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In this work, we examine a third-order nonlinear neutral integro-differential equation with constant delay. By building a Lyapunov functional, we obtain some sufficient criteria that ensure the asymptotic stability and boundedness of solutions for an analyzed equation. We present two examples to demonstrate the applicability of our conclusions, which extend and improve several well-known results in the literature. [ABSTRACT FROM AUTHOR]

Despite the availability of successful vaccines, measles outbreaks have occurred frequently in recent years, presumably due to the lack of proper vaccination implementation. Moreover, measles cases in adult groups, albeit small in number, indicate that the previously neglected adult group may need to be brought into vaccine coverage to achieve WHO's goal of measles eradication from the globe. In this study, we develop a novel transmission dynamics model to describe measles cases in adults and children to evaluate the role of adult infection in persistent measles cases and vaccination programs for eradication. Analysis of our model, validated by measles cases from outbreaks in Nepal, provides the vaccination reproduction number (conditions for measles eradication or persistence) and the role of contact network size. Our results highlight that while children are primary targets for measles outbreaks, a small number of infections in adults may act as a reservoir for measles, causing obstacles to eradication. Furthermore, our model analysis shows that while impactful controls can be achieved by children-focused vaccines, a combined adult-child vaccination program may help assert eradication of the disease.

In this article, we apply the Fourier transform to prove the Hyers-Ulam and Hyers-Ulam-Rassias stability for the first- and second-order nonlinear differential equations with initial conditions. Additionally, we extend the results to investigate the Mittag-Leffler-Hyers-Ulam and Mittag-Leffler-Hyers-Ulam-Rassias stability of these differential equations using the proposed method.

In this study, we are concerned with the existence and exponential stability issue of a delayed differential neoclassical growth model with discontinuous control strategy. By employing the Filippov’s theory and dichotomy theory, together with the Lyapunov functional method, novel criteria on existence and exponential stability are established for the addressed model. The established theoretical results extend and supplement the related results in the existing literature. Moreover, a simulation example is presented to verify the practicability of the proposed results.

This paper explains and applies a numerical technique utilizing the cubic B-spline functions and the mean value theorem (MVT) to solve a general time fractional partial differential equation (FPDE). The MVT for integrals enables us to approximate the time fractional derivatives in an appropriate simple form. We use the cubic B-spline functions to construct the numerical solution and its spatial derivatives. The great advantage of our technique is that it enables us to approximate solutions of many time FPDEs for several choices of F(x, t, u, ux, uxx). Two numerical examples have been included to emphasize the accuracy and efficiency of the method. It is demonstrated that the numerical method is unconditionally stable by employing the Von Neumann method (VNM).

In the present paper, for the first time, we consider the problem of non-fragile finite-time guaranteed cost control for a class of fractional-order singular systems with norm-bounded uncertainties. Firstly, we establish a linear quadratic cost function based on the integral of integer order, which serves practical purposes and acts as a performance measure index for the closed-loop system. Then, leveraging finite-time boundedness theory, inequality techniques, and fractional-order calculus properties, we propose a non-fragile state feedback controller that ensures regularity, impulse-free, and finite-time boundedness in the closed-loop system. Furthermore, we establish an upper limit for the performance index. The resulting findings are reliant on the order of the considered fractional-order systems and are obtained through the use of strict linear matrix inequalities (LMIs), devoid of any equality restrictions. Finally, two simulation examples are provided to validate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we study a nonlinear fractional p-Laplacian boundary value problem containing both left Riemann–Liouville and right Caputo fractional derivatives with initial and integral conditions. Some new results on the existence and uniqueness of a solution for the model are obtained as well as the Ulam stability of the solutions. Two examples are provided to show the applicability of our results. [ABSTRACT FROM AUTHOR]

In this work, we set up a new discrete predator–prey competitive model with time-varying delays and feedback controls. By virtue of the difference inequality knowledge, a sufficient condition which guarantees the permanence of the established discrete predator–prey competitive model with time-varying delays and feedback controls is derived. Under some appropriate parameter conditions, we have proved that the periodic solution of the system without delay exists and globally attractive. To verify the correctness of the derived theoretical fruits, we give two examples and execute computer simulations. Our obtained results are novel and complement previous known results. [ABSTRACT FROM AUTHOR]

