GLOBAL STABILITY OF SIRS EPIDEMIC MODELS WITH A CLASS OF NONLINEAR INCIDENCE RATES AND DISTRIBUTED DELAYS

In this article, we establish the global asymptotic stability of a disease-free equilibrium and an endemic equilibrium of an SIRS epidemic model with a class of nonlin- ear incidence rates and distributed delays. By using strict monotonicity of the incidence function and constructing a Lyapunov functional, we obtain sufficient conditions under which the endemic equilibrium is globally asymptotically stable. When the nonlinear inci- dence rate is a saturated incidence rate, our result provides a new global stability condition for a small rate of immunity loss.

DYNAMICS OF NEW CLASS OF HOPFIELD NEURAL NETWORKS WITH TIME-VARYING AND DISTRIBUTED DELAYS

In this paper, we investigate the dynamics and the global exponential stability of a new class of Hopfield neural network with time-varying and distributed delays. In fact, the properties of norms and the contraction principle are adjusted to ensure the existence as well as the uniqueness of the pseudo almost periodic solution, which is also its derivative pseudo almost periodic. This results are without resorting to the theory of exponential dichotomy. Furthermore, by employing the suitable Lyapunov function, some delayindependent sufficient conditions are derived for exponential convergence. The main originality lies in the fact that spaces considered in this paper generalize the notion of periodicity and almost periodicity. Lastly, two examples are given to demonstrate the validity of the proposed theoretical results.

Global stability of Cohen-Grossberg neural networks with time-varying and distributed delays

In this paper, the global asymptotic stability is investigated for a class of Cohen-Grossberg neural networks with time-varying and distributed delays. By using the Lyapunov-Krasovskii functional and equivalent descriptor form of the considered system, several delay-dependent sufficient conditions are obtained to guarantee the asymptotic stability of the addressed systems. These conditions are dependent on both time-varying and distributed delays and presented in terms of LMIs and therefore, the stability criteria of such systems can be checked readily by resorting to the Matlab LMI toolbox. Finally, an example is given to show the effectiveness and less conservatism of the proposed methods.

Delay-dependent stability of linear multistep methods for differential systems with distributed delays

This paper deals with the stability of linear multistep methods for multidimensional differential systems with distributed delays. The delay-dependent stability of linear multistep methods with compound quadrature rules is studied. Several new sufficient criteria of delay-dependent stability are obtained by means of the argument principle. An algorithm is provided to check delay-dependent stability. An example that illustrates the effectiveness of the derived theoretical results is given.

UNIFORM DECAY IN WEAKLY DISSIPATIVE TIMOSHENKO SYSTEM WITH INTERNAL DISTRIBUTED DELAY FEEDBACKS

In this paper we consider one-dimensional Timoshenko system with linear fric- tional damping and a distributed delay acting on the displacement equation. Under suitable assumptions on the weight of the delay and the wave speeds, we establish the well-posedness of the system and show that the dissipation through the frictional damping is strong enough to uniformly stabilize the system even in the presence of delay.

EXPONENTIAL STABILITY OF A MULTI-PARTICLE SYSTEM WITH LOCAL INTERACTION AND DISTRIBUTED DELAY

For a collective system,the connectedness of the adjacency matrix plays a key role in making the system achieve its emergent feature,such as flocking or multi-clustering.In this paper,we study a nonsymmetric multi-particle system with a constant and local cut-off weight.A distributed communication delay is also introduced into both the velocity adjoint term and the cut-off weight.As a new observation,we show that the desired multi-particle system undergoes both flocking and clustering behaviors when the eigenvalue 1 of the adjacency matrix is semi-simple.In this case,the adjacency matrix may lose the connectedness.In particular,the number of clusters is discussed by using subspace analysis.In terms of results,for both the non-critical and general neighbourhood situations,some criteria of flocking and clustering emergence with an exponential convergent rate are established by the standard matrix analysis for when the delay is free.As a distributed delay is involved,the corresponding criteria are also found,and these small time lags do not change the emergent properties qualitatively,but alter the final value in a nonlinear way.Consequently,some previous works[14]are extended.

The p-moment exponential robust stability for stochastic systems with distributed delays and interval parameters

The p-moment exponential robust stability for stochastic systems with distributed delays and interval parameters is studied. By constructing the Lyapunov- Krasovskii functional and employing the decomposition technique of interval matrix and Ito＇s formula, the delay-dependent criteria for the p-moment exponential robust stability are obtained. Numerical examples show the validity and practicality of the presented criteria.

