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In this paper, we have formulated and analyzed a mathematical model with multiple time delays of the genetic regulatory network of the microRNA‐17‐92 (miR‐17‐92) cluster. We have focused on the effect of the associated transcriptional and translational time delays of both positive and negative feedback loops on the miR‐17‐92 regulatory network. To gain better insight, we have explored the role of individual time delay in controlling the transition of the G1/S cell cycle process. We pay particular attention to the concentration of E2F and Myc, which trigger the protein ON and OFF switches. To analyze the situation mathematically, we employ delay‐dependent stability analysis and bifurcation analysis using delay differential equations and a biological study of the time‐delayed random transitions between the ON and OFF states for protein synthesis. Based on our study, we can conclude that the translational time delay (τ2)$$ \left({\tau}_2\right) $$ can halt the healthy transition of cells in the G1/S cell cycle phase as the small increase in (τ2)$$ \left({\tau}_2\right) $$ can destroy bistability of the system. Simulations are carried out to numerically show the solution trajectories under the combined effect of delays. Finally, we summarize the simulation results and the impact of delays on the dual behavior of miR‐17‐92 as it can act as an oncogene or as a tumor suppressor gene. [ABSTRACT FROM AUTHOR]

This article is devoted towards some necessary conditions for the existence of the mild solution of 'second‐order semilinear stochastic delay differential equations' having a variable delay in the state with the help of Banach fixed point theorem and theory of cosine family which is strongly continuous. In the last section, we have discussed an example to understand the theory of provided results in a better manner. [ABSTRACT FROM AUTHOR]

This paper investigates the analytical stability region and the asymptotic stability of linear fractional neutral delay differential equations. Employing boundary locus techniques, the stability region of this problem is analyzed. Furthermore, we derive the fundamental solution of linear fractional neutral delay differential equations, and prove the exponential boundedness, the asymptotic stability and the algebraic decay rate. Finally, numerical tests are conducted to verify the theoretical results. [ABSTRACT FROM AUTHOR]

This article compared two high-order numerical schemes for convection-diffusion delay differential equation via two fractional operators with singular kernels. The objective is to present two effective schemes that give (3 - α) and second order of accuracy in the time direction when α ∈ (0 , 1) using Caputo and Modified Atangana-Baleanu Caputo derivatives, respectively. We also implemented a trigonometric spline technique in the space direction, giving second order of accuracy. Moreover, meticulous analysis shows these numerical schemes to be unconditionally stable and convergent. The efficiency and reliability of these schemes are illustrated by numerical experiments. The tabulated results obtained from test examples have also shown the comparison of these operators. [ABSTRACT FROM AUTHOR]

In this article, we initially provided the relationship between the RL fractional integral and the Caputo fractional derivative of different orders. Additionally, it is clear from the literature that studies into boundary value problems involving multi‐term operators have been conducted recently, and the aforementioned idea is used in the formulation of several novel models. We offer a unique coupled system of fractional delay differential equations with proper respect for the role that multi‐term operators play in the research of fractional differential equations, taking into account the newly established solution for fractional integral and derivative. We also made the assumptions that connected integral boundary conditions would be added on top of n$$ n $$‐fractional differential derivatives. The requirements for the existence and uniqueness of solutions are also developed using fixed‐point theorems. While analyzing various sorts of Ulam's stability results, the qualitative elements of the underlying model will also be examined. In the paper's final section, an example is given for purposes of demonstration. [ABSTRACT FROM AUTHOR]

This paper deals with the free boundary problem for a double‐layered tumor filled with quiescent cells and proliferating cells, where time delay τ>0$$ \tau >0 $$ in cell proliferation is taken into account. These two types of living cells exhibit different metabolic responses and consume nutrients σ$$ \sigma $$ at different rates λ1$$ {\lambda}_1 $$ and λ2$$ {\lambda}_2 $$ ( λ1⩽λ2$$ {\lambda}_1\leqslant {\lambda}_2 $$). Time delay happens between the time at which a cell commences mitosis and the time at which the daughter cells are produced. The problem is reduced to a delay differential equation on the tumor radius R(t)$$ R(t) $$ over time, and the difficulty arises from the jump discontinuity of the consumption rate function. We give rigorous analysis on this new model and study the dynamical behavior of the global solutions for any initial φ(t)$$ \varphi (t) $$. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of nonlinear differential equations with multiple distributed delays. We reduced the problem of search for periodic solutions of the distributed delay differential equations to that of finding zeros of an operator on some Banach space. Utilizing the equivariant degree theory, we obtain some sufficient conditions to guarantee the existence of multiplicity of periodic solutions. Also, we give some information about the symmetry of the periodic solutions. At last, we give an example to illustrate our result. [ABSTRACT FROM AUTHOR]

