Time-dependent Delay Research Articles

Exploiting delay differential equations solved by Eta functions as suitable mathematical tools for the investigation of thickness controlling in rolling mill

This work is dedicated to introducing the properties and application of Eta functions. We derive the properties of the Eta function, such as the generating function, integral representation, and the Laplace transform. Also, some properties of the Eta-based functions are introduced. To show the advantages of the Eta-based functions in the computational method, we develop a new numerical method to solve the state-dependent and time-dependent neutral delay differential equation based on the Eta-based function. We introduce the operational matrix of derivative for the Eta-base functions to develop the new numerical method. This method uses the operational matrix of derivative and collocation method to convert the delay differential equation to a system of nonlinear algebraic equations. We derive the technique’s error bound and establish the method’s accuracy by solving some examples, which are state-dependent and time-dependent delay differential equations. In the end, we study the model of the metal forming process by rolling the mill using the new numerical method to show the advantages of using the Eta-based function for solving a more practical problem.

Phenanthroindolizidine Alkaloids Isolated from Tylophora ovata as Potent Inhibitors of Inflammation, Spheroid Growth, and Invasion of Triple-Negative Breast Cancer.

Triple-negative breast cancer (TNBC), representing the most aggressive form of breast cancer with currently no targeted therapy available, is characterized by an inflammatory and hypoxic tumor microenvironment. To date, a broad spectrum of anti-tumor activities has been reported for phenanthroindolizidine alkaloids (PAs), however, their mode of action in TNBC remains elusive. Thus, we investigated six naturally occurring PAs extracted from the plant Tylophora ovata: O-methyltylophorinidine (1) and its five derivatives tylophorinidine (2), tylophoridicine E (3), 2-demethoxytylophorine (4), tylophoridicine D (5), and anhydrodehydrotylophorinidine (6). In comparison to natural (1) and for more-in depth studies, we also utilized a sample of synthetic O-methyltylophorinidine (1s). Our results indicate a remarkably effective blockade of nuclear factor kappa B (NFκB) within 2 h for compounds (1) and (1s) (IC50 = 17.1 ± 2.0 nM and 3.3 ± 0.2 nM) that is different from its effect on cell viability within 24 h (IC50 = 13.6 ± 0.4 nM and 4.2 ± 1 nM). Furthermore, NFκB inhibition data for the additional five analogues indicate a structure–activity relationship (SAR). Mechanistically, NFκB is significantly blocked through the stabilization of its inhibitor protein kappa B alpha (IκBα) under normoxic as well as hypoxic conditions. To better mimic the TNBC microenvironment in vitro, we established a 3D co-culture by combining the human TNBC cell line MDA-MB-231 with primary murine cancer-associated fibroblasts (CAF) and type I collagen. Compound (1) demonstrates superiority against the therapeutic gold standard paclitaxel by diminishing spheroid growth by 40% at 100 nM. The anti-proliferative effect of (1s) is distinct from paclitaxel in that it arrests the cell cycle at the G0/G1 state, thereby mediating a time-dependent delay in cell cycle progression. Furthermore, (1s) inhibited invasion of TNBC monoculture spheroids into a matrigel®-based environment at 10 nM. In conclusion, PAs serve as promising agents with presumably multiple target sites to combat inflammatory and hypoxia-driven cancer, such as TNBC, with a different mode of action than the currently applied chemotherapeutic drugs.

Finite-time stability of state-dependent delayed systems and application to coupled neural networks

Finite-time stability and stabilization problems of state-dependent delayed systems are studied in this paper. Different from discrete delays and time-dependent delays which can be well estimated over time, the information of state-dependent delays is usually hard to be estimated, especially when states are unknown or unmeasurable. To guarantee the stability of state-dependent delayed systems in the framework of finite time, a Razumikhin-type inequality is used, following which estimations on the settling time and the region of attraction are proposed. Moreover, the relationship between the variation speed of state-dependent delays and the size of the region of attraction is proposed. Then as an application of the theoretical result, finite-time stabilization is studied for a set of nonlinear coupled neural networks involving state-dependent transmission delay, where the design of memoryless finite-time controllers is addressed. Two numerical examples are given to show the effectiveness of the proposed results.

