Dynamical behavior of a delay virus dynamics model with CTL immune response

In this paper, the dynamical behavior of a virus dynamics model with CTL immune response and time delay is studied. Time delay is used to describe the time between the infected cell and the emission of viral particles on a cellular level. The effect of time delay on stability of the equilibria of the CTL immune response model has been studied and sufficient criteria for local asymptotic stability of the disease-free equilibrium, immune-free equilibrium and endemic equilibrium and global asymptotic stability of the disease-free equilibrium are given. Some conditions for Hopf bifurcation around immune-free equilibrium and endemic equilibrium to occur are also obtained by using the time delay as a bifurcation parameter. Numerical simulation with some hypothetical sets of data has been done to support the analytical findings.     

Dynamics analysis of a delayed viral infection model with immune impairment

In this paper, the dynamical behavior of a delayed viral infection model with immune impairment is studied. It is shown that if the basic reproductive number of the virus is less than one, then the uninfected equilibrium is globally asymptotically stable for both ODE and DDE model. And the effect of time delay on stabilities of the equilibria of the DDE model has been studied. By theoretical analysis and numerical simulations, we show that the immune impairment rate has no effect on the stability of the ODE model, while it has a dramatic effect on the infected equilibrium of the DDE model.     

Dynamics analysis of an HTLV‐1 infection model with mitotic division of actively infected cells and delayed CTL immune response

In this paper, we propose an improved human T‐cell leukemia virus type 1 infection model with mitotic division of actively infected cells and delayed cytotoxic T lymphocyte immune response. By constructing suitable Lyapunov functional and using LaSalle invariance principle, we investigate the global stability of the infection‐free equilibrium of the system. Our results show that the time delay can change stability behavior of the infection equilibrium and lead to the existence of Hopf bifurcations. Finally, numerical simulations are conducted to illustrate the applications of the main results.     

Global stability of the virus dynamics model with intracellular delay and Crowley–Martin functional response

In this paper, a virus dynamics model with intracellular delay and Crowley–Martin functional response is discussed. By constructing suitable Lyapunov functions and using LaSalles invariance principle for delay differential equations, we established the global stability of uninfected equilibrium and infected equilibrium; it is proved that if the basic reproductive number is less than or equal to one, the uninfected equilibrium is globally asymptotically stable; if the basic reproductive number is more than one, the infected equilibrium is globally asymptotically stable. We also discuss the effects of intracellular delay on global dynamical properties by comparing the results with the stability conditions for the model without delay. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

Stability analysis of a virus infection model with humoral immunity response and two time delays

In this paper, we investigate the dynamical properties for a model of delay differential equations, which describes a virus‐immune interaction in vivo. By analyzing corresponding characteristic equations, the local stability of the equilibria for infection‐free, antibody‐free, and antibody response and the existence of Hopf bifurcation with antibody response delay as a bifurcation parameter at the antibody‐activated infection equilibrium are established, respectively. Global stability of the equilibria for infection‐free, antibody‐free, and antibody response, respectively, also are established by applying the Lyapunov functionals method. The numerical simulations are performed in order to illustrate the dynamical behavior of the model. Copyright © 2016 John Wiley & Sons, Ltd.     

Dynamics of the density dependent predator-prey system with Beddington-DeAngelis functional response

Two models of a density dependent predator-prey system with Beddington-DeAngelis functional response are systematically considered. One includes the time delay in the functional response and the other does not. The explorations involve the permanence, local asymptotic stability and global asymptotic stability of the positive equilibrium for the models by using stability theory of differential equations and Lyapunov functions. For the permanence, the density dependence for predators is shown to give some negative effect for the two models. Further the permanence implies the local asymptotic stability for a positive equilibrium point of the model without delay. Also the global asymptotic stability condition, which can be easily checked for the model is obtained. For the model with time delay, local and global asymptotic stability conditions are obtained.     