MAXIMUM PRINCIPLE FOR STOCHASTIC OPTIMAL CONTROL PROBLEM WITH DISTRIBUTED DELAYS

This paper is concerned with a Pontryagin's maximum principle for the stochastic optimal control problem with distributed delays given by integrals of not necessarily linear functions of state or control variables.By virtue of the duality method and the generalized anticipated backward stochastic differential equations,we establish a necessary maximum principle and a sufficient verification theorem.In particular,we deal with the controlled stochastic system where the distributed delays enter both the state and the control.To explain the theoretical results,we apply them to a dynamic advertising problem.

Global Dynamics of a Stochastic Two-Prey One-Predator Model with <i>S</i>-Type Distributed Delays and Lévy Noises

In this paper, a stochastic two-prey one-predator model with S-type distributed time delays and Lévy noises is considered. Using the comparison theorem and Ito’s formula, sufficient conditions of persistence in the mean and extinct for each population are established. Then, conditions of global attractivity and stability in distribution by Barbalat’s conclusion are also obtained. Furthermore, Euler numerical simulation method is given to demonstrate our conclusions.

Improved delay-dependent robust H_∞ control of an uncertain stochastic system with interval time-varying and distributed delays

In this paper, the robust H∞control problem for a class of stochastic systems with interval time-varying and distributed delays is discussed. The system under study involves parameter uncertainty, stochastic disturbance, interval time-varying,and distributed delay. The aim is to design a delay-dependent robust H∞control which ensures the robust asymptotic stability of the given system and to express it in the form of linear matrix inequalities（LMIs）. Numerical examples are given to demonstrate the effectiveness of the proposed method. The results are also compared with the existing results to show its conservativeness.

ONE-PARAMETER FINITE DIFFERENCE METHODS AND THEIR ACCELERATED SCHEMES FOR SPACE-FRACTIONAL SINE-GORDON EQUATIONS WITH DISTRIBUTED DELAY

This paper deals with numerical methods for solving one-dimensional(1D)and twodimensional(2D)initial-boundary value problems(IBVPs)of space-fractional sine-Gordon equations(SGEs)with distributed delay.For 1D problems,we construct a kind of oneparameter finite difference(OPFD)method.It is shown that,under a suitable condition,the proposed method is convergent with second order accuracy both in time and space.In implementation,the preconditioned conjugate gradient(PCG)method with the Strang circulant preconditioner is carried out to improve the computational efficiency of the OPFD method.For 2D problems,we develop another kind of OPFD method.For such a method,two classes of accelerated schemes are suggested,one is alternative direction implicit(ADI)scheme and the other is ADI-PCG scheme.In particular,we prove that ADI scheme can arrive at second-order accuracy in time and space.With some numerical experiments,the computational effectiveness and accuracy of the methods are further verified.Moreover,for the suggested methods,a numerical comparison in computational efficiency is presented.

The Stability of Imitation Dynamics with Continuously Distributed Delays

This paper investigates imitation dynamics with continuously distributed delay.In realistic technological,economic,and social environments,individuals are involved in strategic interactions simultaneously while the influences of their decision-making may not be observable instantaneously.It shows that there exists a time delay effect.Different distributions of delay are further considered to efficiently lucubrate the stability of interior equilibrium in the imitation dynamics with continuous distributions of delay in the two-strategy game contexts.Precisely,when the delay follows the uniform distributions and Gamma distributions,the authors present that interior equilibrium can be asymptotically stable.Furthermore,when the probability density of the delay is general density,the authors also determine a sufficient condition for stability derived from the expected delay.Last but not least,the interested but uncomplicated Snowdrift game is utilized to demonstrate our theoretical results.

