Mackey Glass Model Research Articles

On the analyzing of bifurcation properties of the one‐dimensional Mackey–Glass model by using a generalized approach

The goal of this work is to look at how a nonlinear model describes hematopoiesis and its complexities utilizing commonly used techniques with historical and material links. Based on time delay, the Mackey–Glass model is explored in two instances. To offer a range, the relevance of the parameter impacting stability (bifurcation) is recorded. The power spectrum of the considered model is collected in order to analyze the periodic behavior of a solution in a differential equation. The complex nature of the system is relayed on a parameter which is illustrated in the bifurcation plot. Due to the fact that the considered model is associated with blood‐related diseases, the effect coefficients are effectively captured. The corresponding parameters‐based consequences of the generalized model in different order are deduced. The parametric charts for both examples reveal intriguing results. The current work enables investigations into complex real‐world problems as well as forecasts of essential techniques.

Exponential stability of periodic solutions for a hematopoiesis model with two time delays

In applied mathematics, many practical problems involving heat flow, species interaction, microbiology, neural networks, and more are associated with delay differential equations. To elucidate regulation and control mechanisms in physiological systems, Mackey and Glass introduced new mathematical models as a representation of hematopoiesis (blood-cell formation) in their influential and highly referenced 1977 paper. The presence of multiple delays, rather than a single delay, can lead to a new range of dynamics: an equation that was stable with one delay may become unstable when the delays are not equal, resulting in sustained oscillations. The dynamics of the non-autonomous Mackey-Glass model with two variable delays have not been thoroughly explored yet, which is presented as an open problem by Berezansky and Braverman. This paper focuses on the existence, uniqueness, and exponential stability of positive periodic solutions of a generalized class of the Hematopoiesis Model with two variable delays. Initially, using properties of cones and a fixed point theorem for a concave and increasing operator, we establish conditions ensuring the existence of a unique positive periodic solution for the model with almost periodic coefficients and delays. Subsequently, we employ a suitable Lyapunov functional method and differential inequality techniques to derive sufficient conditions guaranteeing the exponential stability of the unique positive periodic solution of the system. This approach covers a broad range of models that were previously studied. Additionally, an illustrative example with a numerical simulation is provided to validate the efficiency and reliability of the theoretical results. Our approach can be applied to the case of positive almost periodic solutions and positive pseudo almost periodic solutions of the model under consideration.

Global attractivity for reaction–diffusion equations with periodic coefficients and time delays

In this paper, we provide sharp criteria of global attraction for a class of non-autonomous reaction–diffusion equations with delay and Neumann conditions. Our methodology is based on a subtle combination of some dynamical system tools and the maximum principle for parabolic equations. It is worth mentioning that our results are achieved under very weak and verifiable conditions. We apply our results to a wide variety of classical models, including the non-autonomous variants of Nicholson’s equation or the Mackey–Glass model. In some cases, our technique gives the optimal conditions for the global attraction.

Besicovitch almost periodic solutions for a stochastic generalized Mackey-Glass hematopoietic model

<p>This article aimed to investigate the existence and stability of Besicovitch almost periodic ($ B_{ap} $) positive solutions for a stochastic generalized Mackey-Glass hematopoietic model. To begin with, we used stochastic analysis theory, inequality techniques, and fixed point theorems to prove the existence and uniqueness of $ \mathcal{L}^p $-bounded and $ \mathcal{L}^p $-uniformly continuous positive solutions for the model under consideration. Then, we used definitions to prove that this unique positive solution is also a $ B_{ap} $ solution in finite-dimensional distributions. In addition, we established the global exponential stability of the $ B_{ap} $ positive solution using reduction to absurdity. Finally, we provided a numerical example to verify the effectiveness of our conclusions.</p>

Existence of traveling wave fronts for a diffusive Mackey–Glass model with two delays

Based on the diapause effect, a diffusive Mackey–Glass model with distinct diapause and developmental delays is proposed. By constructing appropriate upper and lower solutions and employing the inequality techniques, some sufficient conditions on the existence of traveling wave fronts are obtained. These results not only improve but also supplement some existing ones. Moreover, two numerical examples are provided to demonstrate their reliability and feasibility.