An SIS Epidemic Model with Stage Structure and a Delay

A disease transmission model of SIS type with stage structure and a delay is formulated. Stability of the disease free equilibrium, and existence, uniqueness, and stability of an endemic equilibrium, are investigated for the model. The stability results arc stated in terms of a key threshold parameter. The effects of stage structure and time delay on dynamical behavior of the infectious disease are analyzed. It is shown that stage structure has no effect on the epidemic model and Hopf bifurcation can occur as the time delay increases.     

A delay model for viral infection in toxin producing phytoplankton and zooplankton system

An eco-epidemiological delay model is proposed and analysed for virally infected, toxin producing phytoplankton (TPP) and zooplankton system. It is shown that time delay can destabilize the otherwise stable non-zero equilibrium state. The coexistence of all species is possible through periodic solutions due to Hopf bifurcation. In the absence of infection the delay model may have a complex dynamical behavior which can be controlled by infection. Numerical simulation suggests that the proposed model displays a wide range of dynamical behaviors. Different parameters are identified that are responsible for chaos.     

一类具有时滞的Watt型恒化器模型的稳定性分析

基于"比例依赖"理论,研究了一类具有时滞和Watt型功能反应函数的恒化器模型.详细讨论了正平衡点的局部渐近稳定性,证明了系统在特定的时滞参量值下将产生Hopf分支.利用Lyapunov-LaSalle不变性原理,得到了正平衡点全局渐近稳定的充分条件.     

Stability analysis of time delayed fractional order predator-prey system with Crowley-Martin functional response

In this paper, we present a fractional order predator-prey system with Crowley-Martin functional response. Firstly, we analyze the asymptotic stability of the system. At the same time, some sufficient conditions for the stability of the system are given. Then, we investigate the stability of the corresponding system with time delay and also discuss some sufficient conditions for the equilibrium stability of the system with time delay. In the end, the numerical simulations illustrate the accuracy of our conclusions.     

Asymptotic properties of a HIV-1 infection model with time delay

Based on some important biological meanings, a class of more general HIV-1 infection models with time delay is proposed in the paper. In the HIV-1 infection model, time delay is used to describe the time between infection of uninfected target cells and the emission of viral particles on a cellular level as proposed by Herz et al. [A.V.M. Herz, S. Bonhoeffer, R.M. Anderson, R.M. May, M.A. Nowak, Viral dynamics in vivo: Limitations on estimates of intracellular delay and virus decay, Proc. Natl. Acad. Sci. USA 93 (1996) 7247-7251]. Then, the effect of time delay on stability of the equilibria of the HIV-1 infection model has been studied and sufficient criteria for local asymptotic stability of the infected equilibrium and global asymptotic stability of the viral free equilibrium are given.     

Global stability of humoral immunity virus dynamics models with nonlinear infection rate and removal

In this paper, we investigate the dynamical behavior of two nonlinear models for viral infection with humoral immune response. The first model contains four compartments; uninfected target cells, actively infected cells, free virus particles and B cells. The intrinsic growth rate of uninfected cells, incidence rate of infection, removal rate of infected cells, production rate of viruses, neutralization rate of viruses, activation rate of B cells and removal rate of B cells are given by more general nonlinear functions. The second model is a modification of the first one by including an eclipse stage of infected cells. We assume that the latent-to-active conversion rate is also given by a more general nonlinear function. For each model we derive two threshold parameters and establish a set of conditions on the general functions which are sufficient to determine the global dynamics of the models. By using suitable Lyapunov functions and LaSalle’s invariance principle, we prove the global asymptotic stability of the all equilibria of the models. We perform some numerical simulations for the models with specific forms of the general functions and show that the numerical results are consistent with the theoretical results.     

具有离散和分布时滞的Lotka-Volterra竞争系统

In this paper.the Lotka-Volterra competition system with discrete and distributed time delays is considered.By analyzing the characteristic equation of the linearized system,the local asymptotic stability of the positive equilibrium is investigated.Moreover,we discover the delays don't effect the stability of the equilibrium in the delay system.Finally,we can conclude that the positive equilibrium is global asymptotically stable in the delay system.     

