Infected Equilibrium Research Articles

Global stability of a diffusive virus dynamics model with general incidence function and time delay

In this paper, we propose a model of virus dynamics that includes diffusion, time delay and a general incidence function. By constructing Liapunov functionals, we show that the model has threshold dynamics: if the basic reproduction number R0≤1, then the infection-free equilibrium is globally asymptotically stable; whereas if R0>1, then there exists an infection equilibrium which is globally asymptotically stable. We pay particular attention to demonstrating that solutions are sufficiently bounded away from 0 that the Liapunov functionals are well-defined. Some applications are listed. Our results improve and generalize some known results.

Stability analysis of a computer virus model in latent period

Based on a set of reasonable assumptions, the dynamical features of a novel computer virus model in latent period is proposed in this paper. Through qualitative analysis, we obtain the basic reproduction number R0. Furthermore, it is shown that the model have a infection-free equilibrium and a unique infection equilibrium (positive equilibrium). Using Lyapunov function theory, it is proved that the infection-free equilibrium is globally asymptotically stable if R0<1, implying that the virus would eventually die out. And by means of a classical geometric approach, the infection equilibrium is globally asymptotically stable if R0>1. Finally, the numerical simulations are carried out to illustrate the feasibility of the obtained results.

Stability and Hopf Bifurcation in a Delayed HIV Infection Model with General Incidence Rate and Immune Impairment.

We investigate the dynamical behavior of a delayed HIV infection model with general incidence rate and immune impairment. We derive two threshold parameters, the basic reproduction number R 0 and the immune response reproduction number R 1. By using Lyapunov functional and LaSalle invariance principle, we prove the global stability of the infection-free equilibrium and the infected equilibrium without immunity. Furthermore, the existence of Hopf bifurcations at the infected equilibrium with CTL response is also studied. By theoretical analysis and numerical simulations, the effect of the immune impairment rate on the stability of the infected equilibrium with CTL response has been studied.

Analysis of a Viral Infection Model with Delayed Nonlytic Immune Response

We investigate the dynamical behavior of a virus infection model with delayed nonlytic immune response. By analyzing corresponding characteristic equations, the local stabilities of two boundary equilibria are established. By using suitable Lyapunov functional and LaSalle’s invariance principle, we establish the global stability of the infection-free equilibrium. We find that the infection free equilibriumE0is globally asymptotically stable whenR0⩽1, and the infected equilibrium without immunityE1is local asymptotically stable when1&lt;R0⩽1+bβ/cd. Under the conditionR0&gt;1+bβ/cdwe obtain the sufficient conditions to the local stability of the infected equilibrium with immunityE2. We show that the time delay can change the stability ofE2and lead to the existence of Hopf bifurcations. The stabilities of bifurcating periodic solutions are studied and numerical simulations to our theorems are provided.

Global stability of an age-structured virus dynamics model with Beddington-DeAngelis infection function.

In this paper, we study an age-structured virus dynamics model with Beddington-DeAngelis infection function. An explicit formula for the basic reproductive number R0 of the model is obtained. We investigate the global behavior of the model in terms of R0: if R0 ≤ 1, then the infection-free equilibrium is globally asymptotically stable, whereas if R0 > 1, then the infection equilibrium is globally asymptotically stable. Finally, some special cases, which reduce to some known HIV infection models studied by other researchers, are considered.

Visceral leishmania model

In this work we consider a mathematical model based on a system of ordinary differential equations describing the evolution of population of dogs infected by leishmania diseases. By analyzing the corresponding characteristic equations, the local stability of infection free equilibrium point and infection equilibrium point are discussed. It is shown that if the basic reproduction number R 0 is less than one, the infection free equilibrium is locally asymptotically stable, whereas if the basic reproduction number R 0 is great than one the infection equilibrium point is locally asymptotically stable, and the infection free equilibrium is unstable.

