scholarly journals Dynamical Analysis of a Delayed HIV Virus Dynamic Model with Cell-to-Cell Transmission and Apoptosis of Bystander Cells

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Tongqian Zhang ◽  
Junling Wang ◽  
Yi Song ◽  
Zhichao Jiang
Keyword(s):  
Time Delay ◽  
Basic Reproductive Number ◽  
Dynamical Analysis ◽  
Reproductive Number ◽  
Globally Asymptotically Stable ◽  
Local Hopf Bifurcation ◽  
Hiv Virus ◽  
Bystander Cells ◽  
The Stability ◽  
Disease Free

In this paper, a delayed viral dynamical model that considers two different transmission methods of the virus and apoptosis of bystander cells is proposed and investigated. The basic reproductive number R0 of the model is derived. Based on the basic reproductive number, we prove that the disease-free equilibrium E0 is globally asymptotically stable for R0<1 by constructing suitable Lyapunov functional. For R0>1, by regarding the time delay as bifurcation parameter, the existence of local Hopf bifurcation is investigated. The results show that time delay can change the stability of endemic equilibrium and cause periodic oscillations. Finally, we give some numerical simulations to illustrate the theoretical findings.

scholarly journals Mathematical Analysis and Stochastic Stability of Nonlinear Epidemic Model with Incidence Rate

2020 ◽  
pp. 8-19
Author(s):  
Laid Chahrazed
Keyword(s):  
Incidence Rate ◽  
Epidemic Model ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Globally Asymptotically Stable ◽  
Saturated Incidence Rate ◽  
Saturated Incidence ◽  
The Stability ◽  
Disease Free ◽  
Temporary Immunity

In this work, we consider a nonlinear epidemic model with temporary immunity and saturated incidence rate. Size N(t) at time t, is divided into three sub classes, with N(t)=S(t)+I(t)+Q(t); where S(t), I(t) and Q(t) denote the sizes of the population susceptible to disease, infectious and quarantine members with the possibility of infection through temporary immunity, respectively. We have made the following contributions: The local stabilities of the infection-free equilibrium and endemic equilibrium are; analyzed, respectively. The stability of a disease-free equilibrium and the existence of other nontrivial equilibria can be determine by the ratio called the basic reproductive number, This paper study the reduce model with replace S with N, which does not have non-trivial periodic orbits with conditions. The endemic -disease point is globally asymptotically stable if R0 ˃1; and study some proprieties of equilibrium with theorems under some conditions. Finally the stochastic stabilities with the proof of some theorems. In this work, we have used the different references cited in different studies and especially the writing of the non-linear epidemic mathematical model with [1-7]. We have used the other references for the study the different stability and other sections with [8-26]; and sometimes the previous references.

SVIR epidemic model with age structure in susceptibility, vaccination effects and relapse

2017 ◽  
Vol 82 (5) ◽  
pp. 945-970 ◽  
Author(s):  
Jinliang Wang ◽  
Min Guo ◽  
Shengqiang Liu
Keyword(s):  
Age Structure ◽  
Epidemic Model ◽  
Lyapunov Functional ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Combined Effects ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
The Stability ◽  
Disease Free

Abstract An SVIR epidemic model with continuous age structure in the susceptibility, vaccination effects and relapse is proposed. The asymptotic smoothness, existence of a global attractor, the stability of equilibria and persistence are addressed. It is shown that if the basic reproductive number $\Re_0&lt;1$, then the disease-free equilibrium is globally asymptotically stable. If $\Re_0&gt;1$, the disease is uniformly persistent, and a Lyapunov functional is used to show that the unique endemic equilibrium is globally asymptotically stable. Combined effects of susceptibility age, vaccination age and relapse age on the basic reproductive number are discussed.

scholarly journals Dynamical Behavior of a New Epidemiological Model

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Zizi Wang ◽  
Zhiming Guo
Keyword(s):  
Time Delay ◽  
Dynamical Behavior ◽  
Endemic Equilibrium ◽  
Epidemiological Model ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Sufficient Condition ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Disease Free

A new epidemiological model is introduced with nonlinear incidence, in which the infected disease may lose infectiousness and then evolves to a chronic noninfectious disease when the infected disease has not been cured for a certain timeτ. The existence, uniqueness, and stability of the disease-free equilibrium and endemic equilibrium are discussed. The basic reproductive numberR0is given. The model is studied in two cases: with and without time delay. For the model without time delay, the disease-free equilibrium is globally asymptotically stable provided thatR0≤1; ifR0>1, then there exists a unique endemic equilibrium, and it is globally asymptotically stable. For the model with time delay, a sufficient condition is given to ensure that the disease-free equilibrium is locally asymptotically stable. Hopf bifurcation in endemic equilibrium with respect to the timeτis also addressed.

scholarly journals Stability Analysis of an In-Host Viral Model with Cure of Infected Cells and Humoral Immunity

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Hui Wang ◽  
Rong Wang ◽  
Zhixing Hu ◽  
Fucheng Liao
Keyword(s):  
Time Delay ◽  
Humoral Immunity ◽  
Critical Value ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Numerical Examples ◽  
Infected Cells ◽  
The Stability

An in-host viral model with cure of infected cells and humoral immunity is studied. We prove that the stability is completely determined by the basic reproductive numberR0and show that the infection-free equilibriumE0is globally asymptotically stable if and only ifR0≤1. Moreover, ifR0>1, the infection equilibrium is locally asymptotically stable when the time delayτis small and it loses stability as the length of the time delay increases past a critical valueτ0. Finally, we confirm our analysis by providing several numerical examples.