In this paper, we present sufficient conditions to ensure the stochastic asymptotic stability of the zero solution for a specific type of fourth-order stochastic differential equation (SDE) with constant delay. By reducing the fourth-order SDE to a system of first-order SDEs, we utilize a fourth-order quadratic function to derive an appropriate Lyapunov functional. This functional is then employed to establish standard criteria for the nonlinear functions present in the SDE. The stability result obtained in this study is novel and extends the existing findings on stability in fourth-order differential equations. Additionally, we provide an illustrative example to demonstrate the significance and accuracy of our main result. [ABSTRACT FROM AUTHOR]

This article proves a stability theorem for the disease-free equilibrium of a stochastic differential equations model of malaria disease dynamics. The theorem is formulated in terms of an invariant which is similar to the basic reproduction number of a related deterministic model. Compared to the deterministic model, stability of the disease-free equilibrium holds more generally for the stochastic model. The optimal control theory is applied to the stochastic model, revealing some important new insights. Theoretical results are illustrated by way of simulations. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study a nonlinear fractional p-Laplacian boundary value problem containing both left Riemann–Liouville and right Caputo fractional derivatives with initial and integral conditions. Some new results on the existence and uniqueness of a solution for the model are obtained as well as the Ulam stability of the solutions. Two examples are provided to show the applicability of our results.

This research focuses on the stochastic functional differential equations driven by G-Brownian motion (G-SFDEs) with infinite delay. It is proved that the trivial solution of a G-SFDE with infinite delay is exponentially stable in mean square. An example is also presented to illustrate the effectiveness of the obtained theory.

In this work, we set up a new discrete predator–prey competitive model with time-varying delays and feedback controls. By virtue of the difference inequality knowledge, a sufficient condition which guarantees the permanence of the established discrete predator–prey competitive model with time-varying delays and feedback controls is derived. Under some appropriate parameter conditions, we have proved that the periodic solution of the system without delay exists and globally attractive. To verify the correctness of the derived theoretical fruits, we give two examples and execute computer simulations. Our obtained results are novel and complement previous known results.

This article considers the problems of uniform asymptotic stability and mixed H ∞ and passivity performance for conformable fractional-order uncertain neural networks with time delays. We first present a modified conformable fractional-order Razumikhin theorem for conformable fractional-order systems with time delays. Then, we derive some sufficient conditions in terms of linear matrix inequalities conditions to ensure that the considered system is uniform asymptotic stable with mixed H ∞ and passivity performance levels by employing the conformable fractional-order Razumikhin approach. Two simulation examples confirm the correctness of the theoretical results. [ABSTRACT FROM AUTHOR]

We discuss an analytical method for qualitative investigations of linear fractional delay differential equations. This method originates from the Lambert function technique that is traditionally used in stability analysis of ordinary delay differential equations. Contrary to the existing results based on such a technique, we show that the method can result into fully explicit stability criteria for a linear fractional delay differential equation, supported by a precise description of its asymptotics. As a by-product of our investigations, we also state alternate proofs of some classical assertions that are given in a more lucid form compared to the existing proofs. [ABSTRACT FROM AUTHOR]

In this article, we develop a continuous periodic switching model depicting Wolbachia infection frequency dynamics in mosquito populations by releasing Wolbachia-infected mosquitoes, which is different from the discrete modeling efforts in the literature. We obtain sufficient conditions on the existence of a unique and exactly two periodic solutions and analyze the stability of each periodic solution, respectively. We also provide a brief discussion and several numerical examples to illustrate our theoretical results.

The Laplace transform method is applied in this article to study the semi-Hyers-Ulam-Rassias stability of a Volterra integro-differential equation of order n, with convolution-type kernel. This kind of stability extends the original Hyers-Ulam stability whose study originated in 1940. A general integral equation is formulated first, and then some particular cases (polynomial function and exponential function) for the function from the kernel are considered.

