scholarly journals Global Stability for a Binge Drinking Model with Two Stages

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Hai-Feng Huo ◽  
Na-Na Song
Keyword(s):  
Binge Drinking ◽  
Reproduction Number ◽  
Alcohol Problems ◽  
Global Dynamics ◽  
Direct Transfer ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Drinking Problem ◽  
Mathematical Analyses ◽  
Two Stages

A more realistic two-stage model for binge drinking problem is introduced, where the youths with alcohol problems are divided into those who admit the problem and those who do not admit it. We also consider the direct transfer from the class of susceptible individuals towards the class of admitting drinkers. Mathematical analyses establish that the global dynamics of the model are determined by the basic reproduction number,R0. The alcohol-free equilibrium is globally asymptotically stable, and the alcohol problems are eliminated from the population ifR0<1. A unique alcohol-present equilibrium is globally asymptotically stable ifR0>1. Numerical simulations are also conducted in the analytic results.

scholarly journals Stability of an HIV/AIDS Treatment Model with Different Stages

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Hai-Feng Huo ◽  
Rui Chen
Keyword(s):  
Hiv Infection ◽  
Hiv Positive ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Treatment Model ◽  
Globally Asymptotically Stable ◽  
Asymptomatic Stage ◽  
Mathematical Analyses ◽  
Two Stages ◽  
Hiv Aids

An HIV/AIDS treatment model with different stages is proposed in this paper. The stage of the HIV infection is divided into two stages, that is, HIV-positive in the asymptomatic stage of HIV infection and HIV-positive individuals in the pre-AIDS stage. The fact that some individuals with HIV-positive individuals after the treatment can be transformed into the compartment of HIV-positive individuals in the asymptomatic stage of HIV infection, the compartment of HIV-positive individuals in the pre-AIDS stage, or the compartment of individuals with full-blown AIDS is also considered. Mathematical analyses establish the idea that the global dynamics of the HIV/AIDS model are determined by the basic reproduction numberR0. The disease-free equilibrium is globally asymptotically stable ifR0<1. The endemic equilibrium is globally asymptotically stable ifR0>1for a special case. Numerical simulations are also conducted to support the analytic results.

scholarly journals Global Dynamics of an SIQR Model with Vaccination and Elimination Hybrid Strategies

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 328 ◽  
Author(s):  
Yanli Ma ◽  
Jia-Bao Liu ◽  
Haixia Li
Keyword(s):  
Lyapunov Function ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Endemic Equilibrium ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Disease Free ◽  
The Basic Reproduction Number

In this paper, an SIQR (Susceptible, Infected, Quarantined, Recovered) epidemic model with vaccination, elimination, and quarantine hybrid strategies is proposed, and the dynamics of