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In this paper we highlight a type of hyperbolic equation relating the logarithmic source term with distributed delay and dynamic boundary condition. We get, under comfortable primary data is the weak solution to local existence. The results of the solutions were found using the Faydo–Galerkin method and Schoder’s fixed point theorem. Then, the minimum blow-up result was studied. Our work is an extension of some previous work. [ABSTRACT FROM AUTHOR]

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

This paper dealt with a laminated beam system along with structural damping, past history, distributed delay, and in the presence of both temperatures and micro-temperatures effects. The damping terms left the system dissipative. Employing the semigroup approach, we established the existence and uniqueness of the solution. Additionally, with the help of convenient assumptions on the kernel, we demonstrated a general decay result for the solution of the considered system, with no constraints regarding the speeds of wave propagation. The main aim was to address how specific behaviors of the system were related to memory and delays. We aimed to investigate the joint impact of an infinite memory, distributed delay and micro-temperature effects on the system. We found a new relationship between the decay rate of solution and the growth of g at infinity. The objective was to find studies that use no-trivial results and their applications to relevant problems from mathematical physics. [ABSTRACT FROM AUTHOR]

In this paper, the existence of optimal solutions for problems governed by differential equations involving feedback controls is established for when the problem must account for a Volterra-type distributed delay and is subject to the action of impulsive external forces. The problem is reformulated within the class of impulsive semilinear integro-differential inclusions in Banach spaces and is studied by using topological methods and multivalued analysis. The paper concludes with an application to a population dynamics model. [ABSTRACT FROM AUTHOR]

This research investigates the stability and occurrence of Hopf bifurcation in a credit risk contagion model, which includes distributed delay, using the chain trick method. The model is a generalized version of those previously examined. The model is an expanded version of those previously studied. Comparative analysis showed that unlike earlier models, which only used the nonlinear resistance coefficient to determine the rate of credit risk infection, the credit risk contagion rate is also affected by the weight given to past behaviors of credit risk participants. Therefore, it is recommended to model the transmission of credit risk contagion using dispersed delays. [ABSTRACT FROM AUTHOR]

This article investigates the resilient-based consensus analysis of fractional-order nonlinear leader-following systems with distributed and input lags. To enhance the practicality of the controller design, an incorporation of a disturbance term is proposed. Our modeling framework provides a more precise and flexible approach that considers the memory and heredity aspects of agent dynamics through the utilization of fractional calculus. Furthermore, the leader and follower equations of the system incorporate nonlinear functions to explore the resulting changes. The leader-following system is expressed by a weighted graph, which can be either undirected or directed. Analyzed using algebraic graph theory and the fractional-order Razumikhin technique, the case of leader-following consensus is presented algebraically. To increase robustness in multi-agent systems, input and distributive delays are used to accommodate communication delays and replicate real-time varying environments. This study lays the groundwork for developing control methods that are more robust and flexible in complex networked systems. It does so by advancing our understanding and practical application of fractional-order multi-agent systems. Additionally, experiments were conducted to show the effectiveness of the design in achieving consensus within the system. [ABSTRACT FROM AUTHOR]

This note is a reaction on a recently published sufficient condition for oscillation of all solutions of a neutral delay differential equation. It is shown by a counterexample that the result is not correct and the problem is explained in details. Several, more general, classes of neutral differential equations with time-dependent discrete, distributed as well as mixed delays are considered. New sufficient conditions for oscillation of all their solutions are proved. Applications are given for illustration.

This paper dealt with a laminated beam system along with structural damping, past history, distributed delay, and in the presence of both temperatures and micro-temperatures effects. The damping terms left the system dissipative. Employing the semigroup approach, we established the existence and uniqueness of the solution. Additionally, with the help of convenient assumptions on the kernel, we demonstrated a general decay result for the solution of the considered system, with no constraints regarding the speeds of wave propagation. The main aim was to address how specific behaviors of the system were related to memory and delays. We aimed to investigate the joint impact of an infinite memory, distributed delay and micro-temperature effects on the system. We found a new relationship between the decay rate of solution and the growth of g at infinity. The objective was to find studies that use no- trivial results and their applications to relevant problems from mathematical physics.

This study focuses on investigating a problem involving a coupled nonlocal singular viscoelastic equation with sources and distributed delay terms. We aim to demonstrate, under appropriate conditions, the exponential growth of solutions with positive initial-energy. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we investigate a scenario concerning a coupled nonlocal singular viscoelastic equation with sources and distributed delay terms. By establishing suitable conditions, we have proved that a finite-time blow-up occurs in the solution.