In this paper, a class of nonlinear ψ-Hilfer fractional integrodifferential coupled systems on a bounded domain is investigated. The existence and uniqueness results for the coupled systems are proved based on the contraction mapping principle. Moreover, the Ulam–Hyers–Rassias, Ulam–Hyers, and semi-Ulam–Hyers–Rassias stabilities to the initial value problem are obtained. [ABSTRACT FROM AUTHOR]

We introduce the stability concepts for ψ -Caputo derivative systems in the sense of Lyapunov's idea. Two new Lyapunov theorems are formulated for the stability analysis of such systems. Many new inequalities that involve ψ -Caputo derivative are developed for the application of the proposed Lyapunov theorems. Examples are provided and it is shown that the proposed theorems are very useful for predicting the limiting behavior of the trajectory of such systems starting from any initial condition imposed at any initial time on the real axis. [ABSTRACT FROM AUTHOR]

This paper investigates the problem of finite-time passivity for a class of nonlinear Atangana–Baleanu–Caputo (ABC) fractional-order systems (FOs). Firstly, a new definition of finite-time passivity for the FO nonlinear systems in the frame of the Atangana–Baleanu derivative of Caputo type is introduced. Then several sufficient conditions in the form of linear matrix inequalities (LMIs) are presented to guarantee that such the system is robustly finite-time passive. Finally, two numerical examples are given to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with an epidemic model with quarantine and distributed time delay. We define the basic reproduction number R 0 and show that if R 0 ≤ 1 , then the disease-free equilibrium is globally asymptotically stable, whereas if R 0 > 1 , then it is unstable and there exists a unique endemic equilibrium. We obtain sufficient conditions for a Hopf bifurcation that induces a nontrivial periodic solution which represents recurrent epidemic waves. By numerical simulations, we illustrate stability and instability parameter regions. Our results suggest that the quarantine and time delay play important roles in the occurrence of recurrent epidemic waves. [ABSTRACT FROM AUTHOR]

This paper aims to analyse stability and Hopf bifurcation of the HIV-1 model with immune delay under the functional response of the Holling II type. The global stability analysis has been considered by Lyapunov–LaSalle theorem. And stability and the sufficient condition for the existence of Hopf Bifurcation of the infected equilibrium of the HIV-1 model with immune response are also studied. Some numerical simulations verify the above results. Finally, we propose a novel three dimension system to the future study. [ABSTRACT FROM AUTHOR]

This paper studies the stabilization to an inverted pendulum under a delayed proportional-derivative-acceleration (PDA) feedback, which can be used to understand human balance in quiet standing. The closed-loop system is described by a neutral delay differential equation (NDDE). The optimal feedback gains (OFGs) that make the exponential decaying rate maximized are determined when the characteristic equation of the closed-loop has a repeated real root with multiplicity 4. Such a property is called multiplicity-induced dominancy of time-delay systems, and has been discussed intensively by many authors for retarded delay differential equations (RDDEs). This paper shows that multiplicity-induced dominancy can be achieved in NDDEs. In addition, the OFGs are delay-dependent, and decrease sharply to small numbers correspondingly as the delay increases from zero and varies slowly with respect to moderate delays. Thus, the inverted pendulum can be well-stabilized with moderate delays and relatively small feedback gains. The result might be understandable that the elderly with obvious response delays can be well-stabilized with a delayed PDA feedback controller. [ABSTRACT FROM AUTHOR]

A class of coupled parabolic PDEs with time delay is considered. We treat the problem of boundary exponential stabilization where the goal is to present a state feedback law with actuation on only one end of the domain which provide the exponential stability of the closed-loop system. We consider both the Dirichlet and Neumann boundary conditions. The corresponding proposed control law is given in explicit form. Numerical simulations are carried out to show the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]

The paper studies absolute stability of neutral differential nonlinear systems

The financial system is a complex, nonlinear chaotic dynamic system caused by its operating mechanism. Therefore, the application of previous forecasting models cannot explain the existence of various interference factors in the financial market and the chaotic characteristics of the financial system. With the help of financial market stability, the article establishes a series of differential equation models that reflect changes in interest rates in the financial system. The article introduces the factor of macro-control on the premise of respecting market regulation to regulate and intervene in economic relations and economic operation status. We apply the Logistic model and stability theory to analyse the positive equilibrium point characteristics of the system and obtain the interest rate liquidity equation with a time-lag financial network.