This work introduces a new stochastic volatility model with delay parameters in the volatility process, extending the Barndorff–Nielsen and Shephard model. It establishes an analytical expression for the log price characteristic function, which can be applied to price European options. Empirical analysis on S&P500 European call options shows that adding delay parameters reduces mean squared error. This is the first instance of providing an analytical formula for the log price characteristic function in a stochastic volatility model with multiple delay parameters. We also provide a Monte Carlo scheme that can be used to simulate the model. [ABSTRACT FROM AUTHOR]

The work in this article provides literature with some results concerning the uniform asymptotic stability, the boundedness and the square integrability of solutions for some non autonomous equations of third order supplied with delay and neutral parameters. The main tool used in this work is the second method of Lyapunov. The article is finished by giving a concrete example that ensure the application of the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we propose numerical methods for the Riesz space distributed-order advection–diffusion equation with delay in 2D. We utilize the fractional backward differential formula method of second order (FBDF2), and weighted and shifted Grünwald difference (WSGD) operators to approximate the Riesz fractional derivative and develop the finite difference method for the RFADED. It has been shown that the obtained schemes are unconditionally stable and convergent with the accuracy of O (h 2 + k 2 + κ 2 + σ 2 + ρ 2) , where h, k and κ are space step for x, y and time step, respectively. Also, numerical examples are constructed to demonstrate the effectiveness of the numerical methods, and the results are found to be in excellent agreement with analytic exact solution. [ABSTRACT FROM AUTHOR]

We study certain Volterra integral equations that extend and recover first order ordinary differential equations (ODEs). We formulate the former equations from the latter by replacing classical derivatives with nonlocal integral operators with anti-symmetric kernels. Replacements of spatial derivatives have seen success in fracture mechanics, diffusion, and image processing. In this paper, we consider nonlocal replacements of time derivatives which contain future data. To account for the nonlocal nature of the operators, we formulate initial "volume" problems (IVPs) for these integral equations; the initial data is prescribed on a time interval rather than at a single point. As a nonlocality parameter vanishes, we show that the solutions to these equations recover those of classical ODEs. We demonstrate this convergence with exact solutions of some simple IVPs. However, we find that the solutions of these nonlocal models exhibit several properties distinct from their classical counterparts. For example, the solutions exhibit discontinuities at periodic intervals. In addition, for some IVPs, a continuous initial profile develops a measure-valued singularity in finite time. At subsequent periodic intervals, these solutions develop increasingly higher order distributional singularities. [ABSTRACT FROM AUTHOR]

This paper explores the linearized stability of nonlinear delay differential equations (DDEs) with impulses. The classical results on the existence of periodic solutions are extended from ordinary differential equations (ODEs) to DDEs with impulses. Furthermore, the classical results of linearized stability for nonlinear semigroups are generalized to periodic DDEs with impulses. A significant challenge arises from the need for a discontinuous initial function to obtain periodic solutions. To address this, first-kind discontinuous spaces R ([ a , b ] , R n) are introduced for defining DDEs with impulses, providing key existence and uniqueness results. This study also establishes linear stability results by linearizing the Poincaré operator for DDEs with impulses. Additionally, the stability properties of equilibrium solutions for these equations are analyzed, highlighting their importance due to the wide range of applications in various scientific fields. [ABSTRACT FROM AUTHOR]

The present work focuses on the convergence order of one point large deviations rate functions for the backward Euler method of stochastic delay differential equations (SDDEs) with small noise. The drift and diffusion coefficients of SDDEs are allowed to grow super-linearly with respect to both the state variables and the delay variables. It is shown that the backward Euler method satisfies the one point large deviations principle (LDP) with good rate function. Further, the local uniform convergence orders of the one point large deviations rate function are derived by means of the equivalent characterizations for the original and numerical rate functions. These theoretical results are finally supported by a series of numerical experiments. [ABSTRACT FROM AUTHOR]

We present a Lohner-type algorithm for rigorous integration of systems of delay differential equations (DDEs) with multiple delays, and its application in computation of Poincaré maps, to study the dynamics of some bounded, eternal solutions. The algorithm is based on a piecewise Taylor representation of the solutions in the phase space, and it exploits the smoothing of solutions occurring in DDEs to produce enclosures of solutions of a high order. We apply the topological techniques to prove various kinds of dynamical behaviour, for example, existence of (apparently) unstable periodic orbits in Mackey–Glass equation (in the regime of parameters where chaos is numerically observed) and persistence of symbolic dynamics in a delay-perturbed chaotic ODE (the Rössler system). [ABSTRACT FROM AUTHOR]

In this article, a class of Caputo–Hadamard fractional stochastic differential equation (FSDEs) with time-delays and impulses is considered. With the help of contraction mapping principle, we derive the existence and uniqueness of the solutions to the purposed system. Subsequently, by virtue of the stochastic analysis techniques and generalized Gr o ¨ nwall inequality, the Ulam–Hyers stability (U–Hs) of the addressed system is established. Finally, we present an example to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a class of scalar quartic polynomial delay systems is investigated. We found rich dynamics in this system through numerical simulation, including chaotic attractors, chaotic saddles, and intermittent chaos. Moreover, this chaotic quartic system may serve as an approximation, through Taylor expansion, for a wide class of scalar delay differential equations. Thus, these nonlinear systems may exhibit chaotic behaviors, and the studies in our paper may provide an insight into the emergence of chaos in other time-delay nonlinear systems. We also conduct a detailed theoretical analysis of the system, including the stability of equilibria and Hopf bifurcation analysis based on the theory of normal form and center manifold. Additionally, a numerical analysis is provided to give numerical evidence for the existence of chaos. [ABSTRACT FROM AUTHOR]