Despite the fact that the theory of stability of continuous-time difference equations has a long history, is well developed and very popular in research, there is a simple and clearly formulated problem about the stability of stochastic difference equations with continuous time and distributed delay, which has not been solved for more than 13 years. This paper offers to readers some generalization on this unsolved problem in the hope that it will help move closer to its solution.

In this paper, we investigate a scenario concerning a coupled nonlocal singular viscoelastic equation with sources and distributed delay terms. By establishing suitable conditions, we have proved that a finite-time blow-up occurs in the solution. [ABSTRACT FROM AUTHOR]

In this paper, the stochastic Takagi–Sugeno fuzzy recurrent neural networks (STSFRNNS) with distributed delay is established based on the Takagi–Sugeno (TS) model and the fixed time synchronization problem is investigated. In order to synchronize the networks, we design two kinds of controllers: a feedback controller and an adaptive controller. Then, we obtain the synchronization criteria in a fixed time by combining the Lyapunov method and the related inequality theory of the stochastic differential equation and calculate the stabilization time for the STSFRNNS. In addition, to verify the authenticity of the theoretical results, we use MATLABR2023A to carry out numerical simulation. [ABSTRACT FROM AUTHOR]

In this paper, a two-neuron reaction–diffusion neural network with discrete and distributed delays is proposed, and the state feedback control strategy is adopted to achieve control of its spatiotemporal dynamical behaviours. Adding two virtual neurons, the original system is transformed into a neural network only containing the discrete delay. The conditions under which Hopf bifurcation and Turing instability arise are determined through analysis of the characteristic equation. Additionally, the amplitude equations are derived with the aid of weakly nonlinear analysis, and the selection of the Turing patterns is determined. The simulation results demonstrate that the state feedback controller can delay the onset of Hopf bifurcation and suppress the generation of Turing patterns. [ABSTRACT FROM AUTHOR]

In this article, we consider a swelling porous-heat system with viscous damping and distributed delay term. It's well-known in the literature that the thermal effect only lacks exponential stability. For that, we need to add another damping mechanism to stabilize exponentially the system. However, we prove, based on the energy method, that the combination of viscous damping and thermal effect provokes an exponential stability in the presence of distributed delay irrespective of the wave speeds of the system. [ABSTRACT FROM AUTHOR]

In this paper, we consider one-dimensional nonlinear piezoelectric beams with thermal and magnetic effects in the presence of a distributed delay term acting on the heat equation. First, we show that the system is well-posed in the sense of a semigroup. Through the construction of an appropriate Lyapunov functional, we establish a general decay result for the solutions of the system, for which the exponential and polynomial decays are only special cases, under a suitable assumption on the weight of the delay that the damping effect through heat conduction is strong enough to stabilize the system even in the presence of a time delay. Furthermore, our result does not depend on any relationship between system parameters. [ABSTRACT FROM AUTHOR]

A new adaptive observer is proposed for a class of uncertain time-delay parabolic systems. The uncertainties are located in both the state equation and the output equation which are also subjected to a distributed delay. Our algorithm is based on backstepping-like transformation and a classical persistent excitation condition. To illustrate the performances of this observer, simulation results are provided. [ABSTRACT FROM AUTHOR]

In this paper, a reaction-diffusion population model with Dirichlet boundary condition and a directed movement oriented by a temporally distributed delay is proposed to describe the lasting memory of animals moving over space. The temporal kernel of the memory is taken as Gamma distribution function, among which there are two biologically meaningful cases: one is the weak kernel which implies that animals can immediately acquire knowledge and memory decays over time, the other is the strong kernel by which we assume that animals' memory undergoes learning and memory decay stages. It is shown that the population stabilizes to a positive steady state and aggregates in the interior of the territory when the delay kernel is the weak type. In the strong kernel case, oscillatory patterns can first arise via a Hopf bifurcation with a small memory delay and then vanish when the system undergoes a Hopf bifurcation with a large memory delay, which implies that stability switch occurs and spatial-temporal patterns emerge for intermediate value of delays. [ABSTRACT FROM AUTHOR]