In this paper, we establish the Shehu transform expression for homogeneous and non-homogeneous linear differential equations. With the help of this new integral transform, we solve higher order linear differential equations in the Shehu sense. Moreover, this paper introduced a new concept to find the Hyers-Ulam stability of the differential equation. This is the first attempt to use the Shehu transform to prove the Hyers-Ulam stability of the differential equation. Finally, the applications and remarks are discussed to demonstrate our strategy. Applications of the Shehu transform to fractional differential equations, Newton’s law of cooling, and free undamped motion are also discussed.

Let Bj(t) (j = 1,..., m) and B(t, τ) (t ≥ 0, 0 ≤ τ ≤ 1) be bounded variable operators in a Banach space. We consider the equation u′(t)=∑k=1mBk(t)u(t-hk(t))+∫01B(t,τ)u(t-h0(τ))dτ (t≥0),u'\left( t \right) = \sum\limits_{k = 1}^m {{B_k}\left( t \right)u\left( {t - {h_k}\left( t \right)} \right)} + \int\limits_0^1 {B\left( {t,\tau } \right)u\left( {t - {h_0}\left( \tau \right)} \right)d\tau \,\,\,\,\left( {t \ge 0} \right),} where hk(t) (t ≥ 0; k = 1, ..., m) and h0(τ) are continuous nonnegative bounded functions. Explicit delay-dependent exponential stability conditions for that equation are established. Applications to integro-differential equations with delay are also discussed

The aim of this paper is to analyse the potential impact of multiple current interventions in communities with limited resources in order to obtain optimal control strategies and provide a basis for future predictions of the most effective control measures against the spread of malaria. We developed a population-based model of malaria transmission dynamics to investigate the effectiveness of five different interventions. The model captured both the human and the mosquito compartments. The control interventions considered were: educational campaigns to mobilise people for diagnostic test and treatment and to sleep under bed nets; treatment through mass drug administration; indoor residual spraying(IRS) with insecticide to reduce malaria transmission; insecticide treated net (ITN) to reduce morbidity; and regular destruction of mosquito breeding sites to reduce the number of new mosquito and bites/contact at dusks and dawn. Analysis of the potential impact of the multiple control interventions were carried out and the optimal control strategies that minimized the number of infected human and mosquito and the cost of applying the various control interventions were determined.

This paper deals with the dissipativity analysis problem for a class of fractional-order static neural networks (FOSNNs). We first derive a novel sufficient condition for global asymptotic stability of FOSNNs by constructing novel convex Lyapunov functions and using linear matrix inequality techniques. Then, based on the proposed stable criterion combined with some auxiliary properties of fractional calculus, the dissipative problem for the related system is solved for the first time. In addition, we also extend the obtained results to generalized fractional-order neural networks. Four numerical examples are provided to show the validity and effectiveness of the proposed results. [ABSTRACT FROM AUTHOR]

Under the situation that the nonlinear term is of Carathéodory type and satisfies some weakly compactness condition, controllability outcomes are obtained for a class of multi-term time-fractional random integro-differential equations with state-dependent delay by using Mönch's fixed point theorem combined with a random fixed point theorem with stochastic domain, measure of noncompactness and (β , γ k) -resolvent family theory. An example is included to illustrate the theory. [ABSTRACT FROM AUTHOR]

In the present paper, we investigate the existence of positive periodic solutions for an n-species Lotka-Volterra system with distributed delays and nonlinear impulses. In the process, we convert the given system into an equivalent integral equation. Then, we construct appropriate mappings and use Krasnoselskii's fixed point theorem in a cone of a Banach space to show the existence of a positive periodic solution of the system. Easily verifiable sufficient conditions are established. We discuss our problem in two situations: when impulse functions are subquadratic and when they are sublinear. The technique to deal with the impulsive term is different from earlier approaches. In particular, the results improve some previous ones in the literature. Finally, an example is presented to illustrate the feasibility and effectiveness of the results. [ABSTRACT FROM AUTHOR]

In this manuscript, we first consider the diffusive competition and cooperation system subject to Neumann boundary conditions without delay terms and get the conclusion that the unique positive constant equilibrium is locally asymptotically stable. Then, we study the diffusive delayed competition and cooperation system subject to Neumann boundary conditions, and the existence of Hopf bifurcation at the positive equilibrium is obtained by regarding delay term as the parameter. By the theory of center manifold and normal form, an algorithm for determining the direction and stability of Hopf bifurcation is derived. Finally, some numerical simulations and summarizations are carried out for illustrating the theoretical analytic results.

