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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.

A new class of nonlocal boundary value problems consisting of multi-term delay fractional differential equations and multipoint-integral boundary conditions is studied in this paper. We derive a more general form of the solution for the given problem by applying a fractional integral operator of an arbitrary order $ \beta_{\xi} $ instead of $ \beta_{1} $; for details, see Lemma 2. The given problem is converted into an equivalent fixed-point problem to apply the tools of fixed-point theory. The existence of solutions for the given problem is established through the use of a nonlinear alternative of the Leray-Schauder theorem, while the uniqueness of its solutions is shown with the aid of Banach's fixed-point theorem. We also discuss the stability criteria, icluding Ulam-Hyers, generalized Ulam-Hyers, Ulam-Hyers-Rassias, and generalized Ulam-Hyers-Rassias stability, for solutions of the problem at hand. For illustration of the abstract results, we present examples. Our results are new and useful for the discipline of multi-term fractional differential equations related to hydrodynamics. The paper concludes with some interesting observations.

The mathematical modeling and dynamic analysis of time-delayed enzymatic chemical reactions in biological systems are presented in this research. The objective is to examine the function of time lags in these reactions and to get a complete knowledge of the behavior of biological systems in a reaction to modifications in the quantity present of reactants and products. The model, which is based on delay differential equations, includes a time delay term to account for the lag between changes in the concentration of reactants, reaction rate constants and product responses. The findings give insight into how enzymatic processes behave dynamically and how stability is impacted by time lags, oscillation and general efficiency of the system. These results have significant importance for our comprehension of how biological processes are regulated and for the creation of biological control structures.

In this paper, we constructed a novel fixed point iterative scheme called the Modified-JK iterative scheme. This iteration process is a modification of the JK iterative scheme. Our scheme converged weakly to the fixed point of a nonexpansive mapping and strongly to the fixed point of a mapping satisfying condition (E). We provided some examples to show that the new scheme converges faster than some existing iterations. Stability and data dependence results were proved for this iteration process. To substantiate our results, we applied our results to solving delay differential equations. Furthermore, the newly introduced scheme was applied in approximating the solution of a class of third order boundary value problems (BVPs) by embedding Green's functions. Moreover, some numerical examples were presented to support the application of our results to BVPs via Green's function. Our results extended and generalized other existing results in literature.

In this article, a delay differential equations model is constructed to observe the spread of rabies among human and dog populations by considering two delay effects on incubation period and vaccine efficacy. Other parameters that affect the spread of rabies are also analyzed. Using the basic reproduction number, it is shown that dog populations and the two delays gives a significant effect on the spread of rabies among human and dog populations. The existence of two delays causes the system to experience Transcritical bifurcation instead of Hopf bifurcation. The numerical simulation shows that depending only on one control method is not enough to reduce or eradicate rabies within the dog populations; instead, it requires several combined strategies, such as increasing dog vaccinations, reducing contact with infected dogs, and controlling puppies' birth. The spread within the human population will be reduced if the spread within the dog population is reduced.

In this manuscript, we analyze a new virus model of SARS-CoV-2 infection with immune response. The initial model was proposed by Mochan et al. [23] to describe an experiment made on Macaques. We consider the latent period of newly infected cells by introducing a delay in the model. We fully analyze the quality properties of the model and investigate strategies to reduce secondary infections. Moreover, we investigate the impact of the latent period on the spread of the infection. We observe due to the delay, the possibility to reach an infection-free state when R0>1. This observation is not possible when the delay is not considered. We also demonstrate the occurrence of a Hopf bifurcation when R0>1. We further introduce two control parameters to prevent new infections and inhibit viral production, and we formulate an optimal control problem aiming to minimize infections, virus proliferation and the cost of treatment. We establish the existence of the optimal solution and illustrate the theoretical results numerically. In contrast to Mochan et al. [23] results, our simulations show that increased suppression of viral production can change a lethal or chronic infection to a survivable scenario or acute infection. We also observe that the latency period can cause several states of chronic infection before the host recovers totally. Moreover, the density of viruses increases as the latent period is long. The reason being that the immune system is alerted with a delay (after the latent period), which is an advantage for the replication of viruses.

