Global properties of a class of HIV infection models with Beddington–DeAngelis functional response

In this paper, the global properties of a class of human immunodeficiency virus (HIV) models with Beddington–DeAngelis functional response are investigated. Lyapunov functions are constructed to establish the global asymptotic stability of the uninfected and infected steady states of three HIV infection models. The first model considers the interaction process of the HIV and the CD4 ⁺ T cells and takes into account the latently and actively infected cells. The second model describes two co‐circulation populations of target cells, representing CD4 ⁺ T cells and macrophages. The third model is a two‐target‐cell model taking into account the latently and actively infected cells. We have proven that if the basic reproduction number R₀ is less than unity, then the uninfected steady state is globally asymptotically stable, and if R₀ > 1, then the infected steady state is globally asymptotically stable. Copyright © 2012 John Wiley & Sons, Ltd.     

Global analysis of virus dynamics model with logistic mitosis,cure rate and delay in virus production

In this paper, we study a virus dynamics model with logistic mitosis, cure rate, and intracellular delay. By means of construction of a suitable Lyapunov functionals, obtained by linear combinations of Volterra—type functions, composite quadratic functions and Volterra—type functionals, we provide the global stability for this model. If R₀, the basic reproductive number, satisfies R₀ ≤ 1, then the infection‐free equilibrium state is globally asymptotically stable. Our system is persistent if R₀ > 1. On the other hand, if R₀ > 1, then infection‐free equilibrium becomes unstable and a unique infected equilibrium exists. The local stability analysis is carried out for the infected equilibrium, and it is shown that, if the parameters satisfy a condition, the infected equilibrium can be unstable and a Hopf bifurcation can occur. We also have that if R₀ > 1, then the infected equilibrium state is globally asymptotically stable if a sufficient condition is satisfied. We illustrate our findings with some numerical simulations. Copyright © 2014 John Wiley & Sons, Ltd.     

Global dynamics of an age‐structured virus model with saturation effects

An infection‐age virus dynamics model for human immunodeficiency virus (or hepatitis B virus) infections with saturation effects of infection rate and immune response is investigated in this paper. It is shown that the global dynamics of the model is completely determined by two critical values R ₀, the basic reproductive number for viral infection, and R ₁, the viral reproductive number at the immune‐free infection steady state (R ₁<R ₀). If R ₀<1, the uninfected steady state E ⁰ is globally asymptotically stable; if R ₀>1 > R ₁, the immune‐free infected steady state E ^? is globally asymptotically stable; while if R ₁>1, the antibody immune infected steady state is globally asymptotically stable. Moreover, our results show that ignoring the saturation effects of antibody immune response or infection rate will result in an overestimate of the antibody immune reproductive number. Copyright © 2016 John Wiley & Sons, Ltd.     

Global dynamics of a cell mediated immunity in viral infection models with distributed delays

In this paper, we investigate global dynamics for a system of delay differential equations which describes a virus-immune interaction in vivo. The model has two distributed time delays describing time needed for infection of cell and virus replication. Our model admits three possible equilibria, an uninfected equilibrium and infected equilibrium with or without immune response depending on the basic reproduction number for viral infection R₀ and for CTL response R₁ such that R₁<R₀. It is shown that there always exists one equilibrium which is globally asymptotically stable by employing the method of Lyapunov functional. More specifically, the uninfected equilibrium is globally asymptotically stable if R₀?1, an infected equilibrium without immune response is globally asymptotically stable if R₁?1<R₀ and an infected equilibrium with immune response is globally asymptotically stable if R₁>1. The immune activation has a positive role in the reduction of the infection cells and the increasing of the uninfected cells if R₁>1.     

Stability of an age-structured epidemiological model for hepatitis C

This article introduces an age-structured epidemiological model for the disease transmission dynamics of hepatitis C. We first show that the infection-free steady state is locally and globally asymptotically stable if the basic reproductive number ? ₀ is below one, in this case, the disease always dies out, then we prove that at least one endemic steady state exists when the reproductive number ? ₀ is above one, the stability conditions for the endemic steady states are also given.     

Global dynamics of a viral infection model with a latent period and Beddington-DeAngelis response

In this paper, we study the global dynamics of a viral infection model with a latent period. The model has a nonlinear function which denotes the incidence rate of the virus infection in vivo. The basic reproduction number of the virus is identified and it is shown that the uninfected equilibrium is globally asymptotically stable if the basic reproduction number is equal to or less than unity. Moreover, the virus and infected cells eventually persist and there exists a unique infected equilibrium which is globally asymptotically stable if the basic reproduction number is greater than unity. The basic reproduction number determines the equilibrium that is globally asymptotically stable, even if there is a time delay in the infection.     

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.     

GLOBAL STABILITY IN A VIRAL INFECTION MODEL

This paper investigates the global stability of a viral infection model with lytic immune response. If the basic reproductive ratio of the virus is less than or equal to one, by the LaSalle's invariance principle, the disease-free steady state is globally asymptotically stable. If the basic reproductive ratio of the virus is greater than one but less than or equal to a constant, which is defined by the parameters of the model, then the immune-exhausted steady state is globally asymptotically stable. The endemic steady state is globally asymptotically stable if the inverse is valid.     