The paper studies a class of nonlinear disturbed neutral linear fractional systems with derivatives in the the Riemann–Liouville sense and distributed delays. First, it is proved that the initial problem for these systems with discontinuous initial functions under some natural assumptions possesses a unique solution. The assumptions used for the proof are similar to those used in the case of systems with first-order derivatives. Then, with the obtained result, we derive the existence and uniqueness of a fundamental matrix and a generalized fundamental matrix for the homogeneous system. In the linear case, via these fundamental matrices we obtain integral representations of the solutions of the homogeneous system and the corresponding inhomogeneous system. Furthermore, for the fractional systems with Riemann–Liouville derivatives we introduce a new concept for weighted stabilities in the Lyapunov, Ulam–Hyers, and Ulam–Hyers–Rassias senses, which coincides with the classical stability concepts for the cases of integer-order or Caputo-type derivatives. It is proved that the zero solution of the homogeneous system is weighted stable if and only if all its solutions are weighted bounded. In addition, for the homogeneous system it is established that the weighted stability in the Lyapunov and Ulam–Hyers senses are equivalent if and only if the inequality appearing in the Ulam–Hyers definition possess only bounded solutions. Finally, we derive natural sufficient conditions under which the property of weighted global asymptotic stability of the zero solution of the homogeneous system is preserved under nonlinear disturbances. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and multiplicity of periodic solutions to a class of second-order nonlinear differential equations with distributed delay. Under assumptions that the nonlinearity is odd, differentiable at zero and satisfies the Nagumo condition, by applying the equivariant degree method, we prove that the delay equation admits multiple periodic solutions. The results are then illustrated by an example. [ABSTRACT FROM AUTHOR]

In this study, we address a Cauchy problem within the context of the one-dimensional Timoshenko system, incorporating a distributed delay term. The heat conduction aspect is described by the Lord-Shulman theory. Our demonstration establishes that the dissipation resulting from the coupling of the Timoshenko system with Lord-Shulman’s heat conduction is sufficiently robust to stabilize the system, albeit with a gradual decay rate. To support our findings, we convert the system into a first-order form and, utilizing the energy method in Fourier space, and derive point wise estimates for the Fourier transform of the solution. These estimates, in turn, provide evidence for the slow decay of the solution.

A stochastic Microcystins degradation model with distributed delay is studied in this paper. We first demonstrate the existence and uniqueness of a global positive solution to the stochastic system. Second, we derive a stochastic critical value $ R_0^s $ related to the basic reproduction number $ R_0 $. By constructing suitable Lyapunov function types, we obtain the existence of an ergodic stationary distribution of the stochastic system if $ R_0^s > 1. $ Next, by means of the method developed to solve the general four-dimensional Fokker-Planck equation, the exact expression of the probability density function of the stochastic model around the quasi-endemic equilibrium is derived, which is the key aim of the present paper. In the analysis of statistical significance, the explicit density function can reflect all dynamical properties of a chemostat model. To validate our theoretical conclusions, we present examples and numerical simulations.

In this paper, a periodic stochastic Lotka‐Volterra model with distributed delay and dispersal is investigated. First, by Itô's formula, we show that the solution with any positive initial value is global and positive. Then, we obtain the sufficient conditions guaranteeing the extinction of the system. Furthermore, we discuss the existence of the nontrivial periodic solution by Khasminskii theory. Finally, numerical examples are given to illustrate the results. [ABSTRACT FROM AUTHOR]

In this paper, two prey–predator models with distributed delays are presented based on the growth and loss rates of the predator, which are much smaller than that of the prey, leading to a singular perturbation problem. It is obtained that Hopf bifurcation can occur, where the coexistence equilibrium becomes unstable leading to a stable limit cycle. Subsequently, considering the perturbation parameter 0 < ≪ 1 , the fact that the solution crossing the transcritical point converges to a stable equilibrium is discussed for the model with Holling type I using the linear chain criterion, center-manifold reduction, the geometric singular perturbation theory and entry–exit function. The existence and uniqueness of relaxation oscillation cycle for the model with Holling type II are obtained. In addition, numerical simulations are provided to verify the analytical results. [ABSTRACT FROM AUTHOR]

In this work, we consider the one-dimensional thermoelastic Bresse system by addressing the aspects of nonlinear damping and distributed delay term acting on the first and the second equations. We prove a stability result without the common assumption regarding wave speeds under Neumann boundary conditions. We discover a new relationship between the decay rate of the solution and the growth of ϖ at infinity. Our results were achieved using the multiplier method and the perturbed modified energy, named Lyapunov functions together with some properties of convex functions. [ABSTRACT FROM AUTHOR]