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],Rn) 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.

This study presents a practical method for solving fractional order delay variational problems. The fractional derivative is given in the Caputo sense. The suggested approach is based on the Laplace transform and the shifted Legendre polynomials by approximating the candidate function by the shifted Legendre series with unknown coefficients yet to be determined. The proposed method converts the fractional order delay variational problem into a set of (n ＋ 1) algebraic equations, where the solution to the resultant equation provides us the unknown coefficients of the terminated series that have been utilized to approximate the solution to the considered variational problem. Illustrative examples are given to show that the recommended approach is applicable and accurate for solving such kinds of problems.

The method and the formula of variation of constants for ordinary differential equations (ODEs) is a fundamental tool to analyze the dynamics of an ODE near an equilibrium. It is natural to expect that such a formula works for delay differential equations (DDEs), however, it is well-known that there is a conceptual difficulty in the formula for DDEs. Here we discuss the variation of constants formula for DDEs by introducing the notion of a mild solution, which is a solution under an initial condition having a discontinuous history function. Then the principal fundamental matrix solution is defined as a matrix-valued mild solution, and we obtain the variation of constants formula with this function. This is also obtained in the framework of a Volterra convolution integral equation, but the treatment here gives an understanding in its own right. We also apply the formula to show the principle of linearized stability and the Poincaré–Lyapunov theorem for DDEs, where we do not need to assume the uniqueness of a solution.

Abstract We use an improved technique to establish new sufficient criteria of product type for the oscillation of the delay differential equation x ′ ( t ) + ∑ l = 1 m b l ( t ) x ( σ l ( t ) ) = 0 , t ≥ t 0 , $$\begin{aligned} x'(t)+\sum_{l=1}^{m} b_{l}(t)x\bigl(\sigma _{l}(t)\bigr)= 0,\quad t\geq t_{0}, \end{aligned}$$ with b l , σ l ∈ C ( [ t 0 , ∞ ) , [ 0 , ∞ ) ) $b_{l},\sigma _{l}\in C([t_{0},\infty ),[0,\infty ))$ such that σ l ( t ) ≤ t $\sigma _{l}(t)\leq t$ and lim t → ∞ σ l ( t ) = ∞ $\lim_{t \rightarrow \infty} \sigma _{l}(t)=\infty $ , l = 1 , 2 , … , m $l=1,2,\ldots,m$ . The obtained results are applicable for the nonmonotone delay case. Their strength is supported by a detailed practical example.

A simple-form scalar differential equation with delay and nonlinear negative periodic feedback is considered. The existence of several types of slowly oscillating periodic solutions is shown with the same and double periods of the feedback coefficient. The periodic solutions are built explicitly in the case with piecewise constant nonlinearities involved. The periodic dynamics are shown to persist under small perturbations of the equation, which make it smooth. The theoretical results are verified through extensive numerical simulations.

In this paper, by using the Riccati transformation and integral inequality technique, we establish several oscillation criteria for second-order Emden–Fowler neutral delay differential equations under the canonical case and non-canonical case, respectively. Compared with some recent results reported in the literature, we extend the range of the neutral coefficient. Therefore, our results generalize to some of the results presented in the literature. Furthermore, several examples are provided to illustrate our conclusions.

This paper investigates the asymptotic and oscillatory properties of a distinctive class of third-order linear differential equations characterized by multiple delays in a noncanonical case. Employing the comparative method and the Riccati method, we introduce the novel and rigorous criteria to discern whether the solutions of the examined equation exhibit oscillatory behavior or tend toward zero. Our study contributes to the existing literature by presenting theories that extend and refine the understanding of these properties in the specified context. To validate our findings and demonstrate their applicability in a general setting, we offer two illustrative examples, affirming the robustness and validity of our proposed criteria.