Exponential Stability of Highly Nonlinear Hybrid Differently Structured Neutral Stochastic Differential Equations with Unbounded Delays

This paper mainly studies the exponential stability of the highly nonlinear hybrid neutral stochastic differential equations (NSDEs) with multiple unbounded time-dependent delays and different structures. We prove the existence and uniqueness of the exact global solution of the new stochastic system, and then give several criteria of the exponential stability, including the q1th moment and almost surely exponential stability. Additionally, some numerical examples are given to illustrate the main results. Such systems are widely applied in physics and other fields. For example, a specific case is pantograph dynamics, in which the delay term is a proportional function. These are widely used to determine the motion of a pantograph head on an electric locomotive collecting current from an overhead trolley wire. Compared with the existing works, our results extend the single constant delay of coefficients to multiple unbounded time-dependent delays, which is more general and applicable.

Protective effects of NAMPT or MAPK inhibitors and NaR on Wallerian degeneration of mammalian axons

Wallerian degeneration (WD) is a conserved axonal self-destruction program implicated in several neurological diseases. WD is driven by the degradation of the NAD+ synthesizing enzyme NMNAT2, the buildup of its substrate NMN, and the activation of the NAD+ degrading SARM1, eventually leading to axonal fragmentation. The regulation and amenability of these events to therapeutic interventions remain unclear. Here we explored pharmacological strategies that modulate NMN and NAD+ metabolism, namely the inhibition of the NMN-synthesizing enzyme NAMPT, activation of the nicotinic acid riboside (NaR) salvage pathway and inhibition of the NMNAT2-degrading DLK MAPK pathway in an axotomy model in vitro. Results show that NAMPT and DLK inhibition cause a significant but time-dependent delay of WD. These time-dependent effects are related to NMNAT2 degradation and changes in NMN and NAD+ levels. Supplementation of NAMPT inhibition with NaR has an enhanced effect that does not depend on timing of intervention and leads to robust protection up to 4 days. Additional DLK inhibition extends this even further to 6 days. Metabolite analyses reveal complex effects indicating that NAMPT and MAPK inhibition act by reducing NMN levels, ameliorating NAD+ loss and suppressing SARM1 activity. Finally, the axonal NAD+/NMN ratio is highly predictive of cADPR levels, extending previous cell-free evidence on the allosteric regulation of SARM1. Our findings establish a window of axon protection extending several hours following injury. Moreover, we show prolonged protection by mixed treatments combining MAPK and NAMPT inhibition that proceed via complex effects on NAD+ metabolism and inhibition of SARM1.

A GENERAL AND OPTIMAL DECAY RESULT FOR A VISCOELASTIC EQUATION WITH A STRONG TIME DEPENDENT DELAY

In this paper, we establish an optimal and general decay result for the energy of a viscoelastic equation exhibiting a strong time-dependent delay. This is achieved by considering a minimal condition on the relaxation function g. The exponential and polynomial decay rates are obtained as special cases.The theoretical computations are supported with a numerical analysis of the problem under consideration. This work extends and generalizes some recent results in the literature.

Mittag-Leffler Stability of Fractional-Order Nonlinear Differential Systems With State-Dependent Delays

The stability of fractional-order (FO) nonlinear system with state-dependent delays (SDDs) is investigated. Unlike the usual time-dependent delays, the state-dependent (SD) delays make the size of the delays relative to the states, which makes the system uncertain when historical state information would be used. A Lemma on Riemann-Liouville derivative is first given to ensure the monotonicity of the considered function. Then, based on the Lyapunov method, several sufficient criteria are presented to guarantee the Mittag-Leffler stability of the discussed systems. In the end, three examples are applied to illustrate the correctness and applicability of our theoretical conclusions, including practical applications in submarine positioning models.