In this work, we study a general class of retarded linear systems with distributed delays and variable-order fractional derivatives of Caputo type. We propose an approach consisting of finding an associated one-parameter family of constant-order fractional systems, which is "almost" equivalent to the considered variable-order system in an appropriate sense. This approach allows us to replace the study of the initial problem (IP) for variable-order fractional systems with the study of an IP for these one-parameter families of constant-order fractional systems. We prove that the initial problem for the variable-order fractional system with a discontinuous initial function possesses a unique continuous solution on the half-axis when the function describing the variable order of differentiation is locally bounded, Lebesgue integrable and has an appropriate decomposition similar to the Lebesgue decomposition of functions with bounded variation. The obtained results lead to the existence and uniqueness of a fundamental matrix for the studied variable-order fractional homogeneous system. As an application of the obtained results, we establish an integral representation of the solutions of the studied IP. [ABSTRACT FROM AUTHOR]

As the evolution of species relies on not only the current state but also the past information, it is more reasonable and realistic to take delay into an ecological model. This paper deals with a stochastic predator–prey model that considers the distribution delay and assume that the intrinsic growth rate and the death rate in the model are governed by Ornstein–Uhlenbeck process to simulate the random factors in the environment. Based on the existence and uniqueness of the global solution to the model and the boundedness of the p$$ p $$ order moments of the solution, several conditions are established to analyze the survival of the species. Specifically, a criteria for the existence of the stationary distribution to the stochastic system is established by constructing some suitable Lyapunov functions. And the analytical expression of the probability density function of the model around the quasi‐equilibrium is obtained. Furthermore, the extinction of species in the model is also explored. Finally, numerical simulations are carried out to illustrate the theoretical results obtained in this paper. [ABSTRACT FROM AUTHOR]

The present article is devoted to discuss the pth moment exponential ultimate boundedness for a class of stochastic distributed delay-coupled systems on networks driven by G-Brownian motion. By combining the method of proof by contradiction with Lyapunov method and graph theory, some sufficient condition for both the Lyapunov-type theorem and the coefficients-type theorem are proposed to ensure the pth moment exponential ultimate boundedness of the considered systems. As applications, the boundedness criteria are applied to stochastic distributed delay-coupled neural networks and stochastic distributed delay-coupled control systems in the framework of G-Brownian motion. [ABSTRACT FROM AUTHOR]

This paper aims to analyze the dynamic behavior of a novel stochastic epidemic model (SEM) that incorporates distributed delay and time switching. We prove the existence and uniqueness of a global positive solution and establish the presence of ergodic stationary distribution (ESD) through the construction of appropriate Lyapunov functions. The threshold R 0 S derived from our analysis plays a vital role in this procedure. Additionally, we conduct computer simulations to validate our theoretical discoveries, demonstrating that the incorporation of white noise can induce random fluctuations in the system variables under time switching. [ABSTRACT FROM AUTHOR]

In this work, we consider a one-dimensional laminated Timoshenko beam system with thermoelasticity of type III and distributed delay. Our concern is to investigate the exponential stability of the vibrations in the system without structural damping. By exploiting the perturbed energy method with appropriate assumptions on the delay feedback and speeds of wave propagation, we establish that the unique dissipation through the thermal effect is sufficient for exponential decay of the solution even in the presence of distributed delay. [ABSTRACT FROM AUTHOR]

Inter- and intraspecific competition is most important during the immature life stage for many species of interest, such as multiple coexisting mosquito species that act as vectors of diseases. Mortality caused by competition that occurs during maturation is explicitly modelled in some alternative formulations of the Lotka–Volterra competition model. We generalise this approach by using a distributed delay for maturation time. The kernel of the distributed delay is represented by a truncated Erlang distribution. The shape and rate of the distribution, as well as the position of the truncation, are found to determine the solution at equilibrium. The resulting system of delay differential equations is transformed into a system of ordinary differential equations using the linear chain approximation. Numerical solutions are provided to demonstrate cases where competitive exclusion and coexistence occur. Stability conditions are determined using the nullclines method and local stability analysis. The introduction of a distributed delay promotes coexistence and survival of the species compared to the limiting case of a discrete delay, potentially affecting management of relevant pests and threatened species. [ABSTRACT FROM AUTHOR]