The main result of this paper is a mathematical model that describes the dynamics of the motion of several cars in areas with different speed limits. As such areas, we can consider speed limit zones and speed bumps or uneven road surfaces. The model is a system of differential equations with a delayed argument. The dynamical properties of the model are studied by numerical methods. A computer program has been developed that uses the model to describe the motion of traffic flows in various road situations. The simulation results coincide with the observation data of real traffic flows. [ABSTRACT FROM AUTHOR]

The relaxation self-oscillations of the Mackey–Glass equation are studied under the assumption that the exponent in the nonlinearity denominator is a large parameter. We consider the case where the limit relay equation, which arises as the large parameter tends to infinity, has a periodic solution with the smallest number of breaking points on the period. In this case, we prove the existence of a periodic solution of the Mackey–Glass equation that is asymptotically close to the periodic solution of the limit equation. [ABSTRACT FROM AUTHOR]

The COVID-19 pandemic came with many setbacks, be it to a country's economy or the global missions of organizations like WHO, UNICEF or GTFCC. One of the setbacks is the rise in cholera cases in developing countries due to the lack of cholera vaccination. This model suggested a solution by introducing another public intervention, such as adding Chlorine to water bodies and vaccination. A novel delay differential model of fractional order was recommended, with two different delays, one representing the latent period of the disease and the other being the delay in adding a disinfectant to the aquatic environment. This model also takes into account the population that will receive a vaccination. This study utilized sensitivity analysis of reproduction number to analytically prove the effectiveness of control measures in preventing the spread of the disease. This analysis provided the mathematical evidence for adding disinfectants in water bodies and inoculating susceptible individuals. The stability of the equilibrium points has been discussed. The existence of stability switching curves is determined. Numerical simulation showed the effect of delay, resulting in fluctuations in some compartments. It also depicted the impact of the order of derivative on the oscillations. [ABSTRACT FROM AUTHOR]

This paper trains a multilayered neural network (MLNN) for solving fractional delay differential equations (FDDEs), including nonlinear and singular types. The proposed methodology involves replacing the unknown functions in the equations with a truncated power series expansion. Subsequently, a collection of algebraic equations is solved utilizing an iterative minimization technique that leverages the capabilities of the MLNN architecture. The outcomes demonstrate that the MLNN architecture provides the required accuracy and strong stability compared to several numerical methods. [ABSTRACT FROM AUTHOR]

As delays are common, persistent, and ingrained in daily life, it is imperative to take them into account. In this work, we explore the averaging principle for impulsive Atangana–Baleanu fractional stochastic delay differential equations driven by Lévy noise. The link between the averaged equation solutions and the equivalent solutions of the original equations is shown in the sense of mean square. To achieve the intended outcomes, fractional calculus, semigroup properties, and stochastic analysis theory are used. We also provide an example to demonstrate the practicality and relevance of our research. [ABSTRACT FROM AUTHOR]

This study presents an (ε,μ) -uniform numerical method for a two-parameter singularly perturbed time-delayed parabolic problems. The proposed approach is based on a fitted operator finite difference method. The Crank-Nicolson method is used on a uniform mesh to discretize the time variables initially. Subsequently, the resulting semi-discrete scheme is further discretized in space using Simpson's l/3rd rule. Finally, the finite difference approximation of the first derivatives is applied. The method is unique in that it is not dependent on delay terms, asymptotic expansions, or fitted meshes. The fitting factor's value, which is used to account for abrupt changes in the solution, is calculated using the theory of singular perturbations. The developed scheme is demonstrated to be secondorder accurate and uniformly convergent. The proposed method's applicability is validated by three model examples, which yielded more accurate results than some other methods found in the literature. [ABSTRACT FROM AUTHOR]

