuss the stability through numerical simulations. An important part of the stability study for each model is the examination of the critical case of a zero root of the characteristic equation. The mathematical results are accompanied by biological interpretations. Full article
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14 pages, 1258 KiB  
Article
Numerical Studies of Statistical Management Decisions in Conditions of Stochastic Chaos
by Alexander Musaev and Dmitry Grigoriev
Mathematics 2022, 10(2), 226; https://doi.org/10.3390/math10020226 - 12 Jan 2022
Cited by 13 | Viewed by 2323
Abstract
The research presented in this article is dedicated to analyzing the acceptability of traditional techniques of statistical management decision-making in conditions of stochastic chaos. A corresponding example would be asset management at electronic capital markets. This formulation of the problem is typical for [...] Read more.
The research presented in this article is dedicated to analyzing the acceptability of traditional techniques of statistical management decision-making in conditions of stochastic chaos. A corresponding example would be asset management at electronic capital markets. This formulation of the problem is typical for a large number of applications in which the managed object interacts with an unstable immersion environment. In particular, this issue arises in problems of managing gas-dynamic and hydrodynamic turbulent flows. We highlight the features of observation series of the managed object’s state immersed in an unstable interaction environment. The fundamental difference between observation series of chaotic processes and probabilistic descriptions of traditional models is demonstrated. We also present an additive observation model with a chaotic system component and non-stationary noise which provides the most adequate characterization of the original observation series. Furthermore, we suggest a method for numerically analyzing the efficiency of conventional statistical solutions in the conditions of stochastic chaos. Based on numerical experiments, we establish that techniques of optimal statistical synthesis do not allow for making effective management decisions in the conditions of stochastic chaos. Finally, we propose several versions of compositional algorithms focused on the adaptation of statistical techniques to the non-deterministic conditions caused by the specifics of chaotic processes. Full article
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32 pages, 3719 KiB  
Article
An SIRS Epidemic Model Supervised by a Control System for Vaccination and Treatment Actions Which Involve First-Order Dynamics and Vaccination of Newborns
by Santiago Alonso-Quesada, Manuel De la Sen and Raúl Nistal
Mathematics 2022, 10(1), 36; https://doi.org/10.3390/math10010036 - 23 Dec 2021
Cited by 5 | Viewed by 2309
Abstract
This paper analyses an SIRS epidemic model with the vaccination of susceptible individuals and treatment of infectious ones. Both actions are governed by a designed control system whose inputs are the subpopulations of the epidemic model. In addition, the vaccination of a proportion [...] Read more.
This paper analyses an SIRS epidemic model with the vaccination of susceptible individuals and treatment of infectious ones. Both actions are governed by a designed control system whose inputs are the subpopulations of the epidemic model. In addition, the vaccination of a proportion of newborns is considered. The control reproduction number Rc of the controlled epidemic model is calculated, and its influence in the existence and stability of equilibrium points is studied. If such a number is smaller than a threshold value R¯c, then the model has a unique equilibrium point: the so-called disease-free equilibrium point at which there are not infectious individuals. Furthermore, such an equilibrium point is locally and globally asymptotically stable. On the contrary, if Rc>R¯c, then the model has two equilibrium points: the referred disease-free one, which is unstable, and an endemic one at which there are infectious individuals. The proposed control strategy provides several free-design parameters that influence both values Rc and R¯c. Then, such parameters can be appropriately adjusted for guaranteeing the non-existence of the endemic equilibrium point and, in this way, eradicating the persistence of the infectious disease. Full article
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15 pages, 511 KiB  
Article
Global Stability Analysis of a Five-Dimensional Unemployment Model with Distributed Delay
by Eva Kaslik, Mihaela Neamţu and Loredana Flavia Vesa
Mathematics 2021, 9(23), 3037; https://doi.org/10.3390/math9233037 - 26 Nov 2021
Cited by 5 | Viewed by 1219
Abstract
The present paper proposes a five-dimensional mathematical model for studying the labor market, focusing on unemployment, migration, fixed term contractors, full time employment and the number of available vacancies. The distributed time delay is considered in the rate of change of available vacancies [...] Read more.
The present paper proposes a five-dimensional mathematical model for studying the labor market, focusing on unemployment, migration, fixed term contractors, full time employment and the number of available vacancies. The distributed time delay is considered in the rate of change of available vacancies that depends on the past regular employment levels. The non-dimensional mathematical model is introduced and the existence of the equilibrium points is analyzed. The positivity and boundedness of solutions are provided and global asymptotic stability findings are presented both for the employment free equilibrium and the positive equilibrium. The numerical simulations support the theoretical results. Full article
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20 pages, 454 KiB  
Article
Global Dynamics for an Age-Structured Cholera Infection Model with General Infection Rates
by Xin Jiang
Mathematics 2021, 9(23), 2993; https://doi.org/10.3390/math9232993 - 23 Nov 2021
Cited by 2 | Viewed by 1330
Abstract
This paper studies the global dynamics of a cholera model incorporating age structures and general infection rates. First, we explore the existence and point dissipativeness of the orbit and analyze the asymptotical smoothness. Then, we perform rigorous mathematical analysis on the existence and [...] Read more.
