Chronic Inflammation in the Epidermis: A Mathematical Model by Shinji Nakaoka, Sota Kuwahara, Chang Hyeong Lee, Hyejin Jeon, Junho Lee, Yasuhiro Takeuchi, Yangjin Kim

“…We develop mathematical models (M1 and M2 models) for the inflammation process using ordinary differential equations and delay differential equations. In this paper, we study microbial community dynamics via transcription factors, protease and extracellular cytokines. …”
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Block hybrid methods for numerical treatment of differential equations with applications by Yap, Lee Ken

“…The one-point block hybrid methods are then implemented for numerical solution of first order delay differential equations (DDEs). The Q-stability of the methods is investigated. …”
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Scientific Machine Learning for Dynamical Systems: Theory and Applications to Fluid Flow and Ocean Ecosystem Modeling by Gupta, Abhinav

“…Next, motivated by the presence of inherent delays in real-world systems and the Mori-Zwanzig formulation, we develop a novel delay-differential-equations-based deep learning framework to learn time-delayed closure parameterizations for missing dynamics. …”
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Generalized neural closure models with interpretability by Gupta, Abhinav, Lermusiaux, Pierre F. J.

“…In the present study, we simultaneously address all these challenges by developing the novel and versatile methodology of unified neural partial delay differential equations. We augment existing/low-fidelity dynamical models directly in their partial differential equation (PDE) forms with both Markovian and non-Markovian neural network (NN) closure parameterizations. …”
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A general modeling framework for quantitative tracking, accurate prediction of ICU, and assessing vaccination for COVID-19 in Chile by Patricio Cumsille, Patricio Cumsille, Oscar Rojas-Díaz, Carlos Conca, Carlos Conca

“…Consequently, this article aims to develop a general framework consisting of epidemiological modeling and a practical identifiability approach to assess combined vaccination and non-pharmaceutical intervention (NPI) strategies for the dynamics of any transmissible disease.Materials and methodsEpidemiological modeling of the present work relies on delay differential equations describing time variation and transitions between suitable compartments. …”
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The role of primary and secondary delays in the effective resonance frequency of acoustically interacting microbubbles by Hossein Haghi, Michael C. Kolios

“…Inclusion of the secondary delays modifies the class of the differential equations governing the oscillations of interacting MBs in a cluster from ordinary differential equations to neutral delay differential equations. Previous work has not considered the all the delays associated with the bubble distances when modeling the interactions between bubbles. …”
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Modeling the dynamics of glucose, insulin, and free fatty acids with time delay: The impact of bariatric surgery on type 2 diabetes mellitus by Anarina L. Murillo, Jiaxu Li, Carlos Castillo-Chavez

“…The dynamics of the nonlinear interactions that involve glucose, insulin, and FFA are modeled by incorporating a fixed-time delay with the corresponding delay-differential equations being studied numerically. The results of computational studies, that is, extensive simulations, are compared to the known minimal ordinary differential equations model. …”
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Investigating small non-coding RNA regulating the hallmarks of cancer by Dhawan, A

“…By analysing a system of delay differential equations in the deterministic and stochastic settings, I uncovered the range of potential dynamics that this feedback loop could exhibit. …”

Non-Linear Dynamical Analysis of EEG Time Series Distinguishes Patients with Parkinson’s Disease from Healthy Individuals by Claudia eLainscsek, Claudia eLainscsek, Claudia eLainscsek, Manuel E. Hernandez, Jonathan eWeyhenmeyer, Jonathan eWeyhenmeyer, Jonathan eWeyhenmeyer, Terrence J Sejnowski, Terrence J Sejnowski, Terrence J Sejnowski, Howard ePoizner, Howard ePoizner

“…The pathophysiology of Parkinson’s disease (PD) is known to involve altered patterns of neuronal firing and synchronization in cortical-basal ganglia circuits. We used delay differential equations (DDE) as non-linear time-domain classification tools to analyze electroencephalographic (EEG) recordings from PD patients on and off dopaminergic therapy and healthy individuals. …”
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Stochastic model of cancer growth with the effect of glycosaminoglycans (GAGs) as anticancer therapeutics by Mazma Syahidatul Ayuni, Mazlan

“…It is necessary to model the untreated cancer growth process via stochastic delay differential equations (SDDEs). Nowadays, cancers are traditionally treated with surgery, chemotherapy, and radiotherapy. …”
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Predicting and Reconstructing Aerosol–Cloud–Precipitation Interactions with Physics-Informed Neural Networks by Alice V. Hu, Zbigniew J. Kabala

“…We demonstrate the PINN’s ability to reconstruct the past with limited data for a dynamic system with nonlinear delayed differential equations, such as the Koren–Feingold model, which remains underexplored in the literature. …”
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Bifurcation analysis of multistability and hysteresis in a model of HIV infection by I. V. Mironov, M. Yu. Khristichenk, Yu. M. Nechepurenko, D. S. Grebennikov, G. A. Bocharov