Agricultural Heritage Systems (AHS) are pivotal in preserving rich agricultural production experience and traditional culture, as well as in maintaining biodiversity and promoting sustainable development in agriculture and rural economies. However, climate change poses significant threats to these systems, such as ecological degradation, biodiversity loss, and shifts in agricultural production patterns. This study, grounded in theories of information asymmetry and bounded rationality, constructs evolutionary game models for adaptive management of AHS under market mechanisms and government guidance. By employing stability analysis and numerical simulation with Delay Differential Equations (DDE) that consider historical delays, and through sensitivity analysis, this research delves into the strategic evolutionary outcomes of stakeholders under various scenarios. It aims to provide theoretical insights and policy recommendations for the dynamic protection and adaptive management of AHS in the face of climate change. The findings indicate that the public goods nature of AHS, alongside externalities and information asymmetry, leads to market failure. Sole reliance on autonomous actions by farmers and meteorological departments is insufficient for optimal resource allocation and effective protection. Government intervention, through regulatory and incentive measures, can effectively mitigate market failures and steer adaptive management of AHS towards efficiency and sustainability. Moreover, the study identifies key factors for adaptive management, such as enhancing stakeholders' initial willingness to participate, reducing the costs of adapting to climate change, optimizing cooperative benefit distribution mechanisms, and increasing the profitability of resource cooperation. Sensitivity analysis of government subsidies and penalty mechanisms further reveals the complex and critical role these policy tools play in fostering stakeholder engagement. Based on these findings, the study recommends enhancing initial participation willingness, controlling transformation costs to alleviate economic burdens, optimizing benefit distribution mechanisms to boost cooperative resource profitability, and establishing dynamic subsidy and penalty mechanisms for optimal resource allocation. The theoretical and practical contributions of this research lie in applying theories of information asymmetry and bounded rationality to the adaptive management of AHS under climate change, enriching the theoretical framework in this field, and providing scientific decision-making support for policymakers. By demonstrating an effective path for AHS protection through combined government and market mechanisms in the context of global climate change, this research holds significant theoretical and practical implications for enhancing the efficiency of adaptive management of AHS, protecting, and inheriting valuable agricultural cultural heritage. [ABSTRACT FROM AUTHOR]

AbstractApproximate Bayesian computation (ABC) is a class of Bayesian inference algorithms that targets problems with intractable or unavailable likelihood functions. It uses synthetic data drawn from the simulation model to approximate the posterior distribution. However, ABC is computationally intensive for complex models in which simulating synthetic data is very expensive. In this article, we propose an early rejection Markov chain Monte Carlo (ejMCMC) sampler based on Gaussian processes to accelerate inference speed. We early reject samples in the first stage of the kernel using a discrepancy model, in which the discrepancy between the simulated and observed data is modeled by Gaussian process (GP). Hence, synthetic data is generated only if the parameter space is worth exploring. We demonstrate through theory, simulation experiments, and real data analysis that the new algorithm significantly improves inference efficiency compared to existing early-rejection MCMC algorithms. In addition, we employ our proposed method within an ABC sequential Monte Carlo (SMC) sampler. In our numerical experiments, we use examples of ordinary differential equations, stochastic differential equations, and delay differential equations to demonstrate the effectiveness of the proposed algorithm. We develop an R package that is available at https://github.com/caofff/ejMCMC. [ABSTRACT FROM AUTHOR]

We develop a delay‐adaptive controller for a class of first‐order hyperbolic partial integro‐differential equations with an unknown input delay. By employing a transport PDE to represent delayed actuator states, the system is transformed into a transport partial differential equation with unknown propagation speed cascaded with a PIDE. A parameter update law is designed using a Lyapunov argument and the infinite‐dimensional backstepping technique to establish global stability results. Subsequently, the effectiveness of the proposed approach was validated through numerical simulations. [ABSTRACT FROM AUTHOR]

This work focuses on linear continuous‐time systems with time‐varying delays, in the general case of time‐varying matrices. Initially, we explore the case of positive systems within this class and introduce two distinct delay‐independent conditions of exponential input‐to‐state stability. Moreover, we provide guaranteed exponential convergence rates for such conditions, explicitly stating the ISS gains. Then, we extend our analysis to systems without sign constraints, adopting a state‐bounding approach that takes advantage of the properties of positive systems. Due to the time‐varying nature of the systems, all proposed conditions are in terms of an infinite number of inequalities. Hence, implementation issues are discussed and significant special cases in which the conditions can be cast into a linear programming problem of finite dimension are presented. [ABSTRACT FROM AUTHOR]

In this paper we construct a general observer theory for differential delay systems based on different types of symmetries (exact symmetry or asymptotic symmetry), ending up with a certain number of semi‐global and global observers, with bounded or unbounded system's solutions. We introduce the notions of symmetry for a differential delay system, being inspired by well‐known definitions of symmetry for an ordinary or partial differential system, and variational symmetry for the associated variational differential delay system. We illustrate observer design procedures in details, by proving that the existence of a (variational asymptotic) symmetry with system's detectability in the first approximation have a central role in the design of a state observer. The symmetry is a one‐parameter group of transformations which maps the system into itself (exact symmetry) or into a different system (asymptotic symmetry), approximating the original one with better and better accuracy as the parameter of the symmetry is larger. The types of symmetries we consider here show an important contractive action on the state and input spaces for which the system's solutions are mapped into arbitrarily small neighbourhoods of the origin in which the transformed system can be well‐approximated by its linearization. The parameter of the symmetry may be constant (semiglobal observers) or updated on‐line by a state norm estimator (global observers). [ABSTRACT FROM AUTHOR]