Characterizing multistability regions in the parameter space of the Mackey–Glass delayed system

Proposed to study the dynamics of physiological systems in which the evolution depends on the state in a previous time, the Mackey-Glass model exhibits a rich variety of behaviors including periodic or chaotic solutions in vast regions of the parameter space. This model can be represented by a dynamical system with a single variable obeying a delayed differential equation. Since it is infinite dimensional requires to specify a real function in a finite interval as an initial condition. Here, the dynamics of the Mackey-Glass model is investigated numerically using a scheme previously validated with experimental results. First, we explore the parameter space and describe regions in which solutions of different periodic or chaotic behaviors exist. Next, we show that the system presents regions of multistability, i.e. the coexistence of different solutions for the same parameter values but for different initial conditions. We remark the coexistence of periodic solutions with the same period but consisting of several maximums with the same amplitudes but in different orders. We characterize the multistability regions by introducing families of representative initial condition functions and evaluating the abundance of the coexisting solutions. These findings contribute to describe the complexity of this system and explore the possibility of possible applications such as to store or to code digital information.

About one method of stability investigation for nonlinear stochastic delay differential equations

AbstractStability of equilibria of a nonlinear delay differential equation is investigated under stochastic perturbations. The use of Stiltjes integrals allows to consider both discrete and distributed delays in the investigated equation. Obtained conditions of stability in probability are formulated in terms of linear matrix inequalities. The proposed method of investigation can be applied to a number of very popular in research mathematical models such as Mackey–Glass model, neoclassical growth model and e‐commerce model, Nicholson's blowflies equation, mosquito and glassy‐winged sharpshooter populations, SIR epidemic model, social epidemic models, and many similar other mathematical models which can be considered as particular cases of the investigated nonlinear equation.

On the Weighted Piecewise Pseudo Almost Automorphic Solutions Mackey–Glass Model with Mixed Delays and Harvesting Term

On the Weighted Piecewise Pseudo Almost Automorphic Solutions Mackey–Glass Model with Mixed Delays and Harvesting Term

Time-Delayed Sampled-Data Feedback Control of Differential Systems Undergoing Hopf Bifurcation

In this paper, time-delayed sampled-data feedback control technique is used to asymptotically stabilize a class of unstable delayed differential systems. Through the analysis for the distribution change of eigenvalues, an effective interval of the control parameter is obtained for a given sampling period. Here an indirect strategy is taken. Specifically, the system of continuous-time delayed feedback control is studied first by Hopf bifurcation theory. And then, the result and implicit function theorem are used to analyze the system of time-delayed sampled-data feedback control with a sufficiently small sampling period. Considering the practical criterion for the size of sampling period, the upper bound of sampling period is estimated. Finally, an application example, an unstable Mackey–Glass model, is asymptotically stabilized by introducing a blood transfusion item with time-delayed sampled-data feedback control. The blood transfusion speed and blood collection test period are derived from the main results. Some simulations and comparisons show the correctness and advantages of the main theoretical results.

Does Bitcoin still own the dominant power? An intraday analysis

The study investigates the intraday dynamics and price patterns of the primary cryptocurrencies. The Granger Mackey-Glass (M-G) model is employed to examine the asymmetric and nonlinear dynamic interactions in the first moment using positive and negative returns. The bivariate BEKK-GARCH model is applied to identify cross-market volatility shocks and volatility transmissions in the cryptocurrency market. The intra-cryptocurrency market analysis reveals that Bitcoin contains predictive information that can nonlinearly forecast the performance of other digital currencies when cryptocurrency prices either are rising or declining. The dominant power of Bitcoin is not dismissed using the intraday data. Further, Bitcoin's intraday lagged shocks and volatility induces more rapid and destabilizing effects on the conditional volatility of other currencies than each of the other currencies does on BTC's conditional volatility. The virtual currency markets are dynamically correlated and integrated through first and second-moment spillovers.

A relay Mackey—Glass model with two delays

We study the dynamics of the generalized Mackey—Glass equation with two delays using the large-parameter method. After a special exponential replacement of variables, a singularly perturbed equation is obtained, for which we construct a meaningful limit object, a differential—difference relay equation with two delays. We prove that the relay equation has a simple periodic solution with one interval on the period where the solution is positive. To illustrate the obtained result, we numerically analyze the original singularly perturbed equation, for which we find a solution near the periodic solution of the limit relay equation.

Positive pseudo almost periodic solutions to a class of hematopoiesis model: oscillations and dynamics

This paper presents a new generalized Mackey-Glass model with a non-linear harvesting term and mixed delays. The main purpose of this work is to study the existence and the exponential stability of the pseudo almost periodic solution for the considered model. By using fixed point theorem and under suitable Lyapunov function, sufficient conditions are given to study the pseudo almost periodic solution for the considered model. Moreover, an illustrative example is given to demonstrate the effectiveness of the obtained results.

Dynamics analysis of Mackey-Glass model with two variable delays.

