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Mathematics
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Models of Delay Differential Equations - II
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Models of Delay Differential Equations - II

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: closed (31 January 2023) | Viewed by 9190

Special Issue Editors

Prof. Dr. Francisco Rodríguez

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Guest Editor
Department of Applied Mathematics and Multidisciplinary Institute for Environmental Studies (IMEM), University of Alicante, Apdo. 99, 03080 Alicante, Spain
Interests: delay differential equations; diffusion and heat conduction models with delay; mathematical biology
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Juan Carlos Cortés López

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Guest Editor
Department of Applied Mathematics and Institute for Multidisciplinary Mathematics (im2), Universitat Politècnica de València, 46022 Valencia, Spain
Interests: differential equations with randomness; mathematical modelling
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. María Ángeles Castro

E-Mail Website
Guest Editor
Department of Applied Mathematics, University of Alicante, Apdo. 99, 03080 Alicante, Spain
Interests: delay differential equations; numerical methods; non-Fourier heat conduction models
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Models of differential equations with delay have pervaded many scientific and technical fields in the last decades. The use of delay differential equations (DDE) and partial delay differential equations (PDDE) to model problems with the presence of lags or hereditary effects have demonstrated a valuable balance between realism and tractability. Of special interest in recent years is the development and analysis of models with interactions between delay and random effects, through the use of stochastic and random delay differential equation (SDDE and RDDE). In this Special Issue, we are inviting submissions of original papers dealing with the theory and applications of differential equations with delay (DDE, PDDE, SDDE, and RDDE), including, but not limited to, construction of exact solutions, numerical methods, dynamical properties, and applications to mathematical modeling of phenomena and processes in biology, medicine, economics, engineering, and the social sciences. 

Prof. Dr. Francisco Rodríguez
Prof. Dr. Juan Carlos Cortés López
Prof. Dr. María Ángeles Castro
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

  • Delay differential equations
  • Partial delay differential equations
  • Random and stochastic delay differential equations
  • Numerical methods
  • Exact solutions and dynamical properties
  • Diffusion and heat conduction models with delay
  • Uncertainty quantification with delay differential equations and simulation
  • Models with delay in biology, economics, and engineering

Published Papers (6 papers)