A DNA-structured mathematical model of cell-cycle progression in cyclic hypoxia

New experimental data have shown how the periodic exposure of cells to low oxygen levels (i.e., cyclic hypoxia) impacts their progress through the cell-cycle. Cyclic hypoxia has been detected in tumours and linked to poor prognosis and treatment failure. While fluctuating oxygen environments can be reproduced in vitro, the range of oxygen cycles that can be tested is limited. By contrast, mathematical models can be used to predict the response to a wide range of cyclic dynamics. Accordingly, in this paper we develop a mechanistic model of the cell-cycle that can be combined with in vitro experiments to better understand the link between cyclic hypoxia and cell-cycle dysregulation. A distinguishing feature of our model is the inclusion of impaired DNA synthesis and cell-cycle arrest due to periodic exposure to severely low oxygen levels. Our model decomposes the cell population into five compartments and a time-dependent delay accounts for the variability in the duration of the S phase which increases in severe hypoxia due to reduced rates of DNA synthesis. We calibrate our model against experimental data and show that it recapitulates the observed cell-cycle dynamics. We use the calibrated model to investigate the response of cells to oxygen cycles not yet tested experimentally. When the re-oxygenation phase is sufficiently long, our model predicts that cyclic hypoxia simply slows cell proliferation since cells spend more time in the S phase. On the contrary, cycles with short periods of re-oxygenation are predicted to lead to inhibition of proliferation, with cells arresting from the cell-cycle in the G2 phase. While model predictions on short time scales (about a day) are fairly accurate (i.e, confidence intervals are small), the predictions become more uncertain over longer periods. Hence, we use our model to inform experimental design that can lead to improved model parameter estimates and validate model predictions.

Expression of Scytonemin Biosynthesis Genes under Alternative Stress Conditions in the Cyanobacterium Nostoc punctiforme.

The indole-alkaloid scytonemin is a sunscreen pigment that is widely produced among cyanobacteria as an ultraviolet radiation (UVR) survival strategy. Scytonemin biosynthesis is encoded by two gene clusters that are known to be induced by long-wavelength radiation (UVA). Previous studies have characterized the transcriptome of cyanobacteria in response to a wide range of conditions, but the effect on the expression of scytonemin biosynthesis genes has not been specifically targeted. Therefore, the aim of this study is to determine the variable response of scytonemin biosynthesis genes to a variety of environmental conditions. Cells were acclimated to white light before supplementation with UVA, UVB, high light, or osmotic stress for 48 h. The presence of scytonemin was determined by absorbance spectroscopy and gene expression of representative scytonemin biosynthesis genes was measured using quantitative PCR. Scytonemin genes were up-regulated in UVA, UVB, and high light, although the scytonemin pigment was not detected under high light. There was no scytonemin or upregulation of these genes under osmotic stress. The lack of pigment production under high light, despite increased gene expression, suggests a time-dependent delay for pigment production or additional mechanisms or genes that may be involved in scytonemin production beyond those currently known.

Exponential stability of highly nonlinear hybrid NSDEs with multiple time-Dependent delays and different structures and the Euler-Maruyama method

This study focuses on the highly nonlinear hybrid neutral stochastic differential equations (NSDEs) with multiple time-dependent delays and different structures. The new stochastic system in this paper contains multiple time-dependent delays in the neutral term, and the coefficients are also with different delay functions in drift and diffusion terms respectively. Besides, the set of Markov switching states are divided into two subsets, and the structures of the system are different in each of them. Firstly, the existence, uniqueness and asymptotic boundedness of the global solution of the new NSDE are proved, and then the novel criteria of pth moment and almost surely exponential stability are investigated. All these results are obtained under highly nonlinear coefficients that satisfy local Lipschitz condition and Khasminskii-type conditions, and the Lyapunov function method and the generalized Itô formula are mainly used. Moreover, the Euler-Maruyama approximate solution is established and proved to be convergent in probability to the theoretical solution. Finally, a numerical example is presented together with the corresponding computer simulation to demonstrate the effectiveness of the main results.