It is usually not possible to solve partial differential equations, especially the delay type, with analytical methods. Therefore, in this article, we present an efficient method for solving differential equations of the difference delayed reaction-diffusion type, which can be generalized to other delayed partial differential equations. In the proposed approach, we first convert the delayed equation into an equivalent non-delayed equation by inserting the corresponding delay function with an effective technique. Then, using a pseudo-spectral method, we discretize the obtained equation in the Legendre-Gauss-Lobatto collocation points and present an algebraic system with an equal number of equations and unknowns which can be solved by quasi-Newton methods such as Levenderg-Marquardt algorithm. The approximate solutions can be obtained with exponential accuracy. The convergence analysis of the method is fully discussed and four examples are presented to evaluate the results and compare with one of the conventional methods used to solve partial differential equations, that is, the compact finite difference method. [ABSTRACT FROM AUTHOR]

We consider nonlinear delay differential and renewal equations with infinite delay. We extend the work of Gyllenberg et al. [Appl. Math. Comput., 333 (2018), pp. 490--505] by introducing a unifying abstract framework, and we derive a finite-dimensional approximating system via pseudospectral discretization. For renewal equations, we consider a reformulation in the space of absolutely continuous functions via integration. We prove the one-to-one correspondence of equilibria between the original equation and its approximation, and that linearization and discretization commute. Our most important result is the proof of convergence of the characteristic roots of the pseudospectral approximation of the linear(ized) equations when the collocation nodes are chosen as the family of scaled zeros or extrema of Laguerre polynomials. This ensures that the finite-dimensional system correctly reproduces the stability properties of the original linear equation if the dimension of the approximation is large enough. The result is illustrated with several numerical tests, which also demonstrate the effectiveness of the approach for the bifurcation analysis of equilibria of nonlinear equations. The new approach used to prove convergence also provides the exact location of the spectrum of the differentiation matrices for the Laguerre zeros and extrema, adding new insights into properties that are important in the numerical solution of differential equations by pseudospectral methods. [ABSTRACT FROM AUTHOR]

This mathematical model delves into the dynamics of the immune system during Chronic Myeloid Leukemia (CML) therapy with imatinib. The focus lies in elucidating the allergic reactions induced by imatinib, specifically its impact on T helper (Th) cells and Treg cells. The model integrates cellular interactions, drug pharmacokinetics, and immune responses to unveil the mechanisms underlying the dominance of Th2 over Th1 and Treg cells, leading to allergic manifestations. Through a system of coupled delay differential equations, the interplay between healthy and leukemic cells, the influence of imatinib on T cell dynamics, and the emergence of allergic reactions during CML therapy are explored. [ABSTRACT FROM AUTHOR]

This study is concerned with finding numerical solutions of nonlinear delay differential equations involving extended Mittag-Leffler fractional derivatives of the Caputo-type. The main benefit of the used extension is to address the complexity resulting from the limitations of using fractional derivatives with non-singular Mittag-Leffler kernels. We discussed the existence and uniqueness of solutions for the studied delay models. Next, we modified an Adams-type method to numerically solve fractional delay differential equations combined with Mittag-Leffler kernels. A new type of solution belonging to the L 1 space is presented for the studied models using the proposed scheme. [ABSTRACT FROM AUTHOR]

The problem of stabilizing an unstable highly nonlinear hybrid neutral stochastic functional differential equation with infinite delay (HNSFDEwID) is addressed in this article. The objective of this study is to propose a discrete‐time feedback control (DTFC) approach for stabilizing the original system. Sufficient conditions are derived for the existence and uniqueness of the system's solution, as well as its boundedness, prerequisites for studying the system's dynamic behavior. Four distinct criteria for stabilizing the controlled system are provided, including H∞$$ {H}_{\infty } $$ stabilization, asymptotic stabilization, exponential stabilization, and almost surely exponentially stabilization. Additionally, the upper limit of the time interval for discrete observations is calculated based on these criteria. Finally, the proposed theory is validated through an illustrative example. [ABSTRACT FROM AUTHOR]

This article examines the oscillatory characteristics of a fourth-order Emden–Fowler differential equation, specifically when it includes a sublinear neutral term. Our methodology centers on establishing multiple theorems that introduce innovative conditions to guarantee that there are no positive solutions to the examined equation. Due to the symmetry between non-oscillatory solutions, we obtain oscillation conditions by excluding only positive solutions. We employ the Riccati technique in various ways to achieve this objective. The criteria presented in this study complement and generalize many findings published in the literature. We support the efficiency of our findings by applying them to an example. [ABSTRACT FROM AUTHOR]

In this work, we study the oscillatory features of a class of neutral differential equations with multiple delays. We present novel oscillation conditions for this equation by using a comparison method. We create conditions that exclude the positive solution of the examined equation. Using the feature of symmetry between non-oscillatory solutions (positive and negative), these conditions also exclude negative solutions without adding additional restrictions. In our study, we take into account the even and odd cases of the order of the equation. Furthermore, we study the asymptotic properties in three different cases of functional coefficients. Our results are a fundamental extension and generalization of previous relevant findings, and this fact has been demonstrated through comparisons. [ABSTRACT FROM AUTHOR]

