Discrete Delays Research Articles

Global co-dynamics of viral infections with saturated incidence

<abstract><p>Several mathematical models of two competing viruses (or viral strains) that have been published in the literature assume that the infection rate is determined by bilinear incidence. These models do not show co-existence equilibrium; moreover, they might not be applicable in situations where the virus concentration is high. In this paper, we developed a mathematical model for the co-dynamics of two competing viruses with saturated incidence. The model included the latently infected cells and three types of time delays: discrete (or distributed): (ⅰ) The formation time of latently infected cells; (ⅱ) The activation time of latently infected cells; (ⅲ) The maturation time of newly released virions. We established the mathematical well-posedness and biological acceptability of the model by examining the boundedness and nonnegativity of the solutions. Four equilibrium points were identified, and their stability was examined. Through the application of Lyapunov's approach and LaSalle's invariance principle, we demonstrated the global stability of equilibria. The impact of saturation incidence, latently infected cells, and time delay on the viral co-dynamics was examined. We demonstrated that the saturation could result in persistent viral coinfections. We established conditions under which these types of viruses could coexist. The coexistence conditions were formulated in terms of saturation constants. These findings offered new perspectives on the circumstances under which coexisting viruses (or strains) could live in stable viral populations. It was shown that adding the class of latently infected cells and time delay to the coinfection model reduced the basic reproduction number for each virus type. Therefore, fewer treatment efficacies would be needed to keep the system at the infection-free equilibrium and remove the viral coinfection from the body when utilizing a model with latently infected cells and time delay. To demonstrate the associated mathematical outcomes, numerical simulations were conducted for the model with discrete delays.</p></abstract>

Analysis of finite-time stability in genetic regulatory networks with interval time-varying delays and leakage delay effects

<p>We primarily examined the effect of leakage delays on finite-time stability problems for genetic regulatory networks with interval time-varying delays. Since leakage delays can occur within the negative feedback components of networks and significantly impact their dynamics, they may potentially cause instability or suboptimal performance. The derived criteria encompass both leakage delays and discrete interval time-varying delays through the construction of a Lyapunov-Krasovskii function. We employed the estimation of various integral inequalities and a reciprocally convex technique. Additionally, these models consider lower bounds on delays, which may be either positive or zero, and allow for the derivatives of delays to be either positive or negative. Consequently, new criteria for genetic regulatory networks with interval time-varying delays under the effect of leakage delays are expressed in the form of linear matrix inequalities. Ultimately, a numerical example is presented to show the effect of leakage delays and to emphasize the significance of our theoretical findings.</p>

Exponential Stability of Impulsive Timescale-Type Nonautonomous Neural Networks With Discrete Time-Varying and Infinite Distributed Delays.

Global exponential stability (GES) for impulsive timescale-type nonautonomous neural networks (ITNNNs) with mixed delays is investigated in this article. Discrete time-varying and infinite distributed delays (DTVIDDs) are taken into consideration. First, an improved timescale-type Halanay inequality is proven by timescale theory. Second, several algebraic inequality criteria are demonstrated by constructing impulse-dependent functions and utilizing timescale analytical techniques. Different from the published works, the theoretical results can be applied to GES for ITNNNs and impulsive stabilization design of timescale-type nonautonomous neural networks (TNNNs) with mixed delays. The improved timescale-type Halanay inequality considers time-varying coefficients and DTVIDDs, which improves and extends some existing ones. GES criteria for ITNNNs cover the stability conditions of discrete-time nonautonomous neural networks (NNs) and continuous-time ones, and these theoretical results hold for NNs with discrete-continuous dynamics. The effectiveness of our new theoretical results is verified by two numerical examples in the end.