Stability and Hopf bifurcation of a Nonlinear oscillator with multiple time-delays

This paper examines dynamical behavior of a nonlinear oscillator which models a quarter-car forced by the road profile. The effect of multiple time delays is studied in detail. The focus is on the influence of delay in the system. This naturally gives rise to a delay differential equation (DDE) model of the system. The domain where the control is efficient in reducing the amplitude of vibration is found by the harmonic balance method. Technical stability within definite time and asymptotic stability is derived for selected gain control parameters. The control gain parameters are chosen according to technical and asymptotic stability. The energy analysis is a combination of Lyapunov’s function and the averaging technique, and is used to analyze the Hopf bifurcation.     

Mathematical Analysis of SIR Epidemic Model with Piecewise Infection Rate and Control Strategies

The limitation of contact between susceptible and infected individuals plays an important role in decreasing the transmission of infectious diseases. Prevention and control strategies contribute to minimizing the transmission rate. In this paper, we propose SIR epidemic model with delayed control strategies, in which delay describes the response and effect time. We study the dynamic properties of the epidemic model from three aspects: steady states, stability and bifurcation. By eliminating the existence of limit cycles, we establish the global stability of the endemic equilibrium, when the delay is ignored. Further, we find that the delayed effect on the infection rate does not affect the stability of the disease-free equilibrium, but it can destabilize the endemic equilibrium and bring Hopf bifurcation. Theoretical results show that the prevention and control strategies can effectively reduce the final number of infected individuals in the population. Numerical results corroborate the theoretical ones.     

Global Stability for a Generalized Predator—Prey System with Time Delay In Two-patch Environments

In this article the asymptotic behavior of solutions of a predator—prey system is investigated. The model incorporates time delay due to gestation and assumes that the prey disperses between two patches of a heterogeneous environment with barriers between patches and that the predator disperses between the patches with no barrier. Conditions are derived for the global asymptotic stability of a positive equilibrium.     

Asymptotic behavior of a Lotka–Volterra food chain stochastic model in the Chemostat

This article studies the asymptotic behavior of a stochastic Chemostat model with Lotka–Volterra food chain in which the dilution rate was influenced by white noise. The long-time behavior of the model is studied. Using Lyapunov function and Itô's formula, we show that there is a unique positive solution to the system. Moreover, the sufficient conditions for some population dynamical properties including the boundedness in mean and the stochastically asymptotic stability of the washout equilibrium were obtained. Furthermore, we show how the solutions spiral around the predator-free equilibrium and the positive equilibrium of deterministic system. Besides, the existence of the stationary distribution is proved for the considered model. Numerical simulations are introduced finally to support the obtained results.     

Dynamical analysis of a delay model of phytoplankton-zooplankton interaction

The interaction of toxic-phytoplankton-zooplankton systems and their dynamical behavior will be considered in this paper based upon nonlinear ordinary differential equation model system. We induced a discrete time delay to the both of the consume response function and distribution of toxic substance term to describe the delay in the conversion of nutrient consumed to species and the fact that the time required for the phytoplankton species to mature before they can produce toxic substances. We generalized the model in [1] and explicit results are derived for globally asymptotically stability of the boundary equilibrium. Using numerical simulation method, we determine there is a parameter range for the delay parameter τ where more complicated dynamics occurs, and this appears to be a new result. Significant outcomes of our numerical findings and their interpretations from ecological point of view are provided in this paper.     

Stability and chaos control of regularized Prabhakar fractional dynamical systems without and with delay

In this paper, we studied the stabilization of nonlinear regularized Prabhakar fractional dynamical systems without and with time delay. We establish a Lyapunov stabiliy theorem for these systems and study the asymptotic stability of these systems without design a positive definite function V (without considering the fractional derivative of function V is negative). We design a linear feedback controller to control and stabilize the nonautonomous and autonomous chaotic regularized Prabhakar fractional dynamical systems without and with time delay. By means of the Lyapunov stability, we obtain the control parameters for these type of systems. We further present a numerical method to solve and analyze regularized Prabhakar fractional systems. Furthermore, by employing numerical simulation, we reveal chaotic attractors and asymptotic stability behaviors for four systems to illustrate the presented theorem.