Submit to Special Issue Submit Abstract to Special Issue Review for Mathematics Propose a Special Issue

Journal Menu

Journal Browser

► Journal Browser

Advances in Differential Dynamical Systems with Applications to Economics and Biology, 2nd Edition

Print Special Issue Flyer
Special Issue Editors
Special Issue Information
Keywords
Published Papers

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

Deadline for manuscript submissions: 30 June 2024 | Viewed by 9349

Share This Special Issue

Special Issue Editors

Prof. Dr. Mihaela Neamțu

E-Mail Website
Guest Editor

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
Special Issues, Collections and Topics in MDPI journals

Prof. Dr. Eva Kaslik

E-Mail Website
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
Special Issues, Collections and Topics in MDPI journals

Dr. Anca Rădulescu

E-Mail Website
Guest Editor

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, the 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
Prof. Dr. Eva Kaslik
Dr. Anca Rădulescu
Guest Editors

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

dynamical systems
time delay
stability
bifurcation analysis in economic and biological systems
chaotic behavior
population dynamics
reaction-diffusion systems
economic dynamics
fractional order systems
numerical methods
numerical simulations

Published Papers (11 papers)

Download All Papers

Order results

Result details

Show export options Show export options

Select all

Export citation of selected articles as:

Research

12 pages, 275 KiB

Open AccessFeature PaperArticle

Dynamic Cooperative Oligopolies

by Ferenc Szidarovszky and Akio Matsumoto

Mathematics 2024, 12(6), 891; https://doi.org/10.3390/math12060891 - 18 Mar 2024

Viewed by 467

Abstract

An n-person cooperative oligopoly is considered without product differentiation. It is assumed that the firms know the unit price function but have no access to the cost functions of the competitors. From market data, they have information about the industry output. The firms want to find the output levels that guarantee maximum industry profit. First, the existence of a unique maximizer is proven, which the firms cannot determine directly because of the lack of the knowledge of the cost functions. Instead, a dynamic model is constructed, which is asymptotically stable under realistic conditions, and the state trajectories converge to the optimum output levels of the firms. Three models are constructed: first, no time delay is assumed; second, information delay is considered for the firms on the industry output; and third, in addition, information delay is also assumed about the firms’ own output levels. The stability of the resulting no-delay, one-delay, and two-delay dynamics is examined. Full article

(This article belongs to the Special Issue Advances in Differential Dynamical Systems with Applications to Economics and Biology, 2nd Edition)

► Show Figures

Figure 1

19 pages, 671 KiB

Open AccessArticle

On the Analytical Solution of the SIRV-Model for the Temporal Evolution of Epidemics for General Time-Dependent Recovery, Infection and Vaccination Rates

by Martin Kröger and Reinhard Schlickeiser

Mathematics 2024, 12(2), 326; https://doi.org/10.3390/math12020326 - 19 Jan 2024

Cited by 1 | Viewed by 587

Abstract

The susceptible–infected–recovered/removed–vaccinated (SIRV) epidemic model is an important generalization of the SIR epidemic model, as it accounts quantitatively for the effects of vaccination campaigns on the temporal evolution of epidemic outbreaks. Additional to the time-dependent infection (

a (t)

) and [...] Read more.

a (t)

) and recovery (

μ (t)

) rates, regulating the transitions between the compartments

S \to I

and

I \to R

, respectively, the time-dependent vaccination rate

v (t)

accounts for the transition between the compartments

S \to V

of susceptible to vaccinated fractions. An accurate analytical approximation is derived for arbitrary and different temporal dependencies of the rates, which is valid for all times after the start of the epidemics for which the cumulative fraction of new infections

J (t) ≪ 1

. As vaccination campaigns automatically reduce the rate of new infections by transferring persons from susceptible to vaccinated, the limit

J (t) ≪ 1

is even better fulfilled than in the SIR-epidemic model. The comparison of the analytical approximation for the temporal dependence of the rate of new infections

\overset{˚}{J} (t) = a (t) S (t) I (t)

, the corresponding cumulative fraction

J (t)

, and

V (t)

, respectively, with the exact numerical solution of the SIRV-equations for different illustrative examples proves the accuracy of our approach. The considered illustrative examples include the cases of stationary ratios with a delayed start of vaccinations, and an oscillating ratio of recovery to infection rate with a delayed vaccination at constant rate. The proposed analytical approximation is self-regulating as the final analytical expression for the cumulative fraction

J_{\infty}

after infinite time allows us to check the validity of the original assumption

J (t) \leq J_{\infty} ≪ 1

. Full article

(This article belongs to the Special Issue Advances in Differential Dynamical Systems with Applications to Economics and Biology, 2nd Edition)