On global stability of an HIV pathogenesis model with cure rate

In this paper, applying both Lyapunov function techniques and monotone iterative techniques, we establish new sufficient conditions under which the infected equilibrium of an HIV pathogenesis model with cure rate is globally asymptotically stable. By giving an explicit expression for eventual lower bound of the concentration of susceptible CD4+ T cells, we establish an affirmative partial answer to the numerical simulations investigated in the recent paper [Liu, Wang, Hu and Ma, Global stability of an HIV pathogenesis model with cure rate, Nonlinear Analysis RWA (2011) 12: 2947–2961]. Our monotone iterative techniques are applicable for the small and large growth rate in logistic functions for the proliferation rate of healthy and infected CD4+ T cells. Copyright © 2014 John Wiley &amp; Sons, Ltd.

A generalized HBV model with diffusion and two delays

An hepatitis B virus (HBV) model with spatial diffusion, general incidence rate and time delays subject to homogeneous Neumann boundary conditions is formulated. The stability of the disease-free equilibrium and the chronic infection equilibrium is analyzed by using the linearization method and by constructing appropriate Lyapunov functionals. Furthermore, several diffusive HBV models existing in the literature are generalized and extended.

Dynamic behaviors of a class of HIV compartmental models

Based on heterogeneities in drug efficacy (either spatial or phenotypic), two HIV compartmental models were proposed in Callaway and Perelson (2002) to study the HIV virus dynamics under drug treatment. In this paper, we provide a global analysis on the two models, including the positivity and boundedness of solutions and the global stability of equilibrium solutions. In particular, we show that when the basic reproduction number R0⩽1 (for which the infection equilibrium does not exist), the infection-free equilibrium is globally asymptotically stable; while when R0>1 (for which the infection equilibrium exists), the infection equilibrium is globally asymptotically stable.

Global Stability of an HIV-1 Infection Model with General Incidence Rate and Distributed Delays.

In this work an HIV-1 infection model with nonlinear incidence rate and distributed intracellular delays and with humoral immunity is investigated. The disease transmission function is assumed to be governed by general incidence rate f(T, V)V. The intracellular delays describe the time between viral entry into a target cell and the production of new virus particles and the time between infection of a cell and the emission of viral particle. Lyapunov functionals are constructed and LaSalle invariant principle for delay differential equation is used to establish the global asymptotic stability of the infection-free equilibrium, infected equilibrium without B cells response, and infected equilibrium with B cells response. The results obtained show that the global dynamics of the system depend on both the properties of the general incidence function and the value of certain threshold parameters R 0 and R 1 which depends on the delays.

Stability Analysis of an Improved HBV Model with CTL Immune Response.

To better understand the dynamics of the hepatitis B virus (HBV) infection, we introduce an improved HBV model with standard incidence function, cytotoxic T lymphocytes (CTL) immune response, and take into account the effect of the export of precursor CTL cells from the thymus and the role of cytolytic and noncytolytic mechanisms. The local stability of the disease-free equilibrium and the chronic infection equilibrium is obtained via characteristic equations. Furthermore, the global stability of both equilibria is established by using two techniques, the direct Lyapunov method for the disease-free equilibrium and the geometrical approach for the chronic infection equilibrium.

Analysis of a viral infection model with immune impairment, intracellular delay and general non-linear incidence rate

In this article we study the dynamical behaviour of a intracellular delayed viral infection with immune impairment model and general non-linear incidence rate. Several techniques, including a non-linear stability analysis by means of the Lyapunov theory and sensitivity analysis, have been used to reveal features of the model dynamics. The classical threshold for the basic reproductive number is obtained: if the basic reproductive number of the virus is less than one, the infection-free equilibrium is globally asymptotically stable and the infected equilibrium is globally asymptotically stable if the basic reproductive number is higher than one.

Diabète et cancers

In this paper, we establish new sufficient conditions for the infected equilibrium of a nonresident computer virus model to be globally asymptotically stable. Our results extend two kind of known results in recent literature.

On global stability of a nonresident computer virus model

In this paper, we establish new sufficient conditions for the infected equilibrium of a nonresident computer virus model to be globally asymptotically stable. Our results extend two kind of known results in recent literature.

