Dynamical behavior for an eco-epidemiological model with discrete and distributed delay

In this paper, the dynamical behavior of an eco-epidemiological model with discrete and distributed delay is studied. Sufficient conditions for the local asymptotical stability of the nonnegative equilibria are obtained. We prove that there exists a threshold value of the feedback time delay τ beyond which the positive equilibrium bifurcates towards a periodic solution. Using the normal form theory and center manifold argument, the explicit formulae which determine the stability, the direction and the periodic of bifurcating period solutions are derived. Numerical simulations are carried out to explain the mathematical conclusions.     

Stability and Hopf bifurcation analysis of an eco-epidemic model with a stage structure

In this paper, an eco-epidemiological model with a stage structure is considered. The asymptotical stability of the five equilibria, the existence of stability switches about positive equilibrium, is investigated. It is found that Hopf bifurcation occurs when the delay τ passes though a critical value. Some explicit formulae determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using the normal form theory and center manifold theory. Some numerical simulations for justifying the theoretical analysis are also provided. Finally, biological explanations and main conclusions are given.     

Stability in a diffusive food-chain model with Michaelis–Menten functional response

This paper deals with the behavior of positive solutions to a reaction–diffusion system with homogeneous Neumann boundary conditions describing a three species food chain. A sufficient condition for the local asymptotical stability is given by linearization and also a sufficient condition for the global asymptotical stability is given by a Lyapunov function. Our result shows that the equilibrium solution is globally asymptotically stable if the net birth rate of the first species is big enough and the net death rate of the third species is neither too big nor too small.     

具有常数输入的SEIR模型的稳定性分析

讨论了易感者类和潜伏者类均为常数输入,潜伏期、染病期和恢复期均具有传染力,且传染率为一般传染率的SEIR传染病模型.利用Hurwitz判据证明了地方病平衡点的局部渐近稳定性,进一步利用复合矩阵理论得到了地方病平衡点全局渐近稳定的充分条件.     

Asymptotic behavior of an SEI epidemic model with diffusion

An SEI epidemic model with constant recruitment and infectious force in the latent period is investigated. This model describes the transmission of diseases such as SARS. The behavior of positive solutions to a reaction–diffusion system with homogeneous Neumann boundary conditions are investigated. Sufficient conditions for the local and global asymptotical stability are given by linearization and by the method of upper and lower solutions and its associated monotone iterations. Our result shows that the disease-free equilibrium is globally asymptotically stable if the contact rate is small.     

The dynamic behavior of deterministic and stochastic delayed SIQS model

In this paper, we present the deterministic and stochastic delayed SIQS epidemic models. For the deterministic model, the basic reproductive number $R_{0}$ is given. Moreover, when $R_{0}<1$, the disease-free equilibrium is globally asymptotical stable. When $R_{0}>1$ and additional conditions hold, the endemic equilibrium is globally asymptotical stable. For the stochastic model, a sharp threshold $\overset{\wedge }{R}_{0}$ which determines the extinction or persistence in the mean of the disease is presented. Sufficient conditions for extinction and persistence in the mean of the epidemic are established. Numerical simulations are also conducted in the analytic results.     

Stability properties and Hopf bifurcation of a delayed viral infection model with lytic immune response

A class of more general delayed viral infection model with lytic immune response is proposed based on some important biological meanings. The effect of time delay on stabilities of the equilibria is given. The sufficient criteria for local and global asymptotic stabilities of the viral free equilibrium and the local asymptotic stabilities of the no-immune response equilibrium are given. We also get the sufficient criteria for stability switch of the positive equilibrium. Numerical simulations are carried out to explain the mathematical conclusions.     

Dynamical Analysis of a Delayed SIQS Epidemic Model on Scale-free Networks

In this paper, we establish a novel delayed SIQS epidemic model on scale-free networks, where time delay represents the average quarantine period. Through mathematical analysis, we present the basic reproduction number $R_{0}$. Then, we provide the global asymptotical stability of the disease-free equilibrium and the local asymptotical stability of the endemic equilibrium. Finally, we perform numerical simulations to verify the correctness of the main results and analyze the sensitivity of parameters. Our research shows that when $R_0>1$, lengthening the quarantine period can slow the spread of the disease and reduce the number of infected individuals.     

近岸海域赤潮藻非线性生态模型的建立和动力学分析

基于海洋浮游生态系统特性,以主要赤潮藻-中肋骨条藻为研究对象,建立了中肋骨条藻生物量随氮磷营养盐浓度变化的微分方程,并考虑了养料循环和时滞的影响,利用非线性动力学方法得到了边界平衡点的全局渐近稳定性,正平衡点的局部稳定性和Hopf分支的存在性,以及系统持久的充分条件.     

