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Measles remains one of the leading causes of death among young children globally, even though a safe and cost-effective vaccine is available. Vaccine hesitancy and social response to vaccination continue to undermine efforts to eradicate measles. In this study, we consider data about measles vaccination and measles prevalence in Germany for the years 2008–2012 in 345 districts. In the first part of the paper, we show that the probability of a local outbreak does not significantly depend on the vaccination coverage, but—if an outbreak does take place—the scale of the outbreak depends significantly on the vaccination coverage. Additionally, we show that the willingness to be vaccinated is significantly increased by local outbreaks, with a delay of about one year. In the second part of the paper, we consider a deterministic delay model to investigate the consequences of the statistical findings on the dynamics of the infection. Here, we find that the delay might induce oscillations if the vaccination coverage is rather low and the social response to an outbreak is sufficiently strong. The relevance of our findings is discussed at the end of the paper. [ABSTRACT FROM AUTHOR]

This paper delves into the analysis of oscillation characteristics within third-order quasilinear delay equations, focusing on the canonical case. Novel sufficient conditions are introduced, aimed at discerning the nature of solutions—whether they exhibit oscillatory behavior or converge to zero. By expanding the literature, this study enriches the existing knowledge landscape within this field. One of the foundations on which we rely in proving the results is the symmetry between the positive and negative solutions, so that we can, using this feature, obtain criteria that guarantee the oscillation of all solutions. The paper enhances comprehension through the provision of illustrative examples that effectively showcase the outcomes and implications of the established findings. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the strong feedback limit of two negative feedback schemes which have proven to be efficient for many biological processes (protein synthesis, immune responses, breathing disorders). In this limit, the nonlinear delayed feedback function can be reduced to a function with a threshold nonlinearity. This will considerably help analytical and numerical studies of networks exhibiting different topologies. Mathematically, we compare the bifurcation diagrams for both the delayed and non-delayed feedback functions and show that Hopf classical theory needs to be revisited in the strong feedback limit. [ABSTRACT FROM AUTHOR]

In this paper, singularly perturbed pseudo-parabolic initial-boundary value problems with time-delay parameter are considered by numerically. Initially, the asymptotic properties of the analytical solution are investigated. Then, a discretization with exponential coefficient is suggested on a uniform mesh. The error approximations and uniform convergence of the presented method are estimated in the discrete energy norm. Finally, some numerical experiments are given to clarify the theory. [ABSTRACT FROM AUTHOR]

In this work, a compact finite difference approach is constructed for singularly perturbed parabolic reaction diffusion problems with a retarded term. The time and space derivatives have been discretized using the θ-method and a compact fourth-order finite difference method on a Shishkin mesh, respectively. Parameter uniform error estimates have been calculated in the $ L_\infty $ L ∞ norm. Some numerical examples have been considered to corroborate the theoretical results and compare the numerical results with existing methods in the literature. It is shown that the present approach provides improved results till date for the problem considered in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the asymptotic and oscillatory behavior of quasi-linear delay differential equations of fourth order. We first find new properties for a class of positive solutions of the studied equation, N a . As an extension of the approach taken in [1], we establish a new criterion that guarantees that N a = ∅ . Then, we create a new oscillation criterion. In this paper, we deal with the asymptotic and oscillatory behavior of quasi-linear delay differential equations of fourth order. We first find new properties for a class of positive solutions of the studied equation, . As an extension of the approach taken in [1], we establish a new criterion that guarantees that . Then, we create a new oscillation criterion. [ABSTRACT FROM AUTHOR]

Abstract: Structured compartmental models in mathematical biology track age classes, stage classes, or size classes of a population. Structured modeling becomes important when mechanistic formulations or intraspecific interactions are class‐dependent. The classic derivation of such models from partial differential equations produces time delays in the transition rates between classes. In particular, the transition from juvenile to adult has a delay equal to the maturation period of the organism. In the literature, many structured compartmental models, posed as ordinary differential equations, omit this delay. We reviewed occurrences of continuous‐time compartmental models for age‐ and stage‐structured populations in the recent literature (2000–2016) to discover which papers did so. About half of the 249 papers we reviewed used a maturation delay. Papers with ecological models were more likely to have the delay than papers with disease models, and mathematically focused papers were more likely to have the delay than biologically focused papers. Recommendations for Resource Managers: Interacting populations often are modeled with systems of ordinary differential equations in which the state variables are numbers of individuals of each species and interaction terms depend only on the current state of the system. Single‐population continuous‐time models with age‐ or stage‐structure, in which state variables represent numbers of individuals in classes such as juveniles and adults, often but not always contain maturation time delays in the transition rates between classes. The exclusion of the delay typically changes the model dynamics. Managers should be aware of the maturation delay issue when considering the results of continuous‐time models of structured populations. Discrete‐time models have an inherent time delay, set by the census time step chosen by the modeler, and for that reason are convenient for modeling maturation and other biological delays. [ABSTRACT FROM AUTHOR]