The purpose of this paper is to study the oscillatory properties of solutions to a class of delay differential equations of even order. We focus on criteria that exclude decreasing positive solutions. As in this paper, this type of solution emerges when considering the noncanonical case of even equations. By finding a better estimate of the ratio between the Kneser solution with and without delay, we obtain new constraints that ensure that all solutions to the considered equation oscillate. The new findings improve some previous findings in the literature.

In this article, we study the existence of periodic solutions to a class of distributed delay differential equations. We transform the search for periodic solutions with the special symmetry of a delay differential equation to the problem of finding periodic solutions of an associated Hamiltonian system. Using the critical point theory and the pseudo-index theory, we obtain some sufficient conditions for the multiplicity of periodic solutions. This is the first time that critical point theory has been used to study the existence of periodic solutions to distributed delay differential equations.

The dynamical properties of a class of difference equations with continuous arguments of the form x(t)=g(x(t-r1),x(t-r2)) and its singularly perturbed counterpart ∊dxdt=-x(t)+g(x(t-r1),x(t-r2)) are investigated here. We discuss the effect of the time delays r1 and r2 on the qualitative behavior of the considered dynamical systems. The local stability of the fixed points is studied. It is proved that the systems exhibit Hopf bifurcation which means that periodic orbits can be created from a fixed point by varying the delays. We compare the results of the singularly perturbed equation with those of the associated difference equation with continuous arguments when the perturbation parameter ∊⟶0 and with those of the corresponding delay differential equation when ∊⟶1. By letting the perturbation parameter ∊⟶0, we find that the singularly perturbed equation exhibits the same qualitative behavior as its corresponding difference equation. Furthermore, the singularly perturbed equation behaves qualitatively the same as its corresponding delay differential equation when ∊⟶1. Finally, we discuss that how this work can be generalized in the fractional differential calculus sense.

In this work, we scrutinize the existence and uniqueness of the solution to the Integro differential equations for the Caputo fractional derivative on the time scale. We derive the solution of the neutral fractional differential equations along the finite delay conditions. The fixed point theory is demonstrated, and the solution depends upon the fixed point theorems: Banach contraction principle, nonlinear alternative for Leray-Schauder type, and Krasnoselskii fixed point theorem.

In this paper, we consider a system of one-dimensional hyperbolic delay differential equations (HDDEs) and their corresponding initial conditions. HDDEs are a class of differential equations that involve a delay term, which represents the effect of past states on the present state. The delay term poses a challenge for the application of standard numerical methods, which usually require the evaluation of the differential equation at the current step. To overcome this challenge, various numerical methods and analytical techniques have been developed specifically for solving a system of first-order HDDEs. In this study, we investigate these challenges and present some analytical results, such as the maximum principle and stability conditions. Moreover, we examine the propagation of discontinuities in the solution, which provides a comprehensive framework for understanding its behavior. To solve this problem, we employ the method of lines, which is a technique that converts a partial differential equation into a system of ordinary differential equations (ODEs). We then use the Runge–Kutta method, which is a numerical scheme that solves ODEs with high accuracy and stability. We prove the stability and convergence of our method, and we show that the error of our solution is of the order O(Δt+h¯4), where Δt is the time step and h¯ is the average spatial step. We also conduct numerical experiments to validate and evaluate the performance of our method.

We present a mathematical model that captures the dynamics of the immune system during allergic reactions. Using delay differential equations, we depict the evolution of T cells, APCs, and IL6, considering cell migration between various body compartments. The biological discussions and interpretations within the article revolve around drug desensitization, highlighting one potential application of the model. We conduct stability analysis on certain equilibrium points, demonstrating stability in some cases and only partial stability in others. Numerical simulations validate the theoretical findings.