ARS-CoV-2 can survive in different environments and remain infectious for several days, which presents challenges to eliminating infectious diseases. It encourages researchers to study the effects of SARS-CoV-2 on the environment. In this paper, we formulate an epidemic model for SARS-CoV-2, which focuses on the transmission of the virus under environmental conditions. Two distributed delays are introduced to describe the probability of the exposed and infected individuals in different infection periods based on the transmission of the virus in the environment. The positivity and boundness of solutions of model are derived. The basic reproduction number threshold theory is established and the results demonstrate that the persistence of COVID-19 depends on the basic reproduction number. Numerical simulations are presented to verify the theoretical results. Some measures are proposed to control and eliminate COVID-19 infectious diseases. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Jilin University (Science Edition) / Jilin Daxue Xuebao (Lixue Ban) is the property of Zhongguo Xue shu qi Kan (Guang Pan Ban) Dian zi Za zhi She and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We study the Benjamin-Bona-Mahony model with finite distributed delay in 3D, which depicts the dispersive impact of long waves. Based on the well-posedness of model, the family of pullback attractors for the evolutionary processes generated by a global weak solution has been obtained, which is unique and minimal, via verifying asymptotic compactness in functional space with delay $ C_V $ and topological space $ V\times C_V $, where the energy equation method and a retarded Gronwall inequality are utilized.

In this paper, the existence of optimal solutions for problems governed by differential equations involving feedback controls is established for when the problem must account for a Volterra-type distributed delay and is subject to the action of impulsive external forces. The problem is reformulated within the class of impulsive semilinear integro-differential inclusions in Banach spaces and is studied by using topological methods and multivalued analysis. The paper concludes with an application to a population dynamics model.

This research investigates the stability and occurrence of Hopf bifurcation in a credit risk contagion model, which includes distributed delay, using the chain trick method. The model is a generalized version of those previously examined. The model is an expanded version of those previously studied. Comparative analysis showed that unlike earlier models, which only used the nonlinear resistance coefficient to determine the rate of credit risk infection, the credit risk contagion rate is also affected by the weight given to past behaviors of credit risk participants. Therefore, it is recommended to model the transmission of credit risk contagion using dispersed delays.

This article investigates the resilient-based consensus analysis of fractional-order nonlinear leader-following systems with distributed and input lags. To enhance the practicality of the controller design, an incorporation of a disturbance term is proposed. Our modeling framework provides a more precise and flexible approach that considers the memory and heredity aspects of agent dynamics through the utilization of fractional calculus. Furthermore, the leader and follower equations of the system incorporate nonlinear functions to explore the resulting changes. The leader-following system is expressed by a weighted graph, which can be either undirected or directed. Analyzed using algebraic graph theory and the fractional-order Razumikhin technique, the case of leader-following consensus is presented algebraically. To increase robustness in multi-agent systems, input and distributive delays are used to accommodate communication delays and replicate real-time varying environments. This study lays the groundwork for developing control methods that are more robust and flexible in complex networked systems. It does so by advancing our understanding and practical application of fractional-order multi-agent systems. Additionally, experiments were conducted to show the effectiveness of the design in achieving consensus within the system.

In this study, considering the delays for a susceptible individual becoming an alcoholic and the relapse of a recovered individual back into being an alcoholic, we formulate an epidemic model for alcoholism with distributed delays and relapse. The basic reproduction number R 0 is calculated, and the threshold property of R 0 is established. By analyzing the characteristic equation, we demonstrate the local asymptotic stability of the different equilibria under various conditions: when R 0 < 1 , the alcoholism-free equilibrium is locally asymptotically stable; when R 0 > 1 , the alcoholism equilibrium exists and is locally asymptotically stable. Furthermore, we demonstrate the global asymptotic stability at each equilibrium using a suitable Lyapunov function. Specifically, when R 0 < 1 , the alcoholism-free equilibrium is globally asymptotically stable; when R 0 > 1 , the alcoholism equilibrium is globally asymptotically stable. The sensitivity analysis of R 0 shows that reducing exposure is more effective than treatment in controlling alcoholism. Interestingly, we found that extending the latency delay h 1 and relapse delay h 2 also effectively contribute to the control of the spread of alcoholism. Numerical simulations are also provided to support our theoretical results. [ABSTRACT FROM AUTHOR]

The paper discusses adaptive synchronization for inertial Cohen-Grossberg neural networks (ICGNNs) with distributed delays. The ICGNN is changed into a first-order differential equation model by making use of a variable substitution method. A criterion is proposed to guarantee the adaptive synchronization of the drive-response ICGNNs by means of an appropriate Lyapunov functional and the LaSalle invariant principle. A concrete example is offered to illustrate the validity of the obtained results. [ABSTRACT FROM AUTHOR]

In the present work, we consider a nonlinear Klein–Gordon equation with strong damping, distributed delay and source terms. Under suitable conditions, we prove the blow-up result of solutions. Our result is an extension of many other works in this area. [ABSTRACT FROM AUTHOR]