In the present paper, new oscillation criteria are established for fourth-order delay differential equations of the form (a 3 (t) (a 2 (t) a 1 (t) x ′ (t) ′ ) ′ ) ′ + b (t) x α (σ (t)) = 0 under the assumption (noncanonical) ∫ t 0 ∞ 1 a j (t) d t < ∞ , j = 1 , 2 , 3. We convert the equation into a canonical type, utilize the comparison method, and the Riccati transformation to find sufficient conditions for oscillation of all solutions to the aforementioned problem. This approach greatly simplifies the examination analysis, and provides a substantial improvement of the current results and this is documented by several evidences and illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

In the present paper, we mainly focus on an iterative erythropoiesis model with a nonlinear harvesting term involving a time-varying delay. Under certain conditions and by the Schauder fixed point theorem, we show that the existence of positive periodic solutions can be guaranteed and based on the Banach fixed point theorem, we establish the existence as well as the continuous dependence on parameters theorems of the unique solution. Moreover, two examples are provided to validate the correctness of our outcomes. Our theoretical findings extend and improve several related ones in the existing literature. [ABSTRACT FROM AUTHOR]

This paper delves into the investigation of quasi-linear neutral differential equations in the third-order canonical case. In this study, we refine the relationship between the solution and its corresponding function, leading to improved preliminary results. These enhanced results play a crucial role in excluding the existence of positive solutions to the investigated equation. By building upon the improved preliminary results, we introduce novel criteria that shed light on the nature of these solutions. These criteria help to distinguish whether the solutions exhibit oscillatory behavior or tend toward zero. Moreover, we present oscillation criteria for all solutions. To demonstrate the relevance of our results, we present an illustrative example. This example validates the theoretical framework we have developed and offers practical insights into the behavior of solutions for quasi-linear third-order neutral differential equations. [ABSTRACT FROM AUTHOR]

The asymptotical properties of a special dynamic two-person game are examined under best-response dynamics in both discrete and continuos time scales. The direction of strategy changes by the players depend on the best responses to the strategies of the competitors and on their own strategies. Conditions are given first for the local asymptotical stability of the equilibrium if instantaneous data are available to the players concerning all current strategies. Next, it is assumed that only delayed information is available about one or more strategies. In the discrete case, the presence of delays has an effect on only the order of the governing difference equations. Under continuous scales, several possibilities are considered: each player has a delay in the strategy of its competitor; player 1 has identical delays in both strategies; the players have identical delays in their own strategies; player 1 has different delays in both strategies; and the players have different delays in their own strategies. In all cases, it is assumed that the equilibrium is asymptotically stable without delays, and we examine how delays can make the equilibrium unstable. For small delays, the stability is preserved. In the cases of one-delay models, the critical value of the delay is determined when stability changes to instability. In the cases of two and three delays, the stability-switching curves are determined in the two-dimensional space of the delays, where stability becomes lost if the delay pair crosses this curve. The methodology is different for the one-, two-, and three-delay cases outlined in this paper. [ABSTRACT FROM AUTHOR]

Consider the perturbed nonautonomous linear delay differential equation x(t) = - a(t)x(t-τ) + F(t, x1, t ⩾ 0 where x1(s)=x(t+s) for −δ≤s≤0. Suppose that a(t) ∈ C([0,∞), (0,∞)), τ≥0,F:[0, ∞) x C[−δ,0] → R is a continuous functions and F(t,0) ≡ 0. Here C[−δ,0] is the space of continuous functions Φ: [−δ,0] → R with ∥Φ∥