A delay nonautonomous model for the effects of fear and refuge on predator–prey interactions with water-level fluctuations

The interaction of prey (small fish) and predator (large fish) in lakes/ ponds at temperate and tropical regions varies when water level fluctuates naturally during seasonal time. We relate the perceptible effect of fear and anti-predator behavior of prey with the water-level fluctuations and describe how these are influenced by the seasonal changing of water level. So, we consider these as time-dependent functions to make the system more realistic. Also, we incorporate the time-dependent delay in the negative growth rate of prey in predator–prey model with Crowley–Martin-type functional response. We clearly provide the basic dynamics of the system such as positiveness, permanence and nonpersistence. The existence of positive periodic solution is studied using Continuation theorem, and suffiecient conditions for globally attractivity of positive periodic solution are also derived. To make the system more comprehensive, we establish numerical simulations, and compare the dynamics of autonomous and nonautonomous systems in the absence as well as the presence of time delay. Our results show that seasonality and time delay create the occurrence of complex behavior such as prevalence of chaotic disorder which can be potentially suppressed by the cost of fear and prey refuge. Also, if time delay increases, then system leads a boundary periodic solution. Our findings assert that the predation, fear of predator and prey refuge are correlated with water-level variations, and give some reasonable biological interpretations for persistence as well as extinction of species due to water-level variations.

Boundary stabilization of a flexible structure with dynamic boundary conditions via one time-dependent delayed boundary control

<p style='text-indent:20px;'>This article deals with the dynamic stability of a flexible cable attached at its top end to a cart and a load mass at its bottom end. The model is governed by a system of one partial differential equation coupled with two ordinary differential equations. Assuming that a time-dependent delay occurs in one boundary, the main concern of this paper is to stabilize the dynamics of the cable as well as the dynamical terms related to the cart and the load mass. To do so, we first prove that the problem is well-posed in the sense of semigroups theory provided that some conditions on the delay are satisfied. Thereafter, an appropriate Lyapunov function is put forward, which leads to the exponential decay of the energy as well as an estimate of the decay rate.</p>

Global attractivity for a nonautonomous Nicholson’s equation with mixed monotonicities

We consider a Nicholson’s equation with multiple pairs of time-varying delays and nonlinear terms given by mixed monotone functions. Sufficient conditions for the permanence, local stability and global attractivity of its positive equilibrium K are established. The main novelty here is the construction of a suitable auxiliary difference equation x n+1 = h(x n ) with h having negative Schwarzian derivative, and its application to derive the attractivity of K for a model with one or more pairs of time-dependent delays. Our criteria depend on the size of some delays, improve results in recent literature and provide answers to open problems.

The first positive root of the fundamental solution is an optimal oscillation bound for linear delay differential equations

We consider the nonautonomous linear delay differential equationx′(t)+p(t)x(t−1)=0,t≥t0, where p:[t0,∞)→R is a continuous function such that p(t)≥a>1/e for t≥t0. It is shown that the smallest positive constant ω=ω(a) such that every solution has at least one zero in [t0−1,t0+ω] coincides with the first positive root ζ(a) of the fundamental solution of the equation with constant coefficienty′(t)+ay(t−1)=0. A detailed analysis of the function ζ:(1/e,∞)→(1,∞) is given with the emphasis on the asymptotic description of ζ(a) as a→1/e+. As corollaries new oscillation criteria are obtained which improve some previous results in the literature. With appropriate modifications the method applies to a more general class of linear differential equations with time-dependent delay.

A New Multi-Step Method for Solving Delay Differential Equations using Lagrange Interpolation

This paper presents 2-step p-th order (p = 2,3,4) multi-step methods that are based on the combination of both polynomial and exponential functions for the solution of Delay Differential Equations (DDEs). Furthermore, the delay argument is approximated using the Lagrange interpolation. The local truncation errors and stability polynomials for each order are derived. The Local Grid Search Algorithm (LGSA) is used to determine the stability regions of the method. Moreover, applicability and suitability of the method have been demonstrated by some numerical examples of DDEs with constant delay, time dependent and state dependent delays. The numerical results are compared with the theoretical solution as well as the existing Rational Multi-step Method2 (RMM2).