In this work, we provide for the first time the characterization of delay terms in systems of delay differential equations within the frame of the Network Simulation Method, a procedure that sets a formal equivalence between the system of differential equations and an electrical network. The results were not achieved previously, which strengths the formalism of the model and provides new opportunities for the method. Free circuit software LTspice is employed to conduct the simulations, which requires few simulation rules and can be programmed either by electrical symbol code or text file. Very few devices are needed to design the network model, and the delay terms are easily implemented by voltage-controlled voltage sources. A practical example for delayed adsorption/desorption kinetics is employed to test the methodology, being the results compared with software Mathematica. Additionally, the modelling of pulse width in passively mode-locked quantum dot lasers by the application of a reverse bias voltage is addressed, which constitutes a promising application in communications and advanced sensing. The power, versatility and simplicity of the Network Simulation Method enables it as an exceptional alternative to solve complex systems described by delay differential equations, from both researching and educational points of view.

Chagas disease has been the target of widespread control programs, primarily through residual insecticide treatments. However, in some regions like the Gran Chaco, these efforts have failed to sufficiently curb the disease. Vector reinfestation into homes and vector resistance to insecticides are possible causes of the control failure. This work proposes a mathematical model for the dynamics of Chagas disease in neighboring rural villages of the Gran Chaco region, incorporating human travel between the villages, passive vector migration, and insecticide resistance. Computational simulations across a wide variety of scenarios are presented. The simulations reveal that the effects of human travel and passive vector migration are secondary and unlikely to play a significant role in the overall dynamics, including the number of human infections. The numerical results also show that insecticide resistance causes a notable increase in infections and is an especially important source of reinfestation when spraying stops. The results suggest that control strategies related to migration and travel between the villages are unlikely to yield meaningful benefit and should instead focus on other reinfestation sources like domestic foci that survive insecticide spraying or sylvatic foci.

This paper is devoted to the study of the effect of delays on the asymptotic stability of a linear differential equation with two delays \[x'(t)=-ax(t)-bx(t-\tau)-cx(t-2\tau),\quad t\geq 0,\] where \(a\), \(b\), and \(c\) are real numbers and \(\tau\gt 0\). We establish some explicit conditions for the zero solution of the equation to be asymptotically stable. As a corollary, it is shown that the zero solution becomes unstable eventually after undergoing stability switches finite times when \(\tau\) increases only if \(c-a\lt 0\) and \(\sqrt{-8c(c-a)}\lt |b| \lt a+c\). The explicit stability dependence on the changing \(\tau\) is also described.

In this paper, we investigate the boundedness and uniformly asymptotically stability of the solutions to a certain third order non-autonomous differential equations with bounded delay. By constructing a Lyapunov functional, sufficient conditions for the stability and boundedness of solutions for equations considered are obtained. We used an example to demonstrate the feasibility of our results. The results essentially improve, include, and complement the results in the literature.

We present a systematic introduction to a class of functions that provide fundamental solutions for autonomous linear integer-order and fractional-order delay differential equations. These functions, referred to as delay functions, are defined through power series or fractional power series, with delays incorporated into their series representations. Using this approach, we have defined delay exponential functions, delay trigonometric functions and delay fractional Mittag-Leffler functions, among others. We obtained Laplace transforms of the delay functions and demonstrated how they can be employed in finding solutions to delay differential equations. Our results, which extend and unify previous work, offer a consistent framework for defining and using delay functions.

In the present paper, we prove the existence of unique almost periodic solutions to fuzzy shunting inhibitory cellular neural networks (FSICNN) with several delays. Further, by means of Halanay inequality we analyze the global exponential stability of these solutions and obtain corresponding convergence rate. The results of this paper are new, and they are concluded with numerical simulations confirming them.

This article deals with certain systems of delay differential equations (DDEs) and a system of ordinary differential equations (ODEs). Here, five new theorems are proved on the fundamental properties of solutions of these systems. The results on the properties of solutions consist of sufficient conditions and they dealt with uniformly asymptotically stability (UAS), instability and integrability of solutions of unperturbed systems of DDEs, boundedness of solutions of a perturbed system of DDEs at infinity and exponentially stability (ES) of solutions of a system of nonlinear ODEs. Here, the techniques of proofs depend upon the Lyapunov- Krasovski? functional (LKF) method and Lyapunov function (LF) method. For illustrations, in particular cases, four examples are constructed as applications. Some results of this paper are given at first time in the literature, and the other results generalize and improve some related ones in the literature.