In this paper, a viscoelastic equation with nonlinear distributed delay and nonlinear damping effects is considered. Under very general conditions that do not involve differential inequalities for the kernel function, a general decay criterion is established through the perturbed energy method and properties of convex functions. We extend the stability result of wave equation with linear delay and linear damping to the nonlinear case. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of periodic solutions to the following delay differential system: { x 1 ′ (t) = ∫ − 1 0 f 1 (x 2 (t + θ)) d θ , x 2 ′ (t) = ∫ − 1 0 f 2 (x 1 (t + θ)) d θ. By transforming the problem of searching for periodic solutions of the above system to the problem of finding periodic solutions of an associated ordinary differential system with boundary value conditions and using pseudoindex theory, some sufficient conditions are obtained to guarantee the existence of multiple W -symmetric 2-periodic solutions. We also present three specific examples to illustrate our results. [ABSTRACT FROM AUTHOR]

Of concern is the Hopfield neural network system comprising discrete as well as distributed delays in the form of a convolution. For a desired convergence rate of the solution to the equilibrium state, we establish sufficient conditions on the delay kernels ensuring this matter. Our result improves an existing one in the literature. The adopted approach is completely different. It relies on a judicious choice of a Lyapunov-like function and careful manipulations. [ABSTRACT FROM AUTHOR]

We consider a system of transmission of the wave equation with Neumann feedback control that contains a distributed delay term and that acts on the exterior boundary. We prove under some assumptions that the solutions decay exponentially in an appropriate energy space. To establish this result, we introduce a suitable energy function and use multipliers technique method and compactness-uniqueness argument. [ABSTRACT FROM AUTHOR]

We study several stability properties on a finite or infinite interval of inhomogeneous linear neutral fractional systems with distributed delays and Caputo-type derivatives. First, a continuous dependence of the solutions of the corresponding initial problem on the initial functions is established. Then, with the obtained result, we apply our approach based on the integral representation of the solutions instead on some fixed-point theorems and derive sufficient conditions for Hyers–Ulam and Hyers–Ulam–Rassias stability of the investigated systems. A number of connections between each of the Hyers–Ulam, Hyers–Ulam–Rassias, and finite-time Lyapunov stability and the continuous dependence of the solutions on the initial functions are established. Some results for stability of the corresponding nonlinear perturbed homogeneous fractional linear neutral systems are obtained, too. [ABSTRACT FROM AUTHOR]

In this paper, we construct a general four-dimensional delayed HIV infection model with virus-to-cell infection, cell-to-cell transmission, CTL immune response and parameter perturbations. By substitution, the four-dimensional delayed stochastic differential equations can be transformed into a degenerate eight-dimensional stochastic differential equations. The existence of the global positive solution of the system is obtained rigorously. By constructing appropriate Lyapunov functions, the existence of a stationary Markov process is derived when the stochastic reproduction number R 0 s is greater than one. Finally, we investigate the effects of noise level and the cell-to-cell transmission on the dynamic behavior of the model. Our model is a general model of the existing stochastic virus infection model, and the theoretical results improve and generalize the existing conclusion. [ABSTRACT FROM AUTHOR]

In this paper, synchronization control of a class of neutral type stochastic neural networks with discrete and distributed delays is studied. Firstly, we establish a novel neutral type neural network synchronization error model by employing an improved delayed feedback controller instead of the traditional one. Moreover, despite of the discrete time-delay, the distributed time-delay, the external disturbance and the uncertain parameters are involved in the design of our synchronization model. Secondly, with the aid of proper Lyapunov–Krasovskii functional (LKF), some sufficient conditions for stochastic stability of synchronization error system are derived and the gain of the controller is obtained by using linear matrix inequality (LMI) and proper integral term estimation methods. Finally, we use Euler–Maruyama numerical mechanism to simulate the drive-response systems and the error system, which shows the good convergence performance of our synchronization model. A numerical example in the end of the paper shows the effectiveness of our proposed method. [ABSTRACT FROM AUTHOR]

This work examines a stochastic viral infection model with a general distributed delay. We transform the model with weak kernel case into an equivalent system through the linear chain technique. First, we establish that a global positive solution to the stochastic system exists and is unique. We establish the existence of a stationary distribution of a positive solution under the stochastic condition $ R^s > 0 $, also referred to as a stationary solution, by building appropriate Lyapunov functions. Finally, numerical simulation is proved to verify our analytical result and reveals the impact of stochastic perturbations on disease transmission.