Global properties of delayed models for SARS-CoV-2 infection mediated by ACE2 receptor with humoral immunity

<abstract><p>The coronavirus disease 2019 (COVID-19) is caused by a new coronavirus known as severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). SARS-CoV-2 infects the epithelial (target) cells by binding its spike protein, S, to the angiotensin-converting enzyme 2 (ACE2) receptor on the surface of epithelial cells. During the process of SARS-CoV-2 infection, ACE2 plays an important mediating role. In this work, we develop two models which describe the within-host dynamics of SARS-CoV-2 under the effect of humoral immunity, and considering the role of the ACE2 receptor. We consider two discrete (or distributed) delays: (ⅰ) Delay in the SARS-CoV-2 infection of epithelial cells, and (ⅱ) delay in the maturation of recently released SARS-CoV-2 virions. Five populations are considered in the models: Uninfected epithelial cells, infected cells, SARS-CoV-2 particles, ACE2 receptors and antibodies. We first address the fundamental characteristics of the delayed systems, then find all possible equilibria. On the basis of two threshold parameters, namely the basic reproduction number, $ \Re_{0} $, and humoral immunity activation number, $ \Re_{1} $, we prove the existence and stability of the equilibria. We establish the global asymptotic stability for all equilibria by constructing suitable Lyapunov functions and using LaSalle's invariance principle. To illustrate the theoretical results, we perform numerical simulations. We perform sensitivity analysis and identify the most sensitive parameters. The respective influences of humoral immunity, time delays and ACE2 receptors on the SARS-CoV-2 dynamics are discussed. It is shown that strong stimulation of humoral immunity may prevent the progression of COVID-19. It is also found that increasing time delays can effectively decrease $ \Re_{0} $ and then inhibit the SARS-CoV-2 replication. Moreover, it is shown that $ \Re_{0} $ is affected by the proliferation and degradation rates of ACE2 receptors, and this may provide worthy input for the development of possible receptor-targeted vaccines and drugs. Our findings may thus be helpful for developing new drugs, as well as for comprehending the dynamics of SARS-CoV-2 infection inside the host.</p></abstract>

Dental Developmental Stages and Decayed, Missing, and Restored Teeth in Neurofibromatosis Type 1-affected Children and Adolescents.

Neurofibromatosis type 1 (NF1) is an autosomal dominant inherited tumor predisposition disease with a highly variable phenotype. The influence of the characteristic NF1 tumors (neurofibromas) on dentition has not yet been examined in detail. The aim of the study was to assess the dentition of NF1 children and adolescents, considering the symmetry of tooth development. The panoramic radiographs of 59 patients with a confirmed NF1 diagnosis were compared with 59 age-and-sex-matched controls. The stages of tooth development on the sides of the jaw, added to a score, were assessed. In addition, the number of filled or decayed teeth, and the number of retained or missing teeth were assessed. The tooth development of both study groups is symmetrical for almost all parameters and in the same developmental stage according to the sum score of the tooth development stages. Discrete developmental delays of teeth, in particular in the oral area of facial plexiform neurofibroma (PNF) are noticeable. NF1 patients' teeth showed less decay and more restorations than that of the control group. The facial PNF (FPNF) does not impair emergence of deciduous teeth. Development of dentition of NF1 patients does not differ from the general population. However, FPNF with oral tumor components often prevent mesial movement of permanent molars and premolars, so these teeth do not develop contact (spacing), hardly emerge or may stay retained in bone. Oral PNF may have a low-retarding effect on some tooth root development (e.g., wisdom teeth). This effect is negligible when comparing the affected and unaffected sides of the jaw and is probably non-specific. Key words:Neurofibromatosis type 1, plexiform neurofibroma, dentition, mixed dentition, symmetry, oral health, tooth development.

Analysis of a nonlinear problem involving discrete and proportional delay with application to Houseflies model

<abstract><p>This manuscript established a comprehensive analysis of a general class of fractional order delay differential equations with Caputo-Fabrizio fractional derivative (CFFD). Functional analysis was used to examine the existence and uniqueness of the suggested class and to generate sufficient requirements for Ulam-Hyers (UH) type stability. Further, a numerical method based on Lagrange interpolation is used to compute approximate solution. Then, some applications in physical dynamics including a houseflies model and a Cauchy type problem were discussed to illustrate the established analysis with graphical illustrations.</p></abstract>

Research on the ellipsoidal boundary of reachable sets of neutral systems with bounded disturbances and discrete time delays