Tick infestation and tick-borne disease spread in a region of multiple adjacent patches with different environmental conditions depend heavily on the host mobility and patch-specific suitability for tick growth. Here we introduce a two-patch model where environmental conditions differ in patches and yield different tick developmental delays, and where feeding adult ticks can be dispersed by the movement of larger mammal hosts. We obtain a coupled system of four delay differential equations with two delays, and we examine how the dynamical behaviours depend on patch-specific basic reproduction numbers and host mobility by using singular perturbation analyses and monotone dynamical systems theory. Our theoretical results and numerical simulations provide useful insights for tick population control strategies.

This study aims to propose and analyze a mathematical model of the competitive interaction of the pathogen-immune system. Some effects of the existence of natural delays and the addition of therapeutic proteins are considered in the model. A delay arises from the indirect response of the host body when a pathogen invades. The other comes from the maturation of immune cells to produce immune memory cells since the immune system and antigenic substances responsible for provoking the production of immune memory cells. Analytical investigations suggest several sufficient conditions for the existence of a positive steady-state solution. There is a critical pair of delays at which oscillatory behavior appears around the positive steady-state solution. Numerical simulations were carried out to describe the results of the analysis and show that the proposed model can describe the speed of pathogen eradication due to the addition of therapeutic proteins as antigenic substances.

Delay differential equations (DDEs) are used to model some realistic systems as they provide some information about the past state of the systems in addition to the current state. These DDEs are used to analyze the long-time behavior of the system at both present and past state of such systems. Due to the oscillatory nature of DDEs their explicit solution is not possible and therefore one need to use some numerical approaches. In this article, we developed a higher-order numerical scheme for the approximate solution of higher-order functional differential equations of pantograph type with vanishing proportional delays. Some linear and functional transformations are used to change the given interval [0, T] into standard interval [-1, 1] in order to fully use the properties of orthogonal polynomials. It is assumed that the solution of the equation is smooth on the entire domain of interval of integration. The proposed scheme is employed to the equivalent integrated form of the given equation. A Legendre spectral collocation method relative to Gauss-Legendre quadrature formula is used to evaluate the integral term efficiently. A detail theoretical convergence analysis in L∞ norm is provided. Several numerical experiments were performed to confirm the theoretical results.

Q-switching in a mode-locked laser not only makes the amplitude of every single output pulse unequal, but also limits the time width and peak power of output pulses. This paper investigates the Q-switching in a self-mode-locked semiconductor disk laser numerically and experimentally. By using the delay differential equations for passively mode-locking, conditions of Q-switching in a self-mode-locked semiconductor disk laser are numerically analyzed for the first time. Meanwhile, based on the experimental results, the causes of Q-switching tendency including the change of nonlinear refractive index and the change of soft aperture, are also discussed. Some possible measures to suppress Q-switching instability, i.e., to obtain stable continuous-wave mode-locking in a self-mode-locked semiconductor disk laser are proposed.

This study investigates the oscillatory properties of a fourth-order delay functional differential equation. This study’s methodology is built around two key tenets. First, we propose optimized relationships between the solution and its derivatives by making use of some improved monotonic features. By using a comparison technique to connect the oscillation of the studied equation with some second-order equations, the second aspect takes advantage of the significant progress made in the study of the oscillation of second-order equations. Numerous applications of functional differential equations of the neutral type served as the inspiration for the study of a subclass of these equations.