Dynamics of non-autonomous Mackey-Glass model have not been well documented yet in two variable delays case, which is proposed by Berezansky and Braverman as open problems. This manuscript considers attractivity of all non-oscillating solutions about the positive equilibrium point and the global asymptotical stability of the trivial equilibrium point. Two delay-independent criteria based on the fluctuation lemma and techniques of differential inequality are established. The obtained results improve and complement some published results. Meanwhile, computer simulations of two numerical examples are arranged to illustrate the correctness and effectiveness of the presented results.

Fuzzy delay differential equations under granular differentiability with applications

In this paper, we consider a new setting of fuzzy delay differential equations by employing the concepts of granular differentiability. Firstly, we establish some basic properties of complete granular metric space. Then we study the local and global existence and uniqueness of integral fuzzy solution. For the first task, we use successive approximation technique combined with Gronwall’s lemma. The second task will be solved by Banach contraction principle in weighted metric space defined by granular distance. Additionally, we demonstrate the theoretical results by solving some real-world models in both analytical method and numerical method. The first model is often referred to the exponential growth, namely time-delay growth Malthusian model. The second model is one type of dynamical systems, which is used for studies involving large amounts of tissues, called Ehrlich ascites tumor model with delay. Furthermore, we introduce the Mackey–Glass model which is considered as a useful tool for times series prediction in the neutral network and fuzzy logic fields. Finally, we represent the surface of fuzzy solution for visible identification mark-on.

Dynamics of the Almost Periodic Discrete Mackey–Glass Model

This paper is concerned with a class of the discrete Mackey–Glass model that describes the process of the production of blood cells. Prior to proceeding to the main results, we prove the boundedness and extinction of its solutions. By means of the contraction mapping principle and under appropriate assumptions, we prove the existence of almost periodic positive solutions. Furthermore and by the implementation of the discrete Lyapunov functional, sufficient conditions are established for the exponential convergence of the almost periodic positive solution. Examples, as well as numerical simulations are illustrated to demonstrate the effectiveness of the theoretical findings of the paper. Our results are new and generalize some previously-reported results in the literature.

Stability of Traveling Wavefronts for a Delayed Lattice System with Nonlocal Interaction

In this paper, we mainly investigate exponential stability of traveling wavefronts for delayed $2D$ lattice differential equation with nonlocal interaction. For all non-critical traveling wavefronts with the wave speed $c \gt c_*(\theta)$, where $c_*(\theta) \gt 0$ is the critical wave speed and $\theta$ is the direction of propagation, we prove that these traveling waves are asymptotically stable, when the initial perturbation around the traveling waves decay exponentially at far fields, but can be allowed arbitrarily large in other locations. Our approach adopted in this paper is the weighted energy method and the squeezing technique with the help of Gronwall's inequality. Furthermore, from stability result, we prove the uniqueness (up to shift) of the traveling wavefront. Our results can apply to the discrete diffusive Mackey-Glass model and the dicrete diffusive Nicholson's blowflies model on $2D$ lattices.

Analyse des interdépendances non linéaires des principaux marchés boursiers de la zone euro

Cet article analyse les interdépendances non linéaires susceptibles d’exister au sein de la zone euro. Nous introduisons un modèle dynamique original qui est une combinaison du modèle de Mackey-Glass bivarié et bruité de Kyrtsou et Labys [2006], étendu à un cadre multivarié, avec une spécification GJR-GARCH à corrélations conditionnelles dynamiques. L’application empirique menée sur sept indices boursiers de la zone euro sur une période allant du 28/11/2003 au 25/11/2012 met en exergue plusieurs interdépendances mécaniques. Les relations détectées en moyenne confirment le rôle moteur du couple d’indices franco-allemand, une organisation des pays de l’Europe du Sud autour de l’indice italien et l’isolement des indices hellénique et irlandais. L’interdépendance psychologique, mise en évidence par la spécification DCC-GJR-GARCH, révèle deux groupes particuliers : le premier est composé des indices ATHEX, CAC, ISEQ, IBEX et DAX et le second concerne les indices italien et portugais. Nous constatons de fortes corrélations pour le couple franco-allemand tandis que nous relevons une décorrélation, dans le premier groupe des indices CAC, DAX, ISEQ, IBEX avec l’indice hellénique au lendemain de la crise des dettes souveraines. Les autres indices sont caractérisés par des interdépendances psychologiques instables sur la période d’étude.