In this paper, we analyze a model called the k-nearest neighbor queue with the possibility of having delayed queue length feedback. We prove fluid limits for the stochastic queueing model and show that the fluid limit converges to a system of delay differential equations. Using the properties of circulant matrices, we derive a closed form expression for the value of the critical delay, which governs whether the delayed information will induce oscillations or a Hopf bifurcation in our queueing system. [ABSTRACT FROM AUTHOR]

This paper aims to establish the finite difference scheme on Bakhvalov-type mesh for the singularly perturbed pseudo-parabolic problems with time-delay. Using the energy estimates, the bounds of the solution are investigated. An error analysis is derived in the discrete energy norm and the almost second order convergence is obtained. The reliability of the suggested method is displayed by means of a numerical example. [ABSTRACT FROM AUTHOR]

In this paper, the initial-value problem for a class of first order delay differential equations, which emerges as a model for population dynamics, is considered. To solve this problem numerically, using the finite difference method including interpolating quadrature rules with the basis functions, we construct a fitted difference scheme on a uniform mesh. Although this scheme has the same rate of convergence, it has more efficiency and accuracy compared to the classical Euler scheme. The different models, Nicolson’s blowfly and Mackey–Glass models, in population dynamics are solved by using the proposed method and the classical Euler method. The numerical results obtained from here show that the proposed method is reliable, efficient, and accurate. [ABSTRACT FROM AUTHOR]

The menstrual cycle of fertile female is regulated by multiple organs and hormones, controlling the reproductive health and fertility of females. This paper is dedicated to exploring how two negative feedback loops regulate and influence the dynamic behavior of the menstrual cycle. We simplified the regulation circuit of the menstrual cycle into a nonlinear differential equation with two time delays. This simplified mathematical model integrates the hypothalamus–pituitary–ovary primary negative feedback loop and the negative feedback auxiliary loop between the ovary and the pituitary. We investigated the delay-dependent conditions for the stability of the model’s equilibrium points and also demonstrated the presence of Hopf bifurcations. Theoretical analysis suggests that the primary feedback loop is the major factor driving oscillation in the system. Further numerical simulations indicate that the negative feedback auxiliary loop allows the system to adjust its amplitude while maintaining the robustness of the cycle length. Consequently, the results obtained from this model provide new insights into the regulation of oscillations in menstrual cycle irregularities. [ABSTRACT FROM AUTHOR]

Abstract.This paper chiefly takes into account the singularly perturbed delay Volterra-Fredholm integro-differential equations by numerically. In this context, firstly, priori estimates are given and a new discretization is constructed on piecewise-equidistant mesh by using interpolating quadrature rules [2] and composite integration formulas. Then, the convergence analysis and stability bounds of the presented method are discussed. Finally, numerical results are demonstrated with two test problems. [ABSTRACT FROM AUTHOR]

In this article, we derive an optimized relationship between the solution and its corresponding function for second- and fourth-order neutral differential equations (NDE) in the canonical case. Using this relationship, we obtain new monotonic properties of the second-order equation. The significance of this paper stems from the fact that the asymptotic behavior and oscillation of solutions to NDEs are substantially affected by monotonic features. Based on the new relationships and properties, we obtain oscillation criteria for the studied equations. Finally, we present examples and review some previous theorems in the literature to compare our results with them. [ABSTRACT FROM AUTHOR]

In this paper, a delayed mosquito population suppression model, where the number of sexually active sterile mosquitoes released is regarded as a given nonnegative function, and the birth process is density dependent by considering larvae progression and the intra-specific competition within the larvae, is developed and studied. A threshold value r ∗ for the releases of sterile mosquitoes is determined, and it is proved that the origin is globally asymptotically stable if the number of sterile mosquitoes released is above the threshold value r ∗ . Besides, the case when the number of sterile mosquitoes released stays at a constant level r is also considered. In the special case, it is also proved that the origin is globally asymptotically stable if and only if r > r ∗ and that the model exhibits other complicated dynamics such as bi-stability and semi-stability when r ≤ r ∗ . Numerical examples are also provided to illustrate our main theoretical results. [ABSTRACT FROM AUTHOR]