“…To this end, original computational methods of bifurcation analysis of mathematical models formulated with systems of ordinary differential equations and delay differential equations are used. The macrophage activation rate constant is considered as a bifurcation parameter. …”
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New Oscillation Theorems for Second-Order Differential Equations with Canonical and Non-Canonical Operator via Riccati Transformation by Shyam Sundar Santra, Abhay Kumar Sethi, Osama Moaaz, Khaled Mohamed Khedher, Shao-Wen Yao

“…In this work, we prove some new oscillation theorems for second-order neutral delay differential equations of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mo>(</mo><mi>a</mi><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mrow><mo>(</mo><msup><mrow><mo>(</mo><mi>v</mi><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mo>+</mo><mi>b</mi><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mi>v</mi><mrow><mo>(</mo><mi>ϑ</mi><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>)</mo></mrow><mo>′</mo></msup><mo>)</mo></mrow><mo>)</mo></mrow><mo>′</mo></msup><mo>+</mo><mi>c</mi><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><msub><mi>G</mi><mn>1</mn></msub><mrow><mo>(</mo><mi>v</mi><mrow><mo>(</mo><mi>κ</mi><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>)</mo></mrow><mo>+</mo><mi>d</mi><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><msub><mi>G</mi><mn>2</mn></msub><mrow><mo>(</mo><mi>v</mi><mrow><mo>(</mo><mi>ς</mi><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> under canonical and non-canonical operators, that is, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mo>∫</mo><mrow><msub><mi>ξ</mi><mn>0</mn></msub></mrow><mo>∞</mo></msubsup><mfrac><mrow><mi>d</mi><mi>ξ</mi></mrow><mrow><mi>a</mi><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow></mfrac><mo>=</mo><mo>∞</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mo>∫</mo><mrow><msub><mi>ξ</mi><mn>0</mn></msub></mrow><mo>∞</mo></msubsup><mfrac><mrow><mi>d</mi><mi>ξ</mi></mrow><mrow><mi>a</mi><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow></mfrac><mo><</mo><mo>∞</mo></mrow></semantics></math></inline-formula>. …”
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Oscillation and Asymptotic Properties of Differential Equations of Third-Order by R. Elayaraja, V. Ganesan, Omar Bazighifan, Clemente Cesarano

“…The main purpose of this study is aimed at developing new criteria of the iterative nature to test the asymptotic and oscillation of nonlinear neutral delay differential equations of third order with noncanonical operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mo>(</mo><mi>a</mi><mrow><mo>(</mo><mi>ι</mi><mo>)</mo></mrow><msup><mrow><mo>[</mo><mo>(</mo><mi>b</mi><mrow><mo>(</mo><mi>ι</mi><mo>)</mo></mrow><msup><mfenced separators="" open="[" close="]"><mi>x</mi><mo>(</mo><mi>ι</mi><mo>)</mo><mo>+</mo><mi>p</mi><mo>(</mo><mi>ι</mi><mo>)</mo><mi>x</mi><mo>(</mo><mi>ι</mi><mo>−</mo><mi>τ</mi><mo>)</mo></mfenced><mo>′</mo></msup><mo>)</mo></mrow><mo>′</mo></msup><mo>]</mo></mrow><mi>β</mi></msup><msup><mrow><mo>)</mo></mrow><mo>′</mo></msup></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>+</mo><msubsup><mo>∫</mo><mrow><mi>c</mi></mrow><mi>d</mi></msubsup><mi>q</mi><mrow><mo>(</mo><mi>ι</mi><mo>,</mo><mi>μ</mi><mo>)</mo></mrow><msup><mi>x</mi><mi>β</mi></msup><mrow><mo>(</mo><mi>σ</mi><mrow><mo>(</mo><mi>ι</mi><mo>,</mo><mi>μ</mi><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="0.166667em"></mspace><mi>d</mi><mi>μ</mi><mo>=</mo><mn>0</mn><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ι</mi><mo>≥</mo><msub><mi>ι</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>w</mi><mo>(</mo><mi>ι</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>x</mi><mo>(</mo><mi>ι</mi><mo>)</mo><mo>+</mo><mi>p</mi><mo>(</mo><mi>ι</mi><mo>)</mo><mi>x</mi><mo>(</mo><mi>ι</mi><mo>−</mo><mi>τ</mi><mo>)</mo></mrow></semantics></math></inline-formula>. …”
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Exploring efficient: numerical methods for differential equations by Ismail, Fudziah

“…In this research, it is proven that trigonometrically-fitting the hybrid methods and applying them to solve oscillatory delay differential equations result in better numerical results. …”
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