<abstract><p>This research focuses on the challenge of defining the ellipsoidal boundaries of the reachable set (RS) for neutral-type dynamical systems with time delays. A novel analytical approach is proposed, leveraging the development of new Lyapunov functions and matrix inequality techniques. These methods provide powerful tools for determining the ellipsoidal boundaries of the system's RS. A comparative analysis, supported by numerical examples, demonstrates that the approach outlined in this study can accurately identify smaller yet effective RS boundaries compared to existing literature. This precise boundary determination offers significant theoretical support for state estimation and control design in dynamical systems, thereby enhancing their effectiveness and reliability in real-world applications.</p></abstract>

Chaos detection in predator-prey dynamics with delayed interactions and Ivlev-type functional response

<p>Regarding delay-induced predator-prey systems, extensive research has focused on the phenomenon of delayed destabilization. However, the question of whether delays contribute to stabilizing or destabilizing the system remains a subtle one. In this paper, the predator-prey interaction with discrete delay involving Ivlev-type functional response is studied by theoretical analysis and numerical simulations. The positivity and boundedness of the solution for the delayed model have been discussed. When time delay is accounted as a bifurcation parameter, stability analysis for the coexistence equilibrium is given in theoretical aspect. Supercritical Hopf bifurcation is detected by numerical simulation. Interestingly, by choosing suitable groups of parameter values, the chaotic solutions appear via a cascade of period-doubling bifurcations, which is also detected. The theoretical analysis and numerical conclusions demonstrate that the delay mechanism plays a crucial role in the exploration of chaotic solutions.</p>

Design of infinite horizon LQR controller for discrete delay systems in satellite orbit control: A predictive controller and reduction method approach

In the realm of satellite orbit control, powerful controller design plays a pivotal role in minimizing fuel consumption and ensuring orbit stability. This article introduces an advanced approach to the design of a Linear Quadratic Regulator (LQR) controller with an infinite horizon, tailored for discrete delay systems. The proposed methodology integrates predictive control with a reduction method, aiming for optimality while addressing performance and system constraints. Formulating the control problem as a quadratic program, the predictive control method generates a sequence of control inputs using a reducing horizon strategy. Stability analysis, employing Lyapunov-Krasovsky functions and linear matrix inequalities, yields delay-independent conditions for exponential convergence. A numerical example showcases the controller's effectiveness in maintaining orbit and reducing fuel consumption, underlining its capacity to achieve control objectives despite uncertainties and time delays. This research contributes to robust control strategies in satellite orbit systems, enhancing control performance and operational efficiency.

Turing instability for a space and time discrete delay Lotka-Volterra competitive model with periodic boundary conditions

Turing instability for a space and time discrete delay Lotka-Volterra competitive model with periodic boundary conditions

A mathematical model for tumor-immune competitive system with multiple time delays

A mathematical model is proposed to study the complicated dynamics of tumor-immune interactions with three discrete time delays. The proposed model consists of nine coupled ordinary differential equations (ODEs) whose components are tumor cells, tumor-specific CD8+T cells, dendritic cells, macrophages, regulatory T-cells (Tregs), interleukin-10 (IL-10), IL-12, transforming growth factor-β (TGF-β) and interferon-γ (IFN-γ). By utilizing the quasi-steady-state approximation for cytokine concentrations, we obtain a system of four coupled ODEs. We introduce three discrete time delays into our deterministic system to better understand the tumor-immune dynamics. Basic properties of the system, including existence, uniqueness, positivity, boundedness, and uniform persistence are discussed. Stability analysis of both the delayed and non-delayed system has been performed and the conditions for stability and direction of Hopf bifurcation have been determined. Furthermore, we assess the length of time delay required to maintain the stability of period-1 limit cycle. Parameter estimation techniques are discussed and numerical simulations are provided to support our theoretical analysis. Notably, we observe that time delays do not significantly influence the system behavior for the existing set of parameters value.

Hopf bifurcations in an atypical p53 protein oscillator model with discrete delay or distributed delay

Human cells have a natural cancerous defense system, and p53 protein is at the core. It is well known that p53 relies on oscillatory dynamic behavior to perform tumor suppressor functions. However, the mechanism of its oscillation remains unclear. To reveal the effect of time lag on the onset of p53 oscillations, we analytically and numerically investigate a possible scenario in which p53 oscillations are driven by a time lag. The model has only three nodes, two negative loops, and one delay. We focus on two types of time delays: discrete and distributed, use Hopf bifurcation theory to derive conditions for the occurrence of limit cycle oscillations, and confirm the properties of Hopf bifurcations by normal form and central manifold approaches. The analytical results are supported by numerical results. We suggest that discrete time lag promote p53 oscillations, while distributed time lag have a dual effect on p53 oscillations. These results imply that controlling the time lag in an appropriate range favors the inhibition of p53 on cancer cells.