This paper presents a study of singularly perturbed parabolic convection–diffusion problems with delay. The solution of the problem exhibits a weak interior layer and a strong boundary layer. In these layers, the physical variable changes rapidly over small domains in space and in a short time interval. The solution reveals a multiscale character, which adds stiffness to the problem. We present a second-order accurate difference approximation based on the defect correction method to capture this multiscale phenomenon. The problem is discretized by an implicit Euler scheme in time variable on a uniform mesh and a defect correction method consisting of the upwind scheme and the central difference scheme in space variable on a layer-adapted mesh in such a way that it captures the interior as well as the boundary layers. The proposed method is not only free from directional bias but unconditionally stable and converges uniformly. Test examples and numerical results verify the theoretical predictions and illustrate the efficiency of the method. [ABSTRACT FROM AUTHOR]

In this paper, we aim to study the monotonic properties of the solutions of a class of neutral delay differential equations. The importance of this study lies in the fact that the monotonic properties largely control the study of the oscillation and asymptotic behaviour of the solutions to delay differential equations. Then, by using the new properties, we create improved criteria for testing the oscillation of solutions to the studied equation. We also find new criteria that can be applied more than once. Moreover, we discuss the importance and novelty of the results through the application to a special case of the studied equation. [ABSTRACT FROM AUTHOR]

The problem of finding the oscillation bounds for first-order linear delay differential equations has been in the focus of the oscillation theory for a long time. Although numerous estimates for the oscillation bounds are available in the literature, their explicit values were not known. In this paper, we give the oscillation bounds explicitly in terms of the real branches of the Lambert W function. [ABSTRACT FROM AUTHOR]

In this research paper, we obtain the equivalent symmetries of non-homogeneous second order delay differential equations with variable coefficients. Group methods have been used to do this. The approach followed by us to obtain a Lie type invariance condition for the second order delay differential equation is by using Taylor’s theorem for a function of more than one variable. This Lie type invariance condition established by us in this paper, will be used to obtain the determining equations of the second order delay differential equation. We study certain cases under which the delay differential equation admits infinitesimal generators. Further, by performing symmetry analysis of this delay differential equation, the complete group classification for it has been made. [ABSTRACT FROM AUTHOR]

In this paper, the stability and boundedness analysis of a certain system of two nonlinear delay differential equations with variable delay p(t) is carried out. By using the Lyapunov's second method and Lyapunov-Krasovskii's functional derived from the differential equations describing the system which yielded a better stability and boundedness estimate to establish sufficient conditions for the uniform asymptotic stability of the trivial solution and uniform ultimate boundedness of solution. These new results improve and generalize some results that can be found in the literature. [ABSTRACT FROM AUTHOR]

In this paper, a class of singularly perturbed delay differential equations of convection-diffusion (C–D) type is considered. The solutions for these problems exhibit boundary as well as interior layer due to the presence of delay. A few a priori estimates are given on the exact solution and its derivatives. We propose a numerical technique consisting of hybrid finite difference scheme on a Shishkin-type mesh. The proposed technique is analyzed for its consistency, stability and convergence. To show the effectiveness of the present approach, we conduct some numerical experiments. An almost second-order -uniform convergence is established. [ABSTRACT FROM AUTHOR]

A class of singularly perturbed parabolic partial differential equations with a large delay and an integral boundary condition is studied. The problem's solution features a weak interior layer besides a boundary layer. This paper presents a higher-order accurate numerical method on a specially designed non-uniform mesh. The technique employs the Crank-Nicolson difference scheme in the temporal variable, whereas an upwind difference scheme in space. The proposed method is unconditionally stable and converges uniformly independent of the perturbation parameter. The numerical result for two model problems is presented, which agrees with the theoretical estimates. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of periodic solutions for asymptotically linear second-order delay differential equations. We will establish an index theory for the linear system directly in the sense that we do not need to change the problem of the original linear system into the problem of an associated Hamiltonian system. By using the critical point theory and the index theory, some new existence results are obtained. [ABSTRACT FROM AUTHOR]

Parkinson's disease (PD) is a common movement disorder seen in neurological practice, but the diagnosis and management is challenging. Parkinson's capsules give a variety of signs and symptoms, on occasion, growing the tremor and signs and symptoms of the ailment. In this paper, some mathematical models are presented in order to discuss Parkinson's disease mathematically and biologically. Additionally, the Taylor series is used to transform the models into delay differential equation. The stability conditions are demonstrated using Matlab code ode45, and Matlab code dde23. [ABSTRACT FROM AUTHOR]