► Show Figures

Figure 1

12 pages, 300 KiB

Open AccessArticle

Analyzing the Asymptotic Behavior of an Extended SEIR Model with Vaccination for COVID-19

by Vasileios E. Papageorgiou, Georgios Vasiliadis and George Tsaklidis

Mathematics 2024, 12(1), 55; https://doi.org/10.3390/math12010055 - 23 Dec 2023

Cited by 2 | Viewed by 728

Abstract

Several research papers have attempted to describe the dynamics of COVID-19 based on systems of differential equations. These systems have taken into account quarantined or isolated cases, vaccinations, control measures, and demographic parameters, presenting propositions regarding theoretical results that often investigate the asymptotic behavior of the system. In this paper, we discuss issues that concern the theoretical results proposed in the paper “An Extended SEIR Model with Vaccination for Forecasting the COVID-19 Pandemic in Saudi Arabia Using an Ensemble Kalman Filter”. We propose detailed explanations regarding the resolution of these issues. Additionally, this paper focuses on extending the local stability analysis of the disease-free equilibrium, as presented in the aforementioned paper, while emphasizing the derivation of theorems that validate the global stability of both epidemic equilibria. Emphasis is placed on the basic reproduction number

R_{0}

, which determines the asymptotic behavior of the system. This index represents the expected number of secondary infections that are generated from an already infected case in a population where almost all individuals are susceptible. The derived propositions can inform health authorities about the long-term behavior of the phenomenon, potentially leading to more precise and efficient public measures. Finally, it is worth noting that the examined paper still presents an interesting epidemiological scheme, and the utilization of the Kalman filtering approach remains one of the state-of-the-art methods for modeling epidemic phenomena. Full article

(This article belongs to the Special Issue Advances in Differential Dynamical Systems with Applications to Economics and Biology, 2nd Edition)

40 pages, 719 KiB

Open AccessArticle

Algorithmic Approach for a Unique Definition of the Next-Generation Matrix

by Florin Avram, Rim Adenane, Lasko Basnarkov and Matthew D. Johnston

Mathematics 2024, 12(1), 27; https://doi.org/10.3390/math12010027 - 21 Dec 2023

Viewed by 653

Abstract

The basic reproduction number

R_{0}

is a concept which originated in population dynamics, mathematical epidemiology, and ecology and is closely related to the mean number of children in branching processes (reflecting the fact that the phenomena of interest are well approximated via [...] Read more.

The basic reproduction number

R_{0}

R_{0}

for deterministic epidemic models, we believe there are still aspects which are not fully understood. Foremost is the fact that

R_{0}

is not a function of the original ODE model, unless we also include in it a certain

(F, V)

gradient decomposition, which is not unique. This is related to the specification of the “infected compartments”, which is also not unique. A second interesting question is whether the extinction probabilities of the natural continuous time Markovian chain approximation of an ODE model around boundary points (disease-free equilibrium and invasion points) are also related to the

(F, V)

gradient decomposition. We offer below several new contributions to the literature: (1) A universal algorithmic definition of a

(F, V)

gradient decomposition (and hence of the resulting

R_{0}

). (2) A fixed point equation for the extinction probabilities of a stochastic model associated to a deterministic ODE model, which may be expressed in terms of the

(F, V)

decomposition. Last but not least, we offer Mathematica scripts and implement them for a large variety of examples, which illustrate that our recipe offers always reasonable results, but that sometimes other reasonable

(F, V)

decompositions are available as well. Full article

(This article belongs to the Special Issue Advances in Differential Dynamical Systems with Applications to Economics and Biology, 2nd Edition)

► Show Figures

Figure 1

16 pages, 810 KiB

Open AccessArticle

Asymptotic Behavior for a Coupled Petrovsky–Petrovsky System with Infinite Memories

by Hicham Saber, Mohamed Ferhat, Amin Benaissa Cherif, Tayeb Blouhi, Ahmed Himadan, Tariq Alraqad and Abdelkader Moumen

Mathematics 2023, 11(21), 4457; https://doi.org/10.3390/math11214457 - 27 Oct 2023

Viewed by 591

Abstract

The main goal of this article is to obtain the existence of solutions for a nonlinear system of a coupled Petrovsky–Petrovsky system in the presence of infinite memories under minimal assumptions on the functions

g_{1}, g_{2}

and [...] Read more.