In this paper, we construct a more realistic mathematical model to study toxoplasmosis dynamics. The model considers two discrete time delays. The first delay is related to the latent phase, which is the time lag between when a susceptible cat has effective contact with an oocyst and when it begins to produce oocysts. The second discrete time delay is the time that elapses from when the oocysts become present in the environment to when they are able to infect. The main aim in this paper is to find the conditions under which the toxoplasmosis can disappear from the cat population and to study whether the time delays can affect the qualitative properties of the model. Thus, we investigate the impact of the combination of two discrete time delays on the toxoplasmosis dynamics. Using dynamical systems theory, we are able to find the basic reproduction number R0d that determines the global long-term dynamics of the toxoplasmosis. We prove that, if R0d<1, the toxoplasmosis will be eradicated and that the toxoplasmosis-free equilibrium is globally stable. We design a Lyapunov function in order to prove the global stability of the toxoplasmosis-free equilibrium. We also prove that, if the threshold parameter R0d is greater than one, then there is only one toxoplasmosis-endemic equilibrium point, but the stability of this point is not theoretically proven. However, we obtained partial theoretical results and performed numerical simulations that suggest that, if R0d>1, then the toxoplasmosis-endemic equilibrium point is globally stable. In addition, other numerical simulations were performed in order to help to support the theoretical stability results.

The oscillation of nonlinear differential equations is used in many applications of mathematical physics, biological and medical physics, engineering, aviation, complex networks, sociophysics and econophysics. The goal of this study is to create some new oscillation criteria for fourth-order differential equations with delay and advanced terms (a1(x)(w‴(x))n)′+∑j=1rβj(x)wk(γj(x))=0,{({a}_{1}(x){({w}^{\prime\prime\prime }(x))}^{n})}^{^{\prime} }+\mathop{\sum }\limits_{j=1}^{r}{\beta }_{j}(x){w}^{k}({\gamma }_{j}(x))=0, and (a1(x)(w‴(x))n)′+a2(x)h(w‴(x))+β(x)f(w(γ(x)))=0.{({a}_{1}(x){({w}^{\prime\prime\prime }(x))}^{n})}^{^{\prime} }+{a}_{2}(x)h({w}^{\prime\prime\prime }(x))+\beta (x)f(w(\gamma (x)))=0. The method is based on the use of the comparison technique and Riccati method to obtain these criteria. These conditions complement and extend some of the results published on this topic. Two examples are provided to prove the efficiency of the main results.

The main purpose of this paper is to study the oscillatory properties of solutions of the third-order quasi-linear delay differential equation ℒy(t)+f(t)yβ(σ(t))=0{\cal L}y(t) + f(t){y^\beta }(\sigma (t)) = 0 where ℒy(t) = (b(t)(a(t)(y0(t)) )0)0 is a semi-canonical differential operator. The main idea is to transform the semi-canonical operator into canonical form and then obtain new oscillation results for the studied equation. Examples are provided to illustrate the importance of the main results.

We consider atomistic systems consisting of interacting particles arranged in atomic lattices whose quasi-static evolution is driven by time-dependent boundary conditions. The interaction of the particles is modeled by classical interaction potentials where we implement a suitable irreversibility condition modeling the breaking of atomic bonding. This leads to a delay differential equation depending on the complete history of the deformation at previous times. We prove existence of solutions and provide numerical tests for the prediction of quasi-static crack growth in particle systems.

A numerical method for solving singularly perturbed delay differential equations with a layer or oscillatory behavior for which a small shift is introduced in the reaction term is presented. The stability and convergence of the method have been investigated. To demonstrate the efficiency of the method, two model examples have been presented. Numerical results obtained by the present scheme described that the finding of the present method is accurate than the findings of the previous studies.

This paper aims to establish conditions for the existence, uniqueness and Ulam–Hyers stability of solutions for a coupled system of pantograph problem with three sequential fractional derivatives. Two results on the uniqueness and existence of solutions are proved to utilize the Leray–Schauder and Banach fixed point theorems and positive contraction-type inequalities. Also, the stability in the sense of Ulam–Hyers and Ulam–Hyers–Rassias are studied. An illustrative example with graphical and numerical simulations is also proposed.

Ordinary differential equations (ODE) have long been an important tool for modelling and understanding the dynamics of many real systems. However, mathematical modelling in several areas of the life sciences requires the use of time-delayed differential models (DDEs). The time delays in these models refer to the time required for the manifestation of certain hidden processes, such as the time between the onset of cell infection and the production of new viruses (incubation periods), the infection period, or the immune period. Since real biological systems are always subject to perturbations that are not fully understood or cannot be explicitly modeled, stochastic delay differential systems (SDDEs) provide a more realistic approximation to these models. In this work, we study the predator–prey system considering three time-delay models: one deterministic and two types of stochastic models. Our numerical results allow us to distinguish between different asymptotic behaviours depending on whether the system is deterministic or stochastic, and in particular, when considering stochasticity, we see that both the nature of the stochastic systems and the magnitude of the delay play a crucial role in determining the dynamics of the system.