Global dynamics of a spatial heterogeneous viral infection model with intracellular delay and nonlocal diffusion

In this paper, we propose a spatial heterogeneous viral infection model, where heterogeneous parameters, the intracellular delay and nonlocal diffusion of free virions are considered. The global well-posedness, compactness and asymptotic smoothness of the semiflow generated by the system are established. It is shown that the principal eigenvalue problem of a perturbation of the nonlocal diffusion operator has a principal eigenvalue associated with a positive eigenfunction. The principal eigenvalue plays the same role as the basic reproduction number being defined as the spectral radius of the next generation operator. The existence of the unique chronic-infection steady state is established by the super-sub solution method. Furthermore, the uniform persistence of the model is investigated by using the persistence theory of infinite dimensional dynamical systems. By setting the eigenfunction as the integral kernel of Lyapunov functionals, the global threshold dynamics of the system is established. More precisely, the infection-free steady state is globally asymptotically stable if the basic reproduction number is less than one; while the chronic-infection steady state is globally asymptotically stable if the basic reproduction number is larger than one. Numerical simulations are carried out to illustrate the effects of intracellular delay and diffusion rate on the final concentrations of infected cells and free virions, respectively.     

Analysis of the dynamics of a delayed HIV pathogenesis model

In this paper, considering full Logistic proliferation of CD4⁺ T cells, we study an HIV pathogenesis model with antiretroviral therapy and HIV replication time. We first analyze the existence and stability of the equilibrium, and then investigate the effect of the time delay on the stability of the infected steady state. Sufficient conditions are given to ensure that the infected steady state is asymptotically stable for all delay. Furthermore, we apply the Nyquist criterion to estimate the length of delay for which stability continues to hold, and investigate the existence of Hopf bifurcation by using a delay τ as a bifurcation parameter. Finally, numerical simulations are presented to illustrate the main results.     

Global stability of a virus dynamics model with intracellular delay and CTL immune response

In this paper, the global stability of a virus dynamics model with intracellular delay, Crowley–Martin functional response of the infection rate, and CTL immune response is studied. By constructing suitable Lyapunov functions and using LaSalles invariance principle, the global dynamics is established; it is proved that if the basic reproductive number, R₀, is less than or equal to one, the infection‐free equilibrium is globally asymptotically stable; if R₀ is more than one, and if immune response reproductive number, R⁰, is less than one, the immune‐free equilibrium is globally asymptotically stable, and if R⁰ is more than one, the endemic equilibrium is globally asymptotically stable. Copyright © 2014 John Wiley & Sons, Ltd.     

Global stability of delay multigroup epidemic models with group mixing and nonlinear incidence rates

In this paper, we present a new delay multigroup SEIR model with group mixing and nonlinear incidence rates and investigate its global stability. We establish that the global dynamics of the models are completely determined by the basic reproduction number R₀. It is shown that, if R₀?1, then the disease free equilibrium is globally asymptotically stable and the disease dies out; if R₀>1, there exists a unique endemic equilibrium that is globally asymptotically stable and thus the disease persists in the population. Finally, a numerical example is also discussed to illustrate the effectiveness of the results.     

Global properties for virus dynamics model with Beddington–DeAngelis functional response

This paper investigates the global stability of virus dynamics model with Beddington–DeAngelis infection rate. By constructing Lyapunov functions, the global properties have been analysed. If the basic reproductive ratio of the virus is less than or equal to one, the uninfected steady state is globally asymptotically stable. If the basic reproductive ratio of the virus is more than one, the infected steady state is globally asymptotically stable. The conditions imply that the steady states are always globally asymptotically stable for Holling type II functional response or for a saturation response.     

Bifurcation analysis of HIV infection model with antibody and cytotoxic T‐lymphocyte immune responses and Beddington–DeAngelis functional response

In this paper, a mathematical model for HIV‐1 infection with antibody, cytotoxic T‐lymphocyte immune responses and Beddington–DeAngelis functional response is investigated. The stability of the infection‐free and infected steady states is investigated. The basic reproduction number R₀ is identified for the proposed system. If R₀ < 1, then there is an infection‐free steady state, which is locally asymptotically stable. Further, the infected steady state is locally asymptotically stable for R₀ > 1 in the absence of immune response delay. We use Nyquist criterion to estimate the length of the delay for which stability continues to hold. Also the existence of the Hopf bifurcation is investigated by using immune response delay as a bifurcation parameter. Numerical simulations are presented to justify the analytical results. Copyright © 2014 John Wiley & Sons, Ltd.     

A differential equation model of HIV infection of CD4T-cells with cure rate

A differential equation model of HIV infection of CD4⁺T-cells with cure rate is studied. We prove that if the basic reproduction number R₀<1, the HIV infection is cleared from the T-cell population and the disease dies out; if R₀>1, the HIV infection persists in the host. We find that the chronic disease steady state is globally asymptotically stable if R₀>1. Furthermore, we also obtain the conditions for which the system exists an orbitally asymptotically stable periodic solution. Numerical simulations are presented to illustrate the results.     

Dynamical behavior of computer virus on Internet

In this paper, we presented a computer virus model using an SIRS model and the threshold value R₀ determining whether the disease dies out is obtained. If R₀ is less than one, the disease-free equilibrium is globally asymptotically stable. By using the time delay as a bifurcation parameter, the local stability and Hopf bifurcation for the endemic state is investigated. Numerical results demonstrate that the system has periodic solution when time delay is larger than a critical values. The obtained results may provide some new insight to prevent the computer virus.     

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.