Estimation of Time-Dependent Delay Models at Actuated Traffic Signals in Duhok City

Duhok City, developed rapidly in recent years, leading to an increase in the traffic volume on its streets which are not sufficient for this increase resulting in traffic congestion. Therefore, this paper studied the delay as an important measure of effectiveness at traffic signals. The delay was measured and predicted (for saturated and under saturated conditions) to help in solving some underestimation problems by using the transportation system management (TSM) techniques. Five signalized intersections on Barzani major street in Duhok city was chosen. Vehicle delay was measured in these intersections directly from field, then it was calculated using the time-dependent equations recommended by three different official manuals, Highway Capacity Manual (HCM2000), Canadian Capacity Guide (ITE1995) and Australian Capacity Guide (ARRB1995). After that comparisons between the results of each theoretical method and the field result was made. Finally, the regression analysis technique was used to establish different relationships between the delay data obtained from the field and each of the theoretical methods to find the best applicable model to predict the vehicle delay at signalized intersections in Duhok City. It was proved that the logarithmic relationship between field and theoretical results obtained from the Canadian Capacity Guide is the best

A Taylor method for stochastic differential equations with time-dependent delay via the polynomial condition

The subject of this article is an analytic approximate method for a class of stochastic differential equations with time-dependent delay, with coefficients that do not necessarily satisfy the Lipschitz condition nor the linear growth condition but they satisfy the polynomial condition. More precisely, it is assumed that equations from the observed class have unique solutions with bounded moments, their coefficients satisfy the polynomial condition and some derivatives of the specific order of the drift and diffusion coefficients are uniformly bounded. Approximate equations are defined on partitions of the time interval and their coefficients are Taylor approximations of the coefficients of the initial equation up to arbitrary derivatives. It is shown that the solutions of the approximate equations converge in the sense and almost surely toward the solution of the initial equation and the rate of the convergence is given.

Stage-structured hematopoiesis model with delays in an almost periodic environment

In this article, we focus on stage-structured hematopoiesis model with time-dependent delays in an almost periodic environment. A threshold dynamic is characterized by basic reproduction ratio R0. By making use of skew-product semiflow approach, we show that the population is extinct if R01. The existence of a unique positive almost periodic solution is derived when R0>1, meanwhile, the global stability of the solution is verified under weaker conditions.

Fractional model of stem cell population dynamics

Abstract We develop the fractional model of stem cell population dynamics with state-dependent and time-dependent delays. In this model, the stem cell division rate and self-renewal rate are controlled by an external signal, depending on the effects of the environment’s heterogeneity. We quantify a relationship between the fractional derivative order, which shows the effects of the environment’s heterogeneity and stem cell division and stem cell self-renewal rate. We consider a general form of the fractional neutral delay differential equations with state-dependent and time-dependent delay to study this relationship. First, we show the solution’s existence and uniqueness using the fixed point theorem on the Banach space. We define a completely continuous operator on the non-empty closed convex set to use the fixed point theorem on the Banach space and show this operator has a uniquely defined fixed point. Also, we proof the Ulam–Hyers stability to make sure a close exact solution could be reached using the numerical approximation. Then, we develop a new numerical method based on Jacobi polynomials for solving the fractional neutral delay differential equations with state-dependent and time-dependent delay. We use the least-squares approximation of the candidate function to reduce the solution of fractional neutral delay differential equations to a set of algebraic equations and compare the results obtained by using different collocation points. We evaluate the accuracy of the numerical method, theoretically and numerically. We have used the numerical method to evaluate the fractional model’s behavior of stem cell population dynamics and quantify the relationship between the effects of the environment’s heterogeneity and the rate of stem cell division and stem cell self-renewal.

The truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay

The main goal of this paper is to establish the Lq-convergence of the truncated Euler-Maruyama method for neutral stochastic differential equations with time-dependent delay under the Khasminskii-type condition. Whole consideration is influenced by the presence of the neutral term and delay function. The main theoretical result is illustrated by an example. Since the main result is related to the Lq-convergence of the Euler-Maruyama method under the global Lipschitz condition on the coefficients of the equation under consideration, for completeness of the paper, the appropriate results are given in Appendix.