In this paper, new criteria for oscillation of neutral delay differential equations of second-order are presented. One objective of this study is to complement and extend some well-known related results in the literature. To support our main results, we give illustrating examples.

Somitogenesis is the process by means of which a tissue known as presomitic mesoderm (PSM) is segmented in blocks of cells, called somites, along the anterior-posterior axis of the developing embryo in segmented animals. In vertebrates, somites give rise to axial skeleton, cartilage, tendons, skeletal muscle, and dermis. Somite formation occurs periodically, and this periodicity is driven by a genetic oscillator that operates within PSM cells and is known as the segmentation clock. The correct synchronization of the segmentation clock among PSM cells is essential for somitogenesis to develop normally. When synchronization is disrupted, somites form irregularly and, in consequence, the tissues that originate from them show clear malformations. In this work, based in a model for zebrafish segmentation clock, we investigate by means of a mathematical modeling approach, how PSM-cell synchronization is affected by factors like: the size of PSM-cell networks, the amount of cell-to-cell interactions per PSM cell, the strength of these interactions, and the inherent variability among PSM cells. Interestingly we found that very small PSM-cell networks are unable to synchronize. Moreover, the effect of decreasing the strength of interactions among PSM cells is corrected by increasing the network connectivity-level, and a moderated level of variability among cells can have a positive effect on synchronization, specially in large networks.

In this work, we study the problem of p−th moment global exponential stability for functional differential equations and scalar chaotic delayed equations under random impulsive effects. Meanwhile, the p−th moment global exponential synchronization for the proposed equations is also discussed, whereas the main results are proved by using Lyapunov function and Razumikhin technique. Furthermore, the impact of fixed and random time impulses are presented by applying the results to Mackey Glass blood cell production model and Ikeda bistable resonator model. Finally, the effectiveness of fixed and random impulses are depicted via graphical representations.

The paper is concerned with nonautonomous generalised Nicholson systems under conditions which imply their permanence: by refining the assumptions for permanence, explicit lower and upper uniform bounds for all positive solutions are provided, as well as criteria for the global exponential stability of these systems. In particular, for periodic systems, conditions for the existence of a globally exponentially attractive positive periodic solution are derived.

Abstract We study the oscillation of a first-order linear delay differential equation. A new technique is developed and used to obtain new oscillatory criteria for differential equation with non-monotone delay. Some of these results can improve many previous works. An example is introduced to illustrate the effectiveness and applicability of our results.

This study aims to investigate the asymptotic behavior of a class of third-order delay differential equations. Here, we consider an equation with a middle term and several delays. We obtain an iterative relationship between the positive solution of the studied equation and the corresponding function. Using this new relationship, we derive new criteria that ensure that all non-oscillatory solutions converge to zero. The new findings are an extension and expansion of relevant findings in the literature. We apply our results to a special case of the equation under study to clarify the importance of the new criteria.

The aim of this paper is to investigate the qualitative behavior of the COVID-19 pandemic under an initial vaccination program. We constructed a mathematical model based on a nonlinear system of delayed differential equations. The time delay represents the time that the vaccine takes to provide immune protection against SARS-CoV-2. We investigate the impact of transmission rates, vaccination, and time delay on the dynamics of the constructed system. The model was developed for the beginning of the implementation of vaccination programs to control the COVID-19 pandemic. We perform a stability analysis at the equilibrium points and show, using methods of stability analysis for delayed systems, that the system undergoes a Hopf bifurcation. The theoretical results reveal that under some conditions related to the values of the parameters and the basic reproduction number, the system approaches the disease-free equilibrium point, but if the basic reproduction number is larger than one, the system approaches endemic equilibrium and SARS-CoV-2 cannot be eradicated. Numerical examples corroborate the theoretical results and the methodology. Finally, conclusions and discussions about the results are presented.