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Research

22 pages, 2005 KiB  
Article
Delays and Exposed Populations in Infection Models
by Benito Chen-Charpentier
Mathematics 2023, 11(8), 1919; https://0-doi-org.brum.beds.ac.uk/10.3390/math11081919 - 19 Apr 2023
Cited by 2 | Viewed by 1055
Abstract
Most biological processes take time to occur. In infectious diseases, such as malaria or chikungunya, there is a period of time between when a susceptible individual gets bitten by the vector, and when the individual develops the disease. These times are commonly modeled [...] Read more.
Most biological processes take time to occur. In infectious diseases, such as malaria or chikungunya, there is a period of time between when a susceptible individual gets bitten by the vector, and when the individual develops the disease. These times are commonly modeled by introducing delays or by adding exposed as a new population class. Given a model based on differential equations, delays can be introduced in different forms. In this paper, we study different ways of introducing the delays and, alternatively, using exposed populations. We also analyze their steady solutions and stability, and establish the conditions under which the studied models predict an epidemic. Results and conclusions are presented. Full article
(This article belongs to the Special Issue Models of Delay Differential Equations - II)
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9 pages, 494 KiB  
Article
On the Construction of Exact Numerical Schemes for Linear Delay Models
by Carlos Julio Mayorga, María Ángeles Castro, Antonio Sirvent and Francisco Rodríguez
Mathematics 2023, 11(8), 1836; https://0-doi-org.brum.beds.ac.uk/10.3390/math11081836 - 12 Apr 2023
Cited by 1 | Viewed by 896
Abstract
Exact numerical schemes have previously been obtained for some linear retarded delay differential equations and systems. Those schemes were derived from explicit expressions of the exact solutions, and were expressed in the form of perturbed difference systems, involving the values at previous delay [...] Read more.
Exact numerical schemes have previously been obtained for some linear retarded delay differential equations and systems. Those schemes were derived from explicit expressions of the exact solutions, and were expressed in the form of perturbed difference systems, involving the values at previous delay intervals. In this work, we propose to directly obtain expressions of the same type for the fundamental solutions of linear delay differential equations, by considering vector equations with vector components corresponding to delay-lagged values at previous intervals. From these expressions for the fundamental solutions, exact numerical schemes for arbitrary initial functions can be proposed, and they may also facilitate obtaining explicit exact solutions. We apply this approach to obtain an exact numerical scheme for the first order linear neutral equation x′(t)−γx′(t−τ)=αx(t)+βx(t−τ), with the general initial condition x(t)=φ(t) for −τ≤t≤0. The resulting expression reduces to those previously published for the corresponding retarded equations when γ=0. Full article
(This article belongs to the Special Issue Models of Delay Differential Equations - II)
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19 pages, 352 KiB  
Article
Van der Pol Equation with a Large Feedback Delay
by Sergey Kashchenko
Mathematics 2023, 11(6), 1301; https://0-doi-org.brum.beds.ac.uk/10.3390/math11061301 - 08 Mar 2023
Cited by 4 | Viewed by 843
Abstract
The well-known Van der Pol equation with delayed feedback is considered. It is assumed that the delay factor is large enough. In the study of the dynamics, the critical cases in the problem of the stability of the zero equilibrium state are identified. [...] Read more.
The well-known Van der Pol equation with delayed feedback is considered. It is assumed that the delay factor is large enough. In the study of the dynamics, the critical cases in the problem of the stability of the zero equilibrium state are identified. It is shown that they have infinite dimension. For such critical cases, special local analysis methods have been developed. The main result is the construction of nonlinear evolutionary boundary value problems, which play the role of normal forms. Such boundary value problems can be equations of the Ginzburg–Landau type, as well as equations with delay and special nonlinearity. The nonlocal dynamics of the constructed equations determines the local behavior of the solutions to the original equation. It is shown that similar normalized boundary value problems also arise for the Van der Pol equation with a large coefficient of the delay equation. The important problem of a small perturbation containing a large delay is considered separately. In addition, the Van der Pol equation, in which the cubic nonlinearity contains a large delay, is considered. One of the general conclusions is that the dynamics of the Van der Pol equation in the presence of a large delay is complex and diverse. It fundamentally differs from the dynamics of the classical Van der Pol equation. Full article
(This article belongs to the Special Issue Models of Delay Differential Equations - II)
30 pages, 531 KiB  
Article
Mathematical Modeling of COVID-19 Dynamics under Two Vaccination Doses and Delay Effects
by Gabriel Sepulveda, Abraham J. Arenas and Gilberto González-Parra
Mathematics 2023, 11(2), 369; https://doi.org/10.3390/math11020369 - 10 Jan 2023
Cited by 10 | Viewed by 1791
Abstract
The aim of this paper is to investigate the qualitative behavior of the COVID-19 pandemic under an initial vaccination program. We constructed a mathematical model based on a nonlinear system of delayed differential equations. The time delay represents the time that the vaccine [...] Read more.
The aim of this paper is to investigate the qualitative behavior of the COVID-19 pandemic under an initial vaccination program. We constructed a mathematical model based on a nonlinear system of delayed differential equations. The time delay represents the time that the vaccine takes to provide immune protection against SARS-CoV-2. We investigate the impact of transmission rates, vaccination, and time delay on the dynamics of the constructed system. The model was developed for the beginning of the implementation of vaccination programs to control the COVID-19 pandemic. We perform a stability analysis at the equilibrium points and show, using methods of stability analysis for delayed systems, that the system undergoes a Hopf bifurcation. The theoretical results reveal that under some conditions related to the values of the parameters and the basic reproduction number, the system approaches the disease-free equilibrium point, but if the basic reproduction number is larger than one, the system approaches endemic equilibrium and SARS-CoV-2 cannot be eradicated. Numerical examples corroborate the theoretical results and the methodology. Finally, conclusions and discussions about the results are presented. Full article
(This article belongs to the Special Issue Models of Delay Differential Equations - II)
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13 pages, 11411 KiB  
Article
Dynamic Analysis of a Delayed Carbon Emission-Absorption Model for China’s Urbanization and Population Growth
by Haokun Sui, Leilei Han and Yuting Ding
Mathematics 2022, 10(17), 3072; https://0-doi-org.brum.beds.ac.uk/10.3390/math10173072 - 25 Aug 2022
Cited by 1 | Viewed by 1080
Abstract
The increase of carbon emissions accelerates global warming and causes environmental problems. Therefore, research on emission peak and carbon neutrality is of broad interest. In this paper, based on the relationships among carbon emissions, carbon absorption and urban population growth, a delayed differential [...] Read more.
The increase of carbon emissions accelerates global warming and causes environmental problems. Therefore, research on emission peak and carbon neutrality is of broad interest. In this paper, based on the relationships among carbon emissions, carbon absorption and urban population growth, a delayed differential equations model is established. Then, we analyze the existence and stability of a positive equilibrium point of the system and substitute the data for simulations. Eventually, according to theoretical analysis and numerical simulations, some suggestions for China to achieve the goal of emission peak and carbon neutrality are given. Full article
(This article belongs to the Special Issue Models of Delay Differential Equations - II)
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10 pages, 283 KiB  
Article
Some Unsolved Problems in Stability and Optimal Control Theory of Stochastic Systems
by Leonid Shaikhet
Mathematics 2022, 10(3), 474; https://0-doi-org.brum.beds.ac.uk/10.3390/math10030474 - 01 Feb 2022
Cited by 3 | Viewed by 2521
Abstract
In spite of the fact that the theory of stability and optimal control for different types of stochastic systems is well developed and very popular in research, there are some simply and clearly formulated problems, solutions of which have not been found so [...] Read more.
In spite of the fact that the theory of stability and optimal control for different types of stochastic systems is well developed and very popular in research, there are some simply and clearly formulated problems, solutions of which have not been found so far. To the readers’ attention six open stability problems for stochastic differential equations with delay, for stochastic difference equations with discrete and continuous time and one open optimal control problem for stochastic hyperbolic equation with two-parameter white noise are offered. Full article
(This article belongs to the Special Issue Models of Delay Differential Equations - II)
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