Global Stability of a HBV Infection Model with Saturated Infection Rates and Time Delay

2013 ◽  
Vol 791-793 ◽  
pp. 1314-1317
Author(s):  
Wei Juan Pang ◽  
Zhi Xing Hu ◽  
Fu Cheng Liao
Keyword(s):  
Global Stability ◽  
Sufficient Conditions ◽  
Hbv Infection ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Infection Model ◽  
Asymptotically Stable ◽  
Infection Rates ◽  
Globally Asymptotically Stable ◽  
Viral Infection Model

This paper investigates the global stability of a viral infection model of HBV infection of hepatocytes with saturated infection rate and intracellular delay. we obtain if the basic reproductive number is less than or equal to one, the infection-free equilibrium is globally asymptotically stable. If its greater than one, we obtain the sufficient conditions for the global stability of the infected equilibrium.

scholarly journals Global Stability for a Viral Infection Model with Saturated Incidence Rate

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Huaqin Peng ◽  
Zhiming Guo
Keyword(s):  
Viral Infection ◽  
Global Stability ◽  
Incidence Rate ◽  
Sufficient Conditions ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Infection Model ◽  
Saturated Incidence Rate ◽  
Saturated Incidence ◽  
Viral Infection Model

A viral infection model with saturated incidence rate and viral infection with delay is derived and analyzed; the incidence rate is assumed to be a specific nonlinear formβxv/(1+αv). The existence and uniqueness of equilibrium are proved. The basic reproductive numberR0is given. The model is divided into two cases: with or without delay. In each case, by constructing Lyapunov functionals, necessary and sufficient conditions are given to ensure the global stability of the models.

Stability and Hopf bifurcation of a delayed virus infection model with latently infected cells and Beddington–DeAngelis incidence

2020 ◽  
Vol 13 (05) ◽  
pp. 2050045
Author(s):  
Junxian Yang ◽  
Shoudong Bi
Keyword(s):  
Immune Response ◽  
Hopf Bifurcation ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Infection Model ◽  
Asymptotically Stable ◽  
Response Delay ◽  
Globally Asymptotically Stable ◽  
Latently Infected ◽  
Infected Cells

In this paper, the dynamical behaviors for a five-dimensional virus infection model with Latently Infected Cells and Beddington–DeAngelis incidence are investigated. In the model, four delays which denote the latently infected delay, the intracellular delay, virus production period and CTL response delay are considered. We define the basic reproductive number and the CTL immune reproductive number. By using Lyapunov functionals, LaSalle’s invariance principle and linearization method, the threshold conditions on the stability of each equilibrium are established. It is proved that when the basic reproductive number is less than or equal to unity, the infection-free equilibrium is globally asymptotically stable; when the CTL immune reproductive number is less than or equal to unity and the basic reproductive number is greater than unity, the immune-free infection equilibrium is globally asymptotically stable; when the CTL immune reproductive number is greater than unity and immune response delay is equal to zero, the immune infection equilibrium is globally asymptotically stable. The results show that immune response delay may destabilize the steady state of infection and lead to Hopf bifurcation. The existence of the Hopf bifurcation is discussed by using immune response delay as a bifurcation parameter. Numerical simulations are carried out to justify the analytical results.

Global dynamics of a viral infection model with full logistic terms and antivirus treatments

2016 ◽  
Vol 10 (01) ◽  
pp. 1750012 ◽  
Author(s):  
Lijuan Song ◽  
Cui Ma ◽  
Qiang Li ◽  
Aijun Fan ◽  
Kaifa Wang
Keyword(s):  
Viral Infection ◽  
Global Stability ◽  
Endemic Equilibrium ◽  
Infection Model ◽  
Global Dynamics ◽  
Sufficient Condition ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Influx Rate ◽  
Viral Infection Model

In this paper, mathematical analysis of the global dynamics of a viral infection model in vivo is carried out. Though the model is originally to study hepatitis C virus (HCV) dynamics in patients with high baseline viral loads or advanced liver disease, similar models still hold significance for other viral infection, such as hepatitis B virus (HBV) or human immunodeficiency virus (HIV) infection. By means of Volterra-type Lyapunov functions, we know that the basic reproduction number [Formula: see text] is a sharp threshold para-meter for the outcomes of viral infections. If [Formula: see text], the virus-free equilibrium is globally asymptotically stable. If [Formula: see text], the system is uniformly persistent, the unique endemic equilibrium appears and is globally asymptotically stable under a sufficient condition. Other than that, for the global stability of the unique endemic equilibrium, another sufficient condition is obtained by Li–Muldowney global-stability criterion. Using numerical simulation techniques, we further find that sustained oscillations can exist and different maximum de novo hepatocyte influx rate can induce different global dynamics along with the change of overall drug effectiveness. Finally, some biological implications of our findings are given.

scholarly journals Analysis of a Viral Infection Model with Delayed Nonlytic Immune Response

2015 ◽  
Vol 2015 ◽  
pp. 1-11
Author(s):  
Mengye Chen ◽  
Liang You ◽  
Jie Tang ◽  
Shasha Su ◽  
Ruiming Zhang
Keyword(s):  
Immune Response ◽  
Dynamical Behavior ◽  
Sufficient Conditions ◽  
Infection Model ◽  
Hopf Bifurcations ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
The Stability ◽  
Virus Infection Model ◽  
Viral Infection Model

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<R0⩽1+bβ/cd. Under the conditionR0>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.

scholarly journals Dynamics Analysis and Simulation of a Modified HIV Infection Model with a Saturated Infection Rate