This paper represents a non-integer-order Vallis systems in which we applied the Grünwald-Letnikov tactics with Binomial coefficients in order to realize the numerical simulations to a set of equations. Recently researchers reported in the literature that it is the generalization of integer order dynamical model. Several cases involving non-integer and integer analysis with different values of non-integer order have been applied to Vallis systems to see the behavior of simulations. To visualize the effect of non-integer order approach, the time histories and phase portraits have been plotted. The consequences expose that the non-integer-order Vallis model can reveal a genuine equitable comportment to Vallis systems and might bid greater perceptions towards the understanding of such complex dynamic systems.

We obtain asymptotic result for the solutions of neutral differential equations. Our technique depends on characteristic equations.

We propose two models inspired by the COVID-19 pandemic: a coupled disease-human behaviour (or disease-game theoretic), and a coupled disease-human behaviour-economic model, both of which account for the impact of social-distancing on disease control and economic growth. The models exhibit rich dynamical behaviour including multistable equilibria, a backward bifurcation, and sustained bounded periodic oscillations. Analyses of the first model suggests that the disease can be eliminated if everybody practices full social-distancing, but the most likely outcome is some level of disease coupled with some level of social-distancing. The same outcome is observed with the second model when the economy is weaker than the social norms to follow health directives. However, if the economy is stronger, it can support some level of social-distancing that can lead to disease elimination. [ABSTRACT FROM AUTHOR]

In this work, we establish some necessary results about existence theory to a class of boundary value problems (BVPs) of hybrid fractional differential equations (HFDEs) in the frame of Atangana–Baleanu–Caputo (ABC) fractional derivative. Making use of Krasnoselskii and Banach theorems, we obtain the required conditions. Some appropriate results of Hyers–Ulam (H–U) stability corresponding to the considered problem are also established. Also a pertinent example is given to demonstrate the results. [ABSTRACT FROM AUTHOR]

In this paper, we study a system of nonlinear Riemann–Liouville fractional differential equations with delays. First, we define in an appropriate way initial conditions which are deeply connected with the fractional derivative used. We introduce an appropriate generalization of practical stability which we call practical stability in time. Several sufficient conditions for practical stability in time are obtained using Lyapunov functions and the modified Razumikhin technique. Two types of derivatives of Lyapunov functions are used. Some examples are given to illustrate the introduced definitions and results. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to investigate various types of Ulam stability and Mittag-Leffler stability of linear differential equations of first order with constant coefficients using the Aboodh transform method. We also obtain the Hyers–Ulam stability constants of these differential equations using the Aboodh transform and some examples to illustrate our main results are given. [ABSTRACT FROM AUTHOR]

We present a deterministic SEIR model of the said form. The population in point can be considered as consisting of a local population together with a migrant subpopulation. The migrants come into the local population for a short stay. In particular, the model allows for a constant inflow of individuals into different classes and constant outflow of individuals from the R-class. The system of ordinary differential equations has positive solutions and the infected classes remain above specified threshold levels. The equilibrium points are shown to be asymptotically stable. The utility of the model is demonstrated by way of an application to measles. [ABSTRACT FROM AUTHOR]

Spectral abscissa (SA) is defined as the real part of the rightmost characteristic root(s) of a dynamical system, and it can be regarded as the decaying rate of the system, the smaller the better from the viewpoint of fast stabilization. Based on the Puiseux series expansion of complex-valued functions, this paper shows that the SA can be minimized within a given delay interval at values where the characteristic equation has repeated roots with multiplicity 2 or 3. Four sufficient conditions in terms of the partial derivatives of the characteristic function are established for testing whether the SA is minimized or not, and they can be tested directly and easily. [ABSTRACT FROM AUTHOR]