g_{1}, g_{2}

and

φ_{1}, φ_{2}

. Here,

g_{1}, g_{2}

are relaxation functions and

φ_{1}, φ_{2}

represent the sources. Also, a general decay rate for the associated energy is established. Our work is partly motivated by recent results, with a necessary modification imposed by the nature of our problem. In this work, we limit our results to studying the system in a bounded domain. The case of the entire domain

R^{n}

requires separate consideration. Of course, obtaining such a result will require not only serious technical work but also the use of new techniques and methods. In particular, one of the most significant points in achieving this goal is the use of the perturbed Lyapunov functionals combined with the multiplier method. To the best of our knowledge, there is no result addressing the linked Petrovsky–Petrovsky system in the presence of infinite memory, and we have overcome this lacune. Full article

(This article belongs to the Special Issue Advances in Differential Dynamical Systems with Applications to Economics and Biology, 2nd Edition)

21 pages, 47046 KiB

Open AccessArticle

Mathematical Modeling and Stability Analysis of the Delayed Pine Wilt Disease Model Related to Prevention and Control

by Ruilin Dong, Haokun Sui and Yuting Ding

Mathematics 2023, 11(17), 3705; https://doi.org/10.3390/math11173705 - 28 Aug 2023

Viewed by 716

Abstract

Forest pests and diseases have been seriously threatening ecological security. Effective prevention and control of such threats can extend the growth cycle of forest trees and increase the amount of forest carbon sink, which makes a contribution to achieving China’s goal of “emission peak and carbon neutrality”. In this paper, based on the insect-vector populations (this refers to Monochamus alternatus, which is the main vector in Asia) in pine wilt disease, we establish a two-dimensional delay differential equation model to investigate disease control and the impact of time delay on the effectiveness of it. Then, we analyze the existence and stability of the equilibrium of the system and the existence of Hopf bifurcation, derive the normal form of Hopf bifurcation by using a multiple time scales method, and conduct numerical simulations with realistic parameters to verify the correctness of the theoretical analysis. Eventually, according to theoretical analysis and numerical simulations, some specific suggestions are put forward for prevention and control of pine wilt disease. Full article

(This article belongs to the Special Issue Advances in Differential Dynamical Systems with Applications to Economics and Biology, 2nd Edition)

► Show Figures

Figure 1

21 pages, 1114 KiB

Open AccessArticle

Stability Analysis in a New Model for Desensitization of Allergic Reactions Induced by Chemotherapy of Chronic Lymphocytic Leukemia

by Rawan Abdullah, Irina Badralexi and Andrei Halanay

Mathematics 2023, 11(14), 3225; https://doi.org/10.3390/math11143225 - 22 Jul 2023

Cited by 1 | Viewed by 648

Abstract

We introduce a new model that captures the cellular evolution of patients with chronic lymphocytic leukemia who are receiving chemotherapy. As chemotherapy can induce allergic reactions and tumor lysis syndrome, we took into account the process of desensitization and the number of dead leukemic cells in the body. The mathematical model uses delayed-differential equations. Qualitative properties of the solutions are proved, including partial stability with respect to some variables and to the invariant set of positive initial data. Numerical simulations are also used to complete the description of the interplay between the immune system’s function, the chemotherapeutic activity and the allergic reactions caused by the therapy. Full article

(This article belongs to the Special Issue Advances in Differential Dynamical Systems with Applications to Economics and Biology, 2nd Edition)

► Show Figures

Figure 1

15 pages, 2664 KiB

Open AccessArticle

Semi-Analytical Analysis of Drug Diffusion through a Thin Membrane Using the Differential Quadrature Method

by Abdelfattah Mustafa, Reda S. Salama and Mokhtar Mohamed

Mathematics 2023, 11(13), 2998; https://doi.org/10.3390/math11132998 - 05 Jul 2023

Cited by 2 | Viewed by 811

Abstract

The primary goal of this work is to solve the problem of drug diffusion through a thin membrane using a differential quadrature approach with drastically different shape functions, such as Lagrange interpolation and discrete singular convolution (the delta Lagrange kernel and the regularized Shannon kernel). A nonlinear partial differential equation with two time- and space-dependent variables governs the system. To reduce the two independent variables by one, the partial differential equation is transformed into an ordinary differential equation using a one-parameter group transformation. With the aid of the iterative technique, the differential quadrature methods change this equation into an algebraic equation. Then, using a MATLAB program, a code is created that solves this equation for each shape function. To ensure the validity, efficiency, and accuracy of the developed techniques, the computed results are compared to previous numerical and analytical solutions. In addition, the L∞ error is applied. As a consequence of the numerical outcomes, the differential quadrature method, which is primarily based on a discrete singular convolution shape function, is an effective numerical method that can be used to solve the problem of drug diffusion through a thin membrane, guaranteeing a higher accuracy, faster convergence, and greater reliability than other techniques. Full article