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Qilin Sun ◽  
Lequan Min
Keyword(s):  
Drug Resistance ◽  
Antiretroviral Therapy ◽  
Hiv Infection ◽  
Equilibrium Point ◽  
Infection Rate ◽  
Reproductive Number ◽  
Infection Model ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Hiv Rna

This paper studies a modified human immunodeficiency virus (HIV) infection differential equation model with a saturated infection rate. It is proved that if the basic virus reproductive numberR0of the model is less than one, then the infection-free equilibrium point of the model is globally asymptotically stable; ifR0of the model is more than one, then the endemic infection equilibrium point of the model is globally asymptotically stable. Based on the clinical data from HIV drug resistance database of Stanford University, using the proposed model simulates the dynamics of the two groups of patients’ anti-HIV infection treatment. The numerical simulation results are in agreement with the evolutions of the patients’ HIV RNA levels. It can be assumed that if an HIV infected individual’s basic virus reproductive numberR0<1then this person will recover automatically; if an antiretroviral therapy makes an HIV infected individual’sR0<1, this person will be cured eventually; if an antiretroviral therapy fails to suppress an HIV infected individual’s HIV RNA load to be of unpredictable level, the time that the patient’s HIV RNA level has achieved the minimum value may be the starting time that drug resistance has appeared.

Stability of a CD4+ T cell viral infection model with diffusion

2018 ◽  
Vol 11 (05) ◽  
pp. 1850071 ◽  
Author(s):  
Zhiting Xu ◽  
Youqing Xu
Keyword(s):  
T Cell ◽  
Viral Infection ◽  
Lyapunov Functions ◽  
Threshold Value ◽  
Endemic Equilibrium ◽  
Infection Model ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
The Stability ◽  
Viral Infection Model

This paper is devoted to the study of the stability of a CD[Formula: see text] T cell viral infection model with diffusion. First, we discuss the well-posedness of the model and the existence of endemic equilibrium. Second, by analyzing the roots of the characteristic equation, we establish the local stability of the virus-free equilibrium. Furthermore, by constructing suitable Lyapunov functions, we show that the virus-free equilibrium is globally asymptotically stable if the threshold value [Formula: see text]; the endemic equilibrium is globally asymptotically stable if [Formula: see text] and [Formula: see text]. Finally, we give an application and numerical simulations to illustrate the main results.

Global properties for an age-structured within-host model with Crowley–Martin functional response

2017 ◽  
Vol 10 (02) ◽  
pp. 1750030 ◽  
Author(s):  
Shaoli Wang ◽  
Xinyu Song
Keyword(s):  
Functional Response ◽  
Viral Infections ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Positive Equilibrium ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Multi Scale ◽  
Global Properties ◽  
Age Structured

Based on a multi-scale view, in this paper, we study an age-structured within-host model with Crowley–Martin functional response for the control of viral infections. By means of semigroup and Lyapunov function, the global asymptotical property of infected steady state of the model is obtained. The results show that when the basic reproductive number falls below unity, the infection dies out. However, when the basic reproductive number exceeds unity, there exists a unique positive equilibrium which is globally asymptotically stable. This model can be deduced to different viral models with or without time delay.

Dynamics of a diffusion-driven HBV infection model with capsids and time delay

2017 ◽  
Vol 10 (05) ◽  
pp. 1750062 ◽  
Author(s):  
Kalyan Manna
Keyword(s):  
Steady State ◽  
Time Delay ◽  
Dynamical Behavior ◽  
Hbv Infection ◽  
Infection Model ◽  
Direct Lyapunov Method ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Discrete Time Delay ◽  
Number Formula

In this paper, a diffusive hepatitis B virus (HBV) infection model with a discrete time delay is presented and analyzed, where the spatial mobility of both intracellular capsid covered HBV DNA and HBV and the intracellular delay in the reproduction of infected hepatocytes are taken into account. We define the basic reproduction number [Formula: see text] that determines the dynamical behavior of the model. The local and global stability of the spatially homogeneous steady states are analyzed by using the linearization technique and the direct Lyapunov method, respectively. It is shown that the susceptible uninfected steady state is globally asymptotically stable whenever [Formula: see text] and is unstable whenever [Formula: see text]. Also, the infected steady state is globally asymptotically stable when [Formula: see text]. Finally, numerical simulations are carried out to illustrate the results obtained.

scholarly journals Global Stability of an SEIS Epidemic Model with General Saturation Incidence

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Hui Zhang ◽  
Li Yingqi ◽  
Wenxiong Xu
Keyword(s):  
Latent Period ◽  
Epidemic Model ◽  
Convergence Theorem ◽  
Incidence Rates ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Disease Free

We present an SEIS epidemic model with infective force in both latent period and infected period, which has different general saturation incidence rates. It is shown that the global dynamics are completely determined by the basic reproductive number R0. If R0≤1, the disease-free equilibrium is globally asymptotically stable in T by LaSalle’s Invariance Principle, and the disease dies out. Moreover, using the method of autonomous convergence theorem, we obtain that the unique epidemic equilibrium is globally asymptotically stable in T0, and the disease spreads to be endemic.

scholarly journals Global Dynamics of an HTLV-1 Model with Cell-to-Cell Infection and Mitosis

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Sumei Li ◽  
Yicang Zhou
Keyword(s):  
Chronic Infection ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Cell Infection ◽  
Globally Asymptotically Stable ◽  
Human T Cell

A mathematical model of human T-cell lymphotropic virus type 1 in vivo with cell-to-cell infection and mitosis is formulated and studied. The basic reproductive numberR0is derived. It is proved that the dynamics of the model can be determined completely by the magnitude ofR0. The infection-free equilibrium is globally asymptotically stable (unstable) ifR0<1  (R0>1). There exists a chronic infection equilibrium and it is globally asymptotically stable ifR0>1.

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