This article presents a approximation method for linear neutral delay differential equations using Hermite polynomials. The method ensures the unknown function to be found by approximating the first derivative with the help of the finite Hermite series. This method reduces the problem to a linear algebraic system using the first derivative approach, matrix operations and collocation points. Also error of the solution is estimated by constructing an error problem. Numerical examples are solved to explain the method and error estimation technique. The results demonstrate the effectiveness and accuracy of the current study. Calculations have been made using MATLAB. [ABSTRACT FROM AUTHOR]

Many neurodegenerative diseases (NDs) are characterized by the slow spatial spread of toxic protein species in the brain. The toxic proteins can induce neuronal stress, triggering the Unfolded Protein Response (UPR), which slows or stops protein translation and can indirectly reduce the toxic load. However, the UPR may also trigger processes leading to apoptotic cell death and the UPR is implicated in the progression of several NDs. In this paper, we develop a novel mathematical model to describe the spatiotemporal dynamics of the UPR mechanism for prion diseases. Our model is centered around a single neuron, with representative proteins P (healthy) and S (toxic) interacting with heterodimer dynamics (S interacts with P to form two S’s). The model takes the form of a coupled system of nonlinear reaction–diffusion equations with a delayed, nonlinear flux for P (delay from the UPR). Through the delay, we find parameter regimes that exhibit oscillations in the P- and S-protein levels. We find that oscillations are more pronounced when the S-clearance rate and S-diffusivity are small in comparison to the P-clearance rate and P-diffusivity, respectively. The oscillations become more pronounced as delays in initiating the UPR increase. We also consider quasi-realistic clinical parameters to understand how possible drug therapies can alter the course of a prion disease. We find that decreasing the production of P, decreasing the recruitment rate, increasing the diffusivity of S, increasing the UPR S-threshold, and increasing the S clearance rate appear to be the most powerful modifications to reduce the mean UPR intensity and potentially moderate the disease progression. [ABSTRACT FROM AUTHOR]

A common model for studying pattern formation in large groups of neurons is the neural field. We investigate a neural field with excitatory and inhibitory neurons, like [H. R. Wilson and J. D. Cowan (1972), Biophys. J., 12, pp. 1-24], with transmission delays and gap junctions. We build on the work of [S. Visser, R. Nicks, O. Faugeras, and S. Coombes (2017), Phys. D, 349, pp. 27-45] by investigating pattern formation in these models on the sphere. Specifically, we investigate how gap junctions, modelled by a diffusion term, influence the behavior of the neural field. We look in detail at the periodic orbits that are generated by Hopf bifurcations in the presence of spherical symmetry. To this end, we derive general formulas to compute the normal form coefficients for these bifurcations up to third order and predict the stability of the resulting branches. A novel numerical method to solve delay equations with diffusion on the sphere is formulated and applied in simulations. [ABSTRACT FROM AUTHOR]

We study the oscillation of a first-order delay equation with negative feedback at the critical threshold . We apply a novel center manifold method, proving that the oscillation of the delay equation is equivalent to the oscillation of a -dimensional system of ordinary differential equations (ODEs) on the center manifold. It is well known that the delay equation oscillation is equivalent to the oscillation of a certain second-order ODE, and we furthermore show that the center manifold system is asymptotically equivalent to this same second-order ODE. In addition, the center manifold method has the advantage of being applicable to the case where the parameters oscillate around the critical value , thereby extending and refining previous results in this case. [ABSTRACT FROM AUTHOR]

In this article, we analyze the dynamics of the non-linear tumor-immune delayed (TID) model illustrating the interaction among tumor cells and the immune system (cytotoxic T lymphocytes, T helper cells), where the delays portray the times required for molecule formation, cell growth, segregation, and transportation, among other factors by exploiting the knacks of soft computing paradigm utilizing neural networks with back propagation Levenberg Marquardt approach (NNLMA). The governing differential delayed system of non-linear TID, which comprised the densities of the tumor population, cytotoxic T lymphocytes and T helper cells, is represented by non-linear delay ordinary differential equations with three classes. The baseline data is formulated by exploiting the explicit Runge-Kutta method (RKM) by diverting the transmutation rate of Tc to Th of the Tc population, transmutation rate of Tc to Th of the Th population, eradication of tumor cells through Tc cells, eradication of tumor cells through Th cells, Tc cells' natural mortality rate, Th cells' natural mortality rate as well as time delay. The approximated solution of the non-linear TID model is determined by randomly subdividing the formulated data samples for training, testing, as well as validation sets in the network formulation and learning procedures. The strength, reliability, and efficacy of the designed NNLMA for solving non-linear TID model are endorsed by small/negligible absolute errors, error histogram studies, mean squared errors based convergence and close to optimal modeling index for regression measurements. [ABSTRACT FROM AUTHOR]