(This article belongs to the Special Issue Advances in Differential Dynamical Systems with Applications to Economics and Biology, 2nd Edition)

► Show Figures

Figure 1

24 pages, 517 KiB

Open AccessArticle

Stability and Bifurcations in a Nutrient–Phytoplankton–Zooplankton Model with Delayed Nutrient Recycling with Gamma Distribution

by Mihaela Sterpu, Carmen Rocşoreanu, Raluca Efrem and Sue Ann Campbell

Mathematics 2023, 11(13), 2911; https://doi.org/10.3390/math11132911 - 28 Jun 2023

Viewed by 1323

Abstract

Two nutrient–phytoplankton–zooplankton (NZP) models for a closed ecosystem that incorporates a delay in nutrient recycling, obtained using the gamma distribution function with one or two degrees of freedom, are analysed. The models are described by systems of ordinary differential equations of four and five dimensions. The purpose of this study is to investigate how the mean delay of the distribution and the total nutrients affect the stability of the equilibrium solutions. Local stability theory and bifurcation theory are used to determine the long-time dynamics of the models. It is found that both models exhibit comparable qualitative dynamics. There are a maximum of three equilibrium points in each of the two models, and at most one of them is locally asymptotically stable. The change of stability from one equilibrium to another takes place through a transcritical bifurcation. In some hypotheses on the functional response, the nutrient–phytoplankton–zooplankton equilibrium loses stability via a supercritical Hopf bifurcation, causing the apparition of a stable limit cycle. The way in which the results are consistent with prior research and how they extend them is discussed. Finally, various application-related consequences of the results of the theoretical study are deduced. Full article

(This article belongs to the Special Issue Advances in Differential Dynamical Systems with Applications to Economics and Biology, 2nd Edition)

► Show Figures

Figure 1

29 pages, 576 KiB

Open AccessArticle

Stochastic Dynamics of a Virus Variant Epidemic Model with Double Inoculations

by Hui Chen, Xuewen Tan, Jun Wang, Wenjie Qin and Wenhui Luo

Mathematics 2023, 11(7), 1712; https://doi.org/10.3390/math11071712 - 03 Apr 2023

Viewed by 1056

Abstract

In this paper, we establish a random epidemic model with double vaccination and spontaneous variation of the virus. Firstly, we prove the global existence and uniqueness of positive solutions for a stochastic epidemic model. Secondly, we prove the threshold

R_{0}^{*}

can [...] Read more.

R_{0}^{*}

can be used to control the stochastic dynamics of the model. If

R_{0}^{*} < 0

, the disease will be extinct with probability 1; whereas if

R_{0}^{*} > 0

, the disease can almost certainly continue to exist, and there is a unique stable distribution. Finally, we give some numerical examples to verify our theoretical results. Most of the existing studies prove the stochastic dynamics of the model by constructing Lyapunov functions. However, the construction of a Lyapunov function of higher-order models is extremely complex, so this method is not applicable to all models. In this paper, we use the definition method suitable for more models to prove the stationary distribution. Most of the stochastic infectious disease models studied now are second-order or third-order, and cannot accurately describe infectious diseases. In order to solve this kind of problem, this paper adopts a higher price five-order model. Full article

(This article belongs to the Special Issue Advances in Differential Dynamical Systems with Applications to Economics and Biology, 2nd Edition)

► Show Figures

Figure 1

15 pages, 479 KiB

Open AccessArticle

Dynamics of a Four-Dimensional Economic Model

by Gheorghe Moza, Oana Brandibur and Ariana Găină

Mathematics 2023, 11(4), 797; https://doi.org/10.3390/math11040797 - 04 Feb 2023

Cited by 1 | Viewed by 965

Abstract

The interdependency between interest rates, investment demands and inflation rates in a given economy has a continuous dynamics. We propose a four-dimensional model which describes these interactions by imposing a control law on the interest rate. By a qualitative analysis based on tools from dynamical systems theory, we obtain in the new model that the three economic indicators can be stabilized to three equilibrium states. Full article

(This article belongs to the Special Issue Advances in Differential Dynamical Systems with Applications to Economics and Biology, 2nd Edition)

► Show Figures

Journal Menu

Journal Browser

Advances in Differential Dynamical Systems with Applications to Economics and Biology, 2nd Edition

Share This Special Issue

Special Issue Editors

Special Issue Information

Keywords

Published Papers (11 papers)

Research

Further Information

Guidelines

MDPI Initiatives

Follow MDPI