Mathematical Modelling of Erythropoiesis in Patients with End Stage Renal Disease: New Insights into Marrow Functioning

BACKGROUNDAnemia is a well-recognized complication of chronic kidney disease (CKD). Shortened red blood cell (RBC) survival, uremic inhibitors of erythropoiesis and a relative deficiency of erythropoietin have been proposed as mechanisms of anemia of CKD. In normal individuals a low level of circulating erythropoietin (EPO) produced by the nephrons (10-30 mUnits/mL) is sufficient to maintain erythropoiesis. CKD patients do not have elevations of serum EPO levels that are seen in healthy individuals with comparable degree of anemia. The US FDA approved recombinant EPO in 1989. Since then, treatment with supra-physiologic doses of EPO has become the mainstay of anemia management in CKD. It has been estimated that at any given time 90% of prevalent dialysis patients require erythropoiesis-stimulating agents.Undifferentiated pluripotent stem cells, in the bone marrow are capable of producing committed erythroid colony forming units (CFU-E) under the influence of EPO. These cells ultimately differentiate into RBCs that are released into the circulation. Because of finite non-zero cell maturation and cell replication times there is a delay between EPO administration and RBC production. In healthy individuals the time needed for cell maturation in the marrow ranges from 6-10 days. The maturation of red cells in marrow of CKD patients has not been studied. We hypothesize that this maturation time is prolonged in patients with CKD. By identifying the existence of a delay in cell maturation we can begin to investigate the factors determining the delay. We may also be able to identify new targets for treatments of anemia in kidney disease by targeting cell maturation pathways.METHODSMathematically, erythropoiesis can be formulated as a time model taking the form of a simple differential equation. Delay differential equations can be used to incorporate the delay in cell proliferation depicted by τ.The circulating erythrocytes are removed from the circulation at a rateγ. Therefore, 1/γ represents the life span of the erythrocyte. η and θ are shape parameters in the model and β (rate of RBC production) is a constant. This model was first proposed by Mackey and Glass and is popularly known as the Mackey-Glass Model. Monthly hemoglobin values were collected for 50 end stage renal disease patients on peritoneal dialysis over a period of 24-36 months. Individual models for each patient were developed in Mathematica v. 9.0. To obtain the best fit τ was progressively varied from 1-8 weeks till a good fit was achieved. γ was maintained in the range that represents expected RBC survival in these patients (range 7-14 weeks).RESULTSA representative model is shown in figure 1 below. We found τ to be significantly prolonged up to 6 weeks (mean 4 weeks, range: 1-6 weeks) in patients receiving peritoneal dialysis. This has important implications. A prolonged delay of up to 6 weeks in the effect of EPO can give rise to pseudo EPO-resistance. This phenomenon is clearly observed in clinical practice. Reactionary increase or reduction in EPO doses to apparent lack of response to EPO, leads to fluctuations in hemoglobin, a phenomenon that has been called hemoglobin cycling by clinical epidemiologists and has been related to mortality and cardiovascular outcomes.Recognition of the delay in cell maturation within the marrow which may show wide variation amongst patients (range 1-6 weeks); calls for individualizing drug treatment protocols rather than using population based algorithms.We also found evidence that patients with chronic renal failure are still capable of increasing their rate of red cell production in response to an increase in the rate of cell loss [Fig 2(a)]. Interestingly we found that the delay in cell maturation (τ) is prolonged when the basal erythropoietic rate (β) are increased [Fig 2(b)]. This effect is paradoxical since normal physiologic response to anemia should shorten cell transit times in face of increased demand. This leads us to hypothesize that cells may get arrested during their maturation. Increased levels of several inflammatory mediators may explain the maturation arrest. A lack of clear understanding of the factors affecting the maturation delay makes management of ESA-resistance challenging. New therapeutic approaches that target cellular pathways resulting in arrest of cellular maturation in the marrow may provide an avenue for new drug development. [Display omitted] [Display omitted] DisclosuresNo relevant conflicts of interest to declare.

Global exponential stability for the Mackey-Glass model of respiratory dynamics with a control term

This paper is concerned with the stability of positive periodic solutions for the Mackey–Glass model of respiratory dynamics with a control term. We prove the existence, positivity, and permanence of solutions, which help to deduce the global exponential stability of positive periodic solutions for this model. Our method relies upon a differential inequality technique and a Lyapunov functional. At the end, we give an example with numerical simulations to demonstrate the theoretical results. Copyright © 2015 John Wiley & Sons, Ltd.

Exact Discrete-Time Implementation of the Mackey–Glass Delayed Model

The celebrated Mackey-Glass (MG) model describes the dynamics of physiological delayed systems, in which the actual evolution depends on the values of the variables at some previous times. This kind of system is usually expressed by delayed differential equations, which turn out to be infinite dimensional. In this brief, an electronic implementation mimicking the MG model is proposed. New approaches for both the nonlinear function and the delay block are made. Explicit equations for the actual evolution of the implementation are derived. Simulations of the original equation, the circuit equation, and experimental data show great concordance.