In this paper, we have analysed the influence of viscoelastic and frictional damping on the decay rate of solutions for a Kirchhoff-type viscoelastic wave equation with a distributed delay acting on nonlinear internal damping. Taking the relaxation function of a fairly large class and using the method of energy in which we introduce an adapted Lyapunov functional and by exploiting certain properties of convex functions, under certain assumptions on the constants of system, we obtain the optimal decay rate of energy in the sense that it is compatible with the decay rate of the relaxation function. [ABSTRACT FROM AUTHOR]

The purpose of this work is the study of the qualitative behavior of the homogeneous in space solution of a delay differential equation arising from a model of infection dynamics. This study is mainly based on the monotone dynamical systems theory. Existence and smoothness of solutions are proved, and conditions of asymptotic stability of equilibriums in the sense of monotone dynamical systems are formulated. Then, sufficient conditions of global stability of the nonzero steady state are derived, for the two typical forms of the function f, specifying the efficiency of immune response-mediated virus elimination. Numerical simulations illustrate the analytical results. The obtained theoretical results have been applied, in a context of COVID-19 data calibration, to forecast the immunological behaviour of a real patient. [ABSTRACT FROM AUTHOR]

We provide results of a deterministic approximation for non-Markovian stochastic processes modeling finite populations of individuals who recurrently play symmetric finite games and imitate each other according to payoffs. We show that a system of delay differential equations can be obtained as the deterministic approximation of such a non-Markovian process. We also show that if the initial states of stochastic process and the corresponding deterministic model are close enough, then the trajectory of stochastic process stays close to that of the deterministic model up to any given finite time horizon with a probability exponentially approaching one as the population size increases. We use this result to obtain that the lower bound of the population size on the absorption time of the non-Markovian process is exponentially increasing. Additionally, we obtain the replicator equations with distributed and discrete delay terms as examples and analyze how the memory of individuals can affect the evolution of cooperation in a two-player symmetric snow-drift game. We investigate the stability of the evolutionary stable state of the game when agents have the memory of past population states, and implications of these results are given for the stochastic model. [ABSTRACT FROM AUTHOR]

Human immunodeficiency virus type 1 (HIV-1) gradually destroys the CD4+ T cells leading to immune system dysfunction. HIV-1 can result in acquired immunodeficiency syndrome (AIDS) if antiretroviral drugs are not used. HIV/AIDS patients are more vulnerable to opportunistic infections or cancers. Human herpesvirus 8 (HHV-8) targets B cells and causes an AIDS-related cancer known as kaposi sarcoma (KS). Numerous investigations have demonstrated co-infection instances between HIV-1 and HHV-8. In this research, we investigated the co-dynamics of HIV-1 and HHV-8 in vivo using a system of delay differential equations (DDEs). The model explained the interactions between uninfected CD4+ T cells, latently/actively HIV-1-infected CD4+ T cells, free HIV1 particles, uninfected B cells, latently/actively HHV-8-infected B cells, and free HHV-8 particles. Eight distributed-time delays were incorporated into the model to account for the delays that arose during the generation of both actively and latently infected cells, the activation of latent reservoirs, and the maturation of freshly discharged virions. By examining the nonnegativity and boundedness of the solutions, we demonstrated that the model was both mathematically and biologically well-posed. We calculated the model’s equilibria and threshold numbers. We studied the global asymptotic stability of the model’s equilibria by building appropriate Lyapunov functionals and applying the Lyapunov-LaSalle asymptotic stability theorem. Numerical simulations were used to display the results. For the basic reproduction numbers of HHV-8 single-infection (R1) and HIV-1 single-infection (R2), sensitivity analysis was carried out. Comparing HIV-1 or HHV-8 single infections with co-infections of HHV-8 and HIV-1 was shown. It’s interesting to note that we detected larger amounts of HHV-8 and HIV-1 when they co-infect than when they are infected alone. This outcome aligned with several findings seen in the literature. The effect of antiviral drugs and time delays on the co-dynamics of HIV-1 and HHV-8 was investigated. We found that the delay parameter and drug effectiveness both contributed to a decrease in the basic reproduction numbers, R1 and R2. Less treatment efficacies will be needed to keep the system at the infection-free equilibrium and remove HIV-1 and HHV-8 from the body if a model with time delays is employed. [ABSTRACT FROM AUTHOR]

In this paper, a delay differential equation model is investigated, which describes the biodegradation of microcystins (MCs) by Sphingomonas sp. and its degrading enzymes. First, the local stability of the positive equilibrium and the existence of the Hopf bifurcation are obtained. Second, the global attractivity of the positive equilibrium is obtained by constructing suitable Lyapunov functionals, which implies that the biodegradation of microcystins is sustainable under appropriate conditions. In addition, some numerical simulations of the model are carried out to illustrate the theoretical results. Finally, the parameters of the model are determined from the experimental data and fitted to the data. The results show that the trajectories of the model fit well with the trend of the experimental data. [ABSTRACT FROM AUTHOR]