We study the size of the global attractor for a delay differential equation with unimodal feedback. We are interested in extending and complementing a dichotomy result by Liz and Röst, which assumed that the Schwarzian derivative of the nonlinear feedback is negative in a certain interval. Using recent stability results for difference equations, we obtain a stability dichotomy for the original delay differential equation in the situation wherein the Schwarzian derivative of the nonlinear term may change sign. We illustrate the applicability of our results with several examples.

Immunogenicity is the ability of substances to evoke an immune response such as a therapeutic protein drug that is considered as a foreign object in the human body. The rise of the immune response results in the production of Anti-Drug Antibody (ADA) that requires a certain period to be activated since it is influenced by the number of injected doses of the drug. The entry of ADA from the depot into the plasma also requires a certain period since the ADA must pass through a series of compartments, hence rises a delay. Both processes are considered as a natural process where the system experiences delay with different delay periods. Immunogenicity on therapeutic protein pharmacokinetics is modelled as a nonlinear delay differential system. From the formulated model, one positive equilibrium solution is obtained under some conditions. Qualitative analysis gives a pair of critical delays in terms of a time delay of the accumulation of protein drug injection and a time required by the ADA to enter the plasma and binding the drug in the plasma. Numerical simulations show that the critical delays result in the appearance of oscillatory behavior in the system. For the system to remain stable, the entering process of ADA into the plasma is delayed in accordance with the obtained critical delay. It is intended such that the injected therapeutic protein drugs provide an optimal effect.

Abstract The objective of this study is to establish new sufficient criteria for oscillation of solutions of odd-order nonlinear delay differential equations. Based on creating comparison theorems that compare the odd-order equation with a couple of first-order equations, we improve and complement a number of related ones in the literature. To show the importance of our results, we provide an example.

Western honeybees (Apis Mellifera) serve extremely important roles in our ecosystem and economics as they are responsible for pollinating $ 215 billion dollars annually over the world. Unfortunately, honeybee population and their colonies have been declined dramatically. The purpose of this article is to explore how we should model honeybee population with age structure and validate the model using empirical data so that we can identify different factors that lead to the survival and healthy of the honeybee colony. Our theoretical study combined with simulations and data validation suggests that the proper age structure incorporated in the model and seasonality are important for modeling honeybee population. Specifically, our work implies that the model assuming that (1) the adult bees are survived from the egg population rather than the brood population; and (2) seasonality in the queen egg laying rate, give the better fit than other honeybee models. The related theoretical and numerical analysis of the most fit model indicate that (a) the survival of honeybee colonies requires a large queen egg-laying rate and smaller values of the other life history parameter values in addition to proper initial condition; (b) both brood and adult bee populations are increasing with respect to the increase in the egg-laying rate and the decreasing in other parameter values; and (c) seasonality may promote/suppress the survival of the honeybee colony.

Abstract Background Simeoni and colleagues introduced a compartmental model for tumor growth that has proved quite successful in modeling experimental therapeutic regimens in oncology. The model is based on a system of ordinary differential equations (ODEs), and accommodates a lag in therapeutic action through delay compartments. There is some ambiguity in the appropriate number of delay compartments, which we examine in this note. Methods We devised an explicit delay differential equation model that reflects the main features of the Simeoni ODE model. We evaluated the original Simeoni model and this adaptation with a sample data set of mammary tumor growth in the FVB/N-Tg(MMTVneu)202Mul/J mouse model. Results The experimental data evinced tumor growth heterogeneity and inter-individual diversity in response, which could be accommodated statistically through mixed models. We found little difference in goodness of fit between the original Simeoni model and the delay differential equation model relative to the sample data set. Conclusions One should exercise caution if asserting a particular mathematical model uniquely characterizes tumor growth curve data. The Simeoni ODE model of tumor growth is not unique in that alternative models can provide equivalent representations of tumor growth.