This paper presents the computational analysis of fractional differential equations of variable‐order delay systems. To the proposed problem, the existence of solutions is derived using Arzela‐Ascoli theorem, and the Banach fixed point theorem is used for uniqueness results. To investigate and address the computational solutions, Adams‐Bashforth‐Moulton technique is established. To demonstrate the method's efficiency, computational simulations of chaotic behaviors in several one‐dimensional delayed systems with distinct variable orders are employed. The numerical solution of the proposed problem gives high precision approximations. [ABSTRACT FROM AUTHOR]

In this paper, the pantograph delay differential equation y ′ (t) = a y (t) + b y c t subject to the condition y (0) = λ is reanalyzed for the real constants a, b, and c. In the literature, it has been shown that the pantograph delay differential equation, for λ = 1 , is well-posed if c < 1 , but not if c > 1 . In addition, the solution is available in the form of a standard power series when λ = 1 . In the present research, we are able to determine the solution of the pantograph delay differential equation in a closed series form in terms of exponential functions. The convergence of such a series is analysed. It is found that the solution converges for c ∈ (− 1 , 1) such that b a < 1 and it also converges for c > 1 when a < 0 . For c = − 1 , the exact solution is obtained in terms of trigonometric functions, i.e., a periodic solution with periodicity 2 π b 2 − a 2 when b > a . The current results are introduced for the first time and have not been reported in the relevant literature. [ABSTRACT FROM AUTHOR]

The immune system of the cancer patient's body and the viral lytic cycle play important roles in cancer virotherapy. Most mathematical models for virotherapy do not include these two agents simultaneously. In this paper, based on clinical observations we propose a mathematical model including the time of the viral lytic cycle, the viral burst size, and the immune system response. The proposed model is a nonlinear system of delay differential equations in which the period of the viral lytic cycle is modeled as a delay parameter and is used as the bifurcation parameter. We analyze the stability of equilibrium points and the existence of Hopf bifurcation and obtain some conditions for the stability of equilibrium points in terms of the burst size and delay parameter. Finally, we confirm the results with a numerical example and describe them from a biological point of view. [ABSTRACT FROM AUTHOR]

Many real-world problems have been modeled via delay differential equations. The pantograph delay differential equation y ′ (t) = a y (t) + b y c t belongs to such a set of delay differential equations. To the authors' knowledge, there are no standard methods to solve the delay differential equations, i.e., unlike the ordinary differential equations, for which numerous and standard methods are well-known. In this paper, the Adomian decomposition method is suggested to analyze the pantograph delay differential equation utilizing two different canonical forms. A power series solution is obtained through the first canonical form, while the second canonical form leads to the exponential function solution. The obtained power series solution coincides with the corresponding ones in the literature for special cases. Moreover, several exact solutions are derived from the present power series solution at a specific restriction of the proportional delay parameter c in terms of the parameters a and b. The exponential function solution is successfully obtained in a closed form and then compared with the available exact solutions (derived from the power series solution). The obtained results reveal that the present analysis is efficient and effective in dealing with pantograph delay differential equations. [ABSTRACT FROM AUTHOR]

A stochastic model describing the concentration of the drug in the body during its IV-administration is discussed. The paper compares a deterministic model created with certain simplifications with the stochastic model. Fluctuating and irregular patterns of plasma concentrations of some drugs observed during intravenous infusion are explained. An illustrative example is given with certain values of drug infusion rate and drug elimination rate. [ABSTRACT FROM AUTHOR]

In this paper, we study queueing systems with delayed information that use a generalization of the multinomial logit choice model as its arrival process. Previous literature assumes that the functional form of the multinomial logit model is exponential. However, in this work we generalize this to different functional forms. In particular, we compute the critical delay and analyze how it depends on the choice of the functional form. We highlight how the functional form of the model can be interpreted as an exponential model where the exponential rate parameter is uncertain. Furthermore, the rate parameter distribution is given by the inverse Laplace–Stieltjes transform of the functional form when it exists. We perform numerous numerical experiments to confirm our theoretical insights. [ABSTRACT FROM AUTHOR]