一类带有一般接触率和常数输入的流行病模型的全局分析

借助极限系统理论和构造适当的Liapunov函数,对带有一般接触率和常数输入的SIR型和SIRS型传染病模型进行讨论．当无染病者输入时,地方病平衡点存在的阈值被找到A·D2对相应的SIR模型,关于无病平衡点和地方病平衡点的全局渐近稳定性均得到充要条件;对相应的SIRS模型,得到无病平衡点和地方病平衡点全局渐近稳定的充分条件．当有染病者输入时,模型不存在无病平衡点．对相应的SIR模型,地方病平衡点是全局渐近稳定的;对相应的SIRS模型,得到地方病平衡点全局渐近稳定的充分条件．     

Numerical modelling of an SIR epidemic model with diffusion

A spatial SIR reaction-diffusion model for the transmission disease such as whooping cough is studied. The behaviour of positive solutions to a reaction-diffusion system with homogeneous Neumann boundary conditions are investigated. Sufficient conditions for the local and global asymptotical stability are given by linearization and by using Lyapunov functional. Our result shows that the disease-free equilibrium is globally asymptotically stable if the contact rate is small. These results are verified numerically by constructing, and then simulating, a robust implicit finite-difference method. Furthermore, the new implicit finite-difference method will be seen to be more competitive (in terms of numerical stability) than the standard finite-difference method.     

Dynamics of a Diffusive SIR Epidemic Model with Time Delay

This paper is devoted to a reaction-diffusion system for a SIR epidemic model with time delay and incidence rate. Firstly, the nonnegativity and boundedness of solutions determined by nonnegative initial values are obtained. Secondly, the existence and local stability of the disease-free equilibrium as well as the endemic equilibrium are investigated by analyzing the characteristic equations. Finally, the global asymptotical stability are obtained via Lyapunov functionals.     

A planktonic resource-consumer model with a temporal delay in nutrient recycling

A planktonic resource-consumer model is investigated analytically as well as numerically. A feedback loop and a temporal delay are considered. The local asymptotical stability of the non-trivial equilibrium is discussed. Using Lyapunov functional technique, the global stability of the non-trivial equilibrium is also discussed for a linear uptake term. In conclusion, we show that the introduction of a temporal delay in nutrient recycling can destabilize the system and periodic solution can arise. Finally, numerical simulations are performed to illustrate the dynamical behaviors.     

具扩散的生物控制模型的渐进性

讨论了具有空间扩散的生物控制系统在齐次Neumann边界条件下解的整体性态.通过线性化方法给出了局部渐近稳定条件,通过构造Lyapunov函数给出了全局稳定性的证明.     

Stability and Hopf bifurcation of a delayed viral infection model with logistic growth and saturated immune impairment

In this paper, the stability and Hopf bifurcation of a delayed viral infection model with logistic growth and saturated immune impairment is studied. It is shown that there exist 3 equilibria. The sufficient conditions for local asymptotic stability of the infection‐free equilibrium and no‐immune equilibrium are given. We also discussed the local stability of positive equilibrium and the existence of Hopf bifurcation. Moreover, the direction and stability of Hopf bifurcation is obtained by using standard form theory and the center manifold theorem. Finally, numerical simulations are performed to verify the theoretical conclusions.     

Avian-human influenza epidemic model with diffusion

A diffusive epidemic model is investigated. This model describes the transmission of avian influenza among birds and humans. The behavior of positive solutions to a reaction-diffusion system with homogeneous Neumann boundary conditions are investigated. Sufficient conditions for the local and global asymptotical stability are given by spectral analysis and by using Lyapunov functional. Our result shows that the disease-free equilibrium is globally asymptotically stable, if the contact rate for the susceptible birds and the contact rate for the susceptible humans are small. It suggests that the best policy to prevent the occurrence of a pandemic is not only to exterminate the infected birds with avian influenza but also to reduce the contact rate for susceptible humans with the individuals infected with mutant avian influenza. Numerical simulations are presented to illustrate the main results.     

THE STABILITY IN NEURAL NETWORK WITH DELAY AND DYNAMICAL THRESHOLDEFFECTS

1 IntroductionThe researches on dynamics of neural network systems have been arousinga great deal of interests for more than thirty years. In last ten years, many.researches focused on the study of dynamics of neural network models withdelay. In fact, neural networks often have times delay in reality, for exampledue to the finite switching speed of amplifiers in electronic neural networks.Thus, a large amount of excellent results appeared consecutively [1]--[5]. In thispaper, what we study is…     

半变系数模型的变窗宽和一步局部M-估计

讨论了半变系数模型的变窗宽一步局部M-估计.用一步局部M-估计给出了未知函数的估计,用平均法给出了未知参数的估计,并在其中嵌入一个变窗宽加以提高,得到了未知函数和未知参数的渐近正态性.     

Stability and Hopf bifurcation for a viral infection model with delayed non-lytic immune response

In this paper, a class of more general viral infection model with delayed non-lytic immune response is proposed based on some important biological meanings. The sufficient criteria for local and global asymptotic stabilities of the viral free equilibrium are given. And the stability and Hopf bifurcation of the infected equilibrium have been studied. Numerical simulations are carried out to explain the mathematical conclusions, and the effects of the birth rate of susceptible T cells and the efficacy of the non-lytic component on the stabilities of the positive equilibrium $\bar{E}$ are also studied by numerical simulations.     

一类带有接种的流行病模型的全局稳定性

该文讨论了一类带有接种的流行病模型. 在该模型中假设恢复后的个体与被接种的个体均具有确定的免疫期, 它是一个时滞微分系统. 通过分析, 得到了地方病平衡点存在的阈值, 以及无病平衡点和地方病平衡点局部渐近稳定和全局渐近稳定的充分条件.