Coexistence in two-species competition with delayed maturation

Inter- and intraspecific competition is most important during the immature life stage for many species of interest, such as multiple coexisting mosquito species that act as vectors of diseases. Mortality caused by competition that occurs during maturation is explicitly modelled in some alternative formulations of the Lotka–Volterra competition model. We generalise this approach by using a distributed delay for maturation time. The kernel of the distributed delay is represented by a truncated Erlang distribution. The shape and rate of the distribution, as well as the position of the truncation, are found to determine the solution at equilibrium. The resulting system of delay differential equations is transformed into a system of ordinary differential equations using the linear chain approximation. Numerical solutions are provided to demonstrate cases where competitive exclusion and coexistence occur. Stability conditions are determined using the nullclines method and local stability analysis. The introduction of a distributed delay promotes coexistence and survival of the species compared to the limiting case of a discrete delay, potentially affecting management of relevant pests and threatened species.

General stability for a Cohen–Grossberg neural network system

Of concern is a Cohen–Grossberg neural network (CGNNs) system taking into account distributed and discrete delays. The class of delay kernels ensuring exponential stability existing in the previous papers is enlarged to an extended class of functions guaranteeing more general types of stability. The exponential and polynomial (or power type) type stabilities becomes particular cases of our result. This is achieved using appropriate Lyapunov-type functionals and the characteristics of the considered class.

An Eco-Epidemiological Model with Harvesting and Discrete Delay: A Mathematical Study

An Eco-Epidemiological Model with Harvesting and Discrete Delay: A Mathematical Study

Averaging in multifrequency systems with multi-point conditions and a delay

For multifrequency system of differential equations with a discrete and integral delay we find conditions for the existence and uniqueness of the solution. Linear multipoint conditions are set for the solution. An estimate of the error of the averaging method is obtained, which clearly depends on the small parameter.

Averaging in multifrequency systems with multi-point conditions and a delay

For multifrequency system of differential equations with a discrete and integral delay we find conditions for the existence and uniqueness of the solution. Linear multipoint conditions are set for the solution. An estimate of the error of the averaging method is obtained, which clearly depends on the small parameter.

Analysis of a delayed mathematical model for tumor-immune cell interactions with Holling type II functional response

In this study, we analyzed a three-dimensional nonlinear differential system considering Holling type II functional response that describes the dynamics of tumor cells, cytotoxic T lymphocytes, and helper T cells, with a single interaction delay. The linear stability of the internal equilibrium point and the presence of the Hopf bifurcation are examined, with the discrete time delay serving as the bifurcation parameter. To demonstrate the rich dynamic behavior of the model, we present numerical simulations with various values of the time delay τ and the attack rate of cytotoxic T lymphocytes on tumor cells (α1). These simulations exhibit the presence of periodic oscillations and tumor death with survival of the mentioned immune cells, for all α1 greater than a fixed threshold with or without delay.

Synchronization sampled-data control of uncertain neural networks under an asymmetric Lyapunov–Krasovskii functional method

This paper investigates the synchronization control of neural networks (NNs) considering parameter uncertainties by utilizing an improved asymmetric Lyapunov–Krasovskii functional (ALKF) method. Firstly, an uncertain NNs model with discrete and distributed delays is developed. To synchronize the master–slave system, a memory sampled-data controller is designed. Secondly, an improved ALKF is constructed, in which the positive-definite and symmetric restrictions of matrices are relaxed. Furthermore, improved synchronization criteria are established by linear matrix inequalities (LMIs), which are based on the ALKF method and the integral inequality technique. Finally, two simulation examples are conducted to demonstrate the feasibility and superiority of the established stability conditions in reaching a maximum sampling period.

Complex dynamics of a predator–prey system with fear and memory in the presence of two discrete delays

Complex dynamics of a predator–prey system with fear and memory in the presence of two discrete delays