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.

A MODEL FOR VECTOR TRANSMITTED DISEASES WITH SATURATION INCIDENCE

2001 ◽  
Vol 09 (04) ◽  
pp. 235-245 ◽  
Author(s):  
LOURDES ESTEVA ◽  
MARIANO MATIAS
Keyword(s):  
Endemic Equilibrium ◽  
Saturation Effect ◽  
Basic Reproductive Number ◽  
Infected Host ◽  
Reproductive Number ◽  
The Stability ◽  
Stability Results ◽  
Disease Free

A model for a disease that is transmitted by vectors is formulated. All newborns are assumed susceptible, and human and vector populations are assumed to be constant. The model assumes a saturation effect in the incidences due to the response of the vector to change in the susceptible and infected host densities. Stability of the disease free equilibrium and existence, uniqueness and stability of the endemic equilibrium is investigated. The stability results are given in terms of the basic reproductive number R0.

Analysis of a sex-structured HIV/AIDS model with the effect of screening of infectives

2014 ◽  
Vol 07 (05) ◽  
pp. 1450054 ◽  
Author(s):  
S. Athithan ◽  
Mini Ghosh
Keyword(s):  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Asymptotically Stable ◽  
Epidemic Threshold ◽  
Heterosexual Contact ◽  
Globally Asymptotically Stable ◽  
Transmission Of Disease ◽  
Globally Stable ◽  
Disease Free ◽  
Hiv Aids

This paper presents a nonlinear sex-structured mathematical model to study the spread of HIV/AIDS by considering transmission of disease by heterosexual contact. The epidemic threshold and equilibria for the model are determined, local stability and global stability of both the "Disease-Free Equilibrium" (DFE) and "Endemic Equilibrium" (EE) are discussed in detail. The DFE is shown to be locally and globally stable when the basic reproductive number ℛ0 is less than unity. We also prove that the EE is locally and globally asymptotically stable under some conditions. Finally, numerical simulations are reported to support the analytical findings.

Dynamical analysis for delayed virus infection models with cell-to-cell transmission and density-dependent diffusion

2020 ◽  
Vol 13 (07) ◽  
pp. 2050060
Author(s):  
Shaoli Wang ◽  
Achun Zhang ◽  
Fei Xu
Keyword(s):  
Neumann Boundary ◽  
Basic Reproductive Number ◽  
Dynamical Analysis ◽  
Reproductive Number ◽  
Asymptotically Stable ◽  
Cell Infection ◽  
Globally Asymptotically Stable ◽  
Single Strain ◽  
Density Dependent ◽  
Number Formula

In this paper, certain delayed virus dynamical models with cell-to-cell infection and density-dependent diffusion are investigated. For the viral model with a single strain, we have proved the well-posedness and studied the global stabilities of equilibria by defining the basic reproductive number [Formula: see text] and structuring proper Lyapunov functional. Moreover, we found that the infection-free equilibrium is globally asymptotically stable if [Formula: see text], and the infection equilibrium is globally asymptotically stable if [Formula: see text]. For the multi-strain model, we found that all viral strains coexist if the corresponding basic reproductive number [Formula: see text], while virus will extinct if [Formula: see text]. As a result, we found that delay and the density-dependent diffusion does not influence the global stability of the model with cell-to-cell infection and homogeneous Neumann boundary conditions.

scholarly journals Stability Analysis of an HIV/AIDS Dynamics Model with Drug Resistance

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Qianqian Li ◽  
Shengshan Cao ◽  
Xiao Chen ◽  
Guiquan Sun ◽  
Yunxi Liu ◽  
...  
Keyword(s):  
Drug Resistance ◽  
Threshold Value ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Asymptotically Stable ◽  
Dynamics Model ◽  
Globally Asymptotically Stable ◽  
Antiretroviral Drug ◽  
Disease Free ◽  
Hiv Aids

A mathematical model of HIV/AIDS transmission incorporating treatment and drug resistance was built in this study. We firstly calculated the threshold value of the basic reproductive number (R0) by the next generation matrix and then analyzed stability of two equilibriums by constructing Lyapunov function. WhenR0<1, the system was globally asymptotically stable and converged to the disease-free equilibrium. Otherwise, the system had a unique endemic equilibrium which was also globally asymptotically stable. While an antiretroviral drug tried to reduce the infection rate and prolong the patients’ survival, drug resistance was neutralizing the effects of treatment in fact.

GLOBAL STABILITY OF AN HIV/AIDS MODEL WITH STOCHASTIC PERTURBATION

2011 ◽  
Vol 04 (02) ◽  
pp. 349-358 ◽  
Author(s):  
Junyuan Yang ◽  
Xiaoyan Wang ◽  
Xuezhi Li
Keyword(s):  
Stochastic Perturbation ◽  
Endemic Equilibrium ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Asymptotically Stable ◽  
Ode Model ◽  
Globally Asymptotically Stable ◽  
Stochastic Perturbations ◽  
Hiv Model ◽  
Disease Free

In this paper, we investigate the dynamic behavior of an HIV model with stochastic perturbation. Firstly, in ODE model, the disease-free equilibrium E0 is globally asymptotically stable if the basic reproductive number R0 < 1. When R0 > 1, the endemic equilibrium E* is globally asymptotically stable. Secondly, the criterion for robustness of the system is established under stochastic perturbations. The conditions of stochastic stability of the endemic equilibrium E* are obtained. Finally, we simulate our analytical results.

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