A class of anomalous diffusion epidemic models based on CTRW and distributed delay

In recent years,the epidemic model with anomalous diffusion has gained popularity in the literature.However,when introducing anomalous diffusion into epidemic models,they frequently lack physical explanation,in contrast to the traditional reaction-diffusion epidemic models.The point of this paper is to guarantee that anomalous diffusion systems on infectious disease spreading remain physically reasonable.Specifically,based on the continuous-time random walk(CTRW),starting from two stochastic processes of the waiting time and the step length,time-fractional space-fractional diffusion,timefractional reaction-diffusion and fractional-order diffusion can all be naturally introduced into the SIR(S:susceptible,I:infectious and R:recovered)epidemic models,respectively.The three models mentioned above can also be applied to create SIR epidemic models with generalized distributed time delays.Distributed time delay systems can also be reduced to existing models,such as the standard SIR model,the fractional infectivity model and others,within the proper bounds.Meanwhile,as an application of the above stochastic modeling method,the physical meaning of anomalous diffusion is also considered by taking the SEIR(E:exposed)epidemic model as an example.Similar methods can be used to build other types of epidemic models,including SIVRS(V:vaccine),SIQRS(Q:quarantined)and others.Finally,this paper describes the transmission of infectious disease in space using the real data of COVID-19.

Global Existence of Positive Periodic Solutions for a Distributed Delay Competition Model

By using the continuation theorem of Mawhin’s coincidence degree theory, a sufficient condition is derived for the existence of positive periodic solutions for a distributed delay competition model , where r 1 and r 2 are continuous ω-periodic functions in R + = [0,∞) with are positive continuous ω-periodic functions in R + = [0,∞), b i (i = 1, 2) is nonnegative continuous ω-periodic function in R + = [0,∞), ω and T are positive constants, and . Some known results are improved and extended.

Global stability of bidirectional associative memory neural networks with continuously distributed delays

Global asymptotic stability of the equilibrium point of bidirectional associative memory (BAM) neural networks with continuously distributed delays is studied. Under two mild assumptions on the activation functions, two sufficient conditions ensuring global stability of such networks are derived by utilizing Lyapunov functional and some inequality analysis technique. The results here extend some previous results. A numerical example is given showing the validity of our method.

Improved Stability Criteria on Discrete-time Systems with Time-varying and Distributed Delays

This work was supported by National Natural Science Foun- dation of China （Nos. 60905009, 61004032, 61104119, 61174076, and 61172135）, and Jiangsu Province Natural Science Foundation （Nos. SBK201240801 and BK2012384.）

Sliding mode control of uncertain systems with distributed time-delay: parameter-dependent Lyapunov functional approach

The robust stability and robust sliding mode control problems are studied for a class of linear distributed time-delay systems with polytopic-type uncertainties by applying the parameter-dependent Lyapunov functional approach combining with a new method of introducing some relaxation matrices and tuning parameters, which can be chosen properly to lead to a less conservative result. First, a sufficient condition is proposed for robust stability of the autonomic system; next, the sufficient conditions of the robust stabilization controller and the existence condition of sliding mode are developed. The results are given in terms of linear matrix inequalities （LMIs）, which can be solved via efficient interior-point algorithms. A numerical example is presented to illustrate the feasibility and advantages of the proposed design scheme.

Dynamics of stochastic predator-prey models with distributed delay and stage structure for prey

In this paper,we analyze two stochastic predator-prey models with distributed delay and stage structure for prey.For the nonautonomous periodic case of the model,by using Khasminskii’s theory of periodic solution,we show that the system has at least one positive T-periodic solution.For the model which is disturbed by both white and telegraph noises,we obtain sufficient criteria for positive recurrence of the solutions to the model by constructing a suitable stochastic Lyapunov function with regime switching.The positive recurrence implies that both prey and predator populations will be persistent in the long term.

New type of adaptive control for a class of distributed time-delay systems with adaptation regard to delay parameter

The problem on stabilization for the system with distributed delays is researched. The distributed time-delay under consideration is assumed to be a constant time-delay, but not known exactly. A design method is proposed for a memory proportional and integral （PI） feedback controller with adaptation to distributed time-delay. The feedback controller with memory simultaneously contains the current state and the past distributed information of the addressed systems. The design for adaptation law to distributed delay is very concise. The controller can be derived by solving a set of linear matrix inequalities （LMIs）. Two numerical examples are given to illustrate the effectiveness of the design method.

Controllability of nonlinear stochastic fractional systems with distributed delays in control

In this paper we study the controllability of linear and nonlinear stochastic fractional systems with bounded operator having distributed delay in control.The necessary and sufficient conditions for controllability of the linear system is obtained.Also,the nonlinear system is shown controllable under the assumption that the corresponding linear system is controllable and using the Banach contraction principle.