Global stability and periodic oscillations for an SIV infection model with immune response and intracellular delays

In this paper, we consider the combined effects of cytotoxic T lymphocyte (CTL) responses on the competition dynamics of two Simian immunodeficiency virus (SIV) strains model. One of strains concerns a relatively slowly replicating and mildly cytopathic virus in the early infection (SIVMneCL8), the other is faster replicating and more cytopathic virus at later stages of the infection (SIVMne170). It is shown that the global dynamics of the ordinary differential equations can be determined by several threshold parameters, and we prove the global stability of the equilibria by rigorous mathematical analysis. To account for a series of infection mechanism leading to viral production, we incorporate time delays in the infection term. Using the methods of constructing suitable Lyapunov functionals and LaSalle’s invariance principle, we obtain the sufficient conditions for the global attractiveness of infection-free equilibrium with both virus strains going extinct, single-infection equilibrium with one of two virus strains out-competing the other one and the two strains coexisting infection equilibrium. We establish that the intracellular delays can destabilize the single-infection equilibrium leading to Hopf bifurcation and periodic oscillations. We show that introduction of immune responses is responsible for the coexistence of two virus strains and the intracellular delays may alter the two-strain competition results. Numerical simulations are presented to illustrate the theoretical conclusions.

The stability analysis of a general viral infection model with distributed delays and multi-staged infected progression

We investigate an in-host model with general incidence and removal rate, as well as distributed delays in virus infections and in productions. By employing Lyapunov functionals and LaSalle’s invariance principle, we define and prove the basic reproductive number R0 as a threshold quantity for stability of equilibria. It is shown that if R0>1, then the infected equilibrium is globally asymptotically stable, while if R0⩽1, then the infection free equilibrium is globally asymptotically stable under some reasonable assumptions. Moreover, n+1 distributed delays describe (i) the time between viral entry and the transcription of viral RNA, (ii) the n-1-stage time needed for activated infected cells between viral RNA transcription and viral release, and (iii) the time necessary for the newly produced viruses to be infectious (maturation), respectively. The model can describe the viral infection dynamics of many viruses such as HIV-1, HCV and HBV.

A note of a staged progression HIV model with imperfect vaccine

In this note, we investigate the global stability for a staged progression HIV model with imperfect vaccine. It is shown that the global dynamics of the system can be completely determined by the reproduction number, that is, the infected endemic equilibrium of the system is globally asymptotically stable whenever it exists. The result improves the corresponding conclusions presented in Gumel et al. (2006) 1].

Dynamics of an HIV-1 infection model with cell mediated immunity

In this paper, we study the dynamics of an improved mathematical model on HIV-1 virus with cell mediated immunity. This new 5-dimensional model is based on the combination of a basic 3-dimensional HIV-1 model and a 4-dimensional immunity response model, which more realistically describes dynamics between the uninfected cells, infected cells, virus, the CTL response cells and CTL effector cells. Our 5-dimensional model may be reduced to the 4-dimensional model by applying a quasi-steady state assumption on the variable of virus. However, it is shown in this paper that virus is necessary to be involved in the modeling, and that a quasi-steady state assumption should be applied carefully, which may miss some important dynamical behavior of the system. Detailed bifurcation analysis is given to show that the system has three equilibrium solutions, namely the infection-free equilibrium, the infectious equilibrium without CTL, and the infectious equilibrium with CTL, and a series of bifurcations including two transcritical bifurcations and one or two possible Hopf bifurcations occur from these three equilibria as the basic reproduction number is varied. The mathematical methods applied in this paper include characteristic equations, Routh–Hurwitz condition, fluctuation lemma, Lyapunov function and computation of normal forms. Numerical simulation is also presented to demonstrate the applicability of the theoretical predictions.

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

AbstractIn 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 &gt; 1. On the other hand, if R0 &gt; 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 &gt; 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 &amp; Sons, Ltd.

A MATHEMATICAL MODEL WITH OPTIMAL CONTROLS FOR CELLULAR IMMUNOLOGY OF TUBERCULOSIS

In this paper we propose a system of ordinary differential equations to model the interaction among non-infected macrophages, infected macrophages, T cells and Mtb bacilli. Model analysis reveals the existence of infection-free equilibrium and the endemically infected equilibrium. And we analyze the dynamics of this model, characterize the optimal controls related to drug therapy, and discuss a quadratic control and a linear control. The quadratic control allows for a weaker treatment that more effectively than the linear control.