We generalize the model proposed in [Adimy, Babin, Pujo-Menjouet, SIAM Journal on Applied Dynamical Systems (2022)] for prion infection to a network of neurons. We do so by applying a so-called multigroup approach to the system of Delay Differential Equations (DDEs) proposed in the aforementioned paper. We derive the classical threshold quantity $ \mathcal{R}_0 $, i.e. the basic reproduction number, exploiting the fact that the DDEs of our model qualitatively behave like Ordinary Differential Equations (ODEs) when evaluated at the Disease Free Equilibrium. We prove analytically that the disease naturally goes extinct when $ \mathcal{R}_0<1 $, whereas it persists when $ \mathcal{R}_0>1 $. We conclude with some selected numerical simulations of the system, to illustrate our analytical results. [ABSTRACT FROM AUTHOR]

Autoregressive Markov switching (ARMS) time series models are used to represent real-world signals whose dynamics may change over time. They have found application in many areas of the natural and social sciences, as well as in engineering. In general, inference in this kind of systems involves two problems: (a) detecting the number of distinct dynamical models that the signal may adopt and (b) estimating any unknown parameters in these models. In this paper, we introduce a new class of nonlinear ARMS time series models with delays that includes, among others, many systems resulting from the discretisation of stochastic delay differential equations (DDEs). Remarkably, this class includes cases in which the discretisation time grid is not necessarily aligned with the delays of the DDE, resulting in discrete-time ARMS models with real (non-integer) delays. The incorporation of real, possibly long, delays is a key departure compared to typical ARMS models in the literature. We describe methods for the maximum likelihood detection of the number of dynamical modes and the estimation of unknown parameters (including the possibly non-integer delays) and illustrate their application with a nonlinear ARMS model of El Niño–southern oscillation (ENSO) phenomenon. [ABSTRACT FROM AUTHOR]

In this paper, we address the effects of two fiscal policy delays on the dynamical analysis of macroeconomics. First, a time gap between the accrual of taxes and their payment is considered. Second, the time spent between the purchasing decisions and the actual expenditure is also taken into consideration. Since both these delays are significant in controlling macroeconomic conditions, this paper incorporates aforementioned delays into the IS–LM model. At first, a mathematical model is developed using delayed differential equations. Then a unique steady state solution is obtained. Around the equilibrium point, linear stability analysis is done. Also, the occurance of Hopf bifurcation is observed when delay crosses a critical point and switches in stability are also detected. Properties of Hopf bifurcation using center manifold theorem are discussed. Lastly, numerical simulations are run to verify our analysis. In this work, we considered a case study to perform simulation wherein GDP of India for last ten years is recorded for estimating some parameters. In different investment scenarios, numerical simulations corroborate the analytical findings of the model. Furthermore, rigorous analysis shows that adding the right mix of delays can help in maintaining/ regaining the stability after periods of instability, or even gaining stability in the long run. [ABSTRACT FROM AUTHOR]

In this work, we proposed some generalized Darbo type fixed point theorem by employing the concept of generalized operators and measure of non-compactness within the framework of partial order Banach spaces. Some fixed point theorems and their consequences are presented as part of our exploration. These theorems encompass the presentation of various fixed point theorems and the subsequent implications they entail. Further, we demonstrate the practical application of our findings by establishing the existence of solutions for delay differential equations. To substantiate our conclusions, we provide numerical estimations based on a real-world example. This research contributes to the advancement of fixed point theory in the context of partial order Banach spaces and showcases its practical relevance through a concrete application and empirical support. [ABSTRACT FROM AUTHOR]

Herein, we present an algorithm for handling fractional delay differential equations (FDDEs). Chebyshev polynomials (CPs) class of the seventh kind is a subclass of the generalized Gegenbauer (ultraspherical) polynomials. The members of this class make up the basis functions in this paper. Our suggested numerical algorithm is derived using new theoretical findings about these polynomials and their shifted counterparts. We will use the Tau method to convert the FDDE with the governing conditions into a linear algebraic system, which can then be solved numerically using a suitable procedure. We will give a detailed discussion of the convergence and error analysis of the shifted Chebyshev expansion. Lastly, some numerical examples are provided to verify the precision and applicability of the proposed strategy. [ABSTRACT FROM AUTHOR]

This paper focuses on using piecewise derivatives to simulate the dynamic behavior and investigate the crossover effect within the coupled fractional system with delays by dividing the study interval into two subintervals. We establish and prove significant lemmas concerning piecewise derivatives. Furthermore, we extend and develop the necessary conditions for the existence and uniqueness of solutions, while also investigating the Hyers–Ulam stability results of the proposed system. The results are derived using the Banach contraction principle and the Leary–Schauder alternative fixed-point theorem. Additionally, we employ a numerical method based on Newton’s interpolation polynomials to compute approximate solutions for the considered system. Finally, we provide an illustrative example demonstrating our theoretical conclusions’ practical application. [ABSTRACT FROM AUTHOR]