This paper studies the global dynamics of a cholera model incorporating age structures and general infection rates. First, we explore the existence and point dissipativeness of the orbit and analyze the asymptotical smoothness. Then, we perform rigorous mathematical analysis on the existence and local stability of equilibria. Based on the uniform persistence, we further investigate the global behavior of the cholera infection model. The results of theoretical analysis are well confirmed by numerical simulations. This research generalizes some known results and provides deeper insights into the dynamics of cholera propagation. Full article
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13 pages, 1124 KiB  
Article
Graph, Spectra, Control and Epidemics: An Example with a SEIR Model
by Giacomo Aletti, Alessandro Benfenati and Giovanni Naldi
Mathematics 2021, 9(22), 2987; https://doi.org/10.3390/math9222987 - 22 Nov 2021
Cited by 3 | Viewed by 1914
Abstract
Networks and graphs offer a suitable and powerful framework for studying the spread of infection in human and animal populations. In the case of a heterogeneous population, the social contact network has a pivotal role in the analysis of directly transmitted infectious diseases. [...] Read more.
Networks and graphs offer a suitable and powerful framework for studying the spread of infection in human and animal populations. In the case of a heterogeneous population, the social contact network has a pivotal role in the analysis of directly transmitted infectious diseases. The literature presents several works where network-based models encompass realistic features (such as contacts networks or host–pathogen biological data), but analytical results are nonetheless scarce. As a significant example, in this paper, we develop a multi-group version of the epidemiological SEIR population-based model. Each group can represent a social subpopulation with the same habits or a group of geographically localized people. We consider also heterogeneity in the weighting of contacts between two groups. As a simple application, we propose a simple control algorithm in which we optimize the connection weights in order to minimize the combination between an economic cost and a social cost. Some numerical simulations are also provided. Full article
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12 pages, 6898 KiB  
Article
A Comparative Numerical Study and Stability Analysis for a Fractional-Order SIR Model of Childhood Diseases
by Mohamed M. Mousa and Fahad Alsharari
Mathematics 2021, 9(22), 2847; https://doi.org/10.3390/math9222847 - 10 Nov 2021
Cited by 2 | Viewed by 1252
Abstract
The objective of this work is to examine the dynamics of a fractional-order susceptible-infectious-recovered (SIR) model that simulate epidemiological diseases such as childhood diseases. An effective numerical scheme based on Grünwald–Letnikov fractional derivative is suggested to solve the considered model. A stability analysis [...] Read more.
The objective of this work is to examine the dynamics of a fractional-order susceptible-infectious-recovered (SIR) model that simulate epidemiological diseases such as childhood diseases. An effective numerical scheme based on Grünwald–Letnikov fractional derivative is suggested to solve the considered model. A stability analysis is performed to qualitatively examine the dynamics of the SIR model. The reliability and robustness of the proposed scheme is demonstrated by comparing obtained results with results obtained from a fourth order Runge–Kutta built-in Maple syntax when considering derivatives of integer order. Graphical illustrations of the numerical results are given. The inaccuracy of some results presented in two studies exist in the literature have been clearly explained. Generalizing of the cases examined in another study, by considering a model with fraction-order derivatives, is another objective of this work as well. Full article
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16 pages, 3309 KiB  
Article
Computational Study on the Dynamics of a Consumer-Resource Model: The Influence of the Growth Law in the Resource
by Luis M. Abia, Óscar Angulo, Juan Carlos López-Marcos and Miguel Ángel López-Marcos
Mathematics 2021, 9(21), 2746; https://doi.org/10.3390/math9212746 - 29 Oct 2021
Cited by 1 | Viewed by 1177
Abstract
The dynamics of a specific consumer-resource model for Daphnia magna is studied from a numerical point of view. In this study, Malthusian, chemostatic, and Gompertz growth laws for the evolution of the resource population are considered, and the resulting global dynamics of the [...] Read more.
The dynamics of a specific consumer-resource model for Daphnia magna is studied from a numerical point of view. In this study, Malthusian, chemostatic, and Gompertz growth laws for the evolution of the resource population are considered, and the resulting global dynamics of the model are compared as different parameters involved in the model change. In the case of Gompertz growth law, a new complex dynamic is found as the carrying capacity for the resource population increases. The numerical study is carried out with a second-order scheme that approximates the size-dependent density function for individuals in the consumer population. The numerical method is well adapted to the situation in which the growth rate for the consumer individuals is allowed to change the sign and, therefore, individuals in the consumer population can shrink in size as time evolves. The numerical simulations confirm that the shortage of the resource has, as a biological consequence, the effective shrink in size of individuals of the consumer population. Moreover, the choice of the growth law for the resource population can be selected by how the dynamics of the populations match with the qualitative behaviour of the data. Full article
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13 pages, 1178 KiB  
Article
Integrable Deformations and Dynamical Properties of Systems with Constant Population
by Cristian Lăzureanu
Mathematics 2021, 9(12), 1378; https://doi.org/10.3390/math9121378 - 14 Jun 2021
Cited by 1 | Viewed by 1362
Abstract
In this paper we consider systems of three autonomous first-order differential equations x˙=f(x),x=(x,y,z),f=(f1,f2,f3) such [...] Read more.
In this paper we consider systems of three autonomous first-order differential equations x˙=f(x),x=(x,y,z),f=(f1,f2,f3) such that x(t)+y(t)+z(t) is constant for all t. We present some Hamilton–Poisson formulations and integrable deformations. We also analyze the case of Kolmogorov systems. We study from some standard and nonstandard Poisson geometry points of view the three-dimensional Lotka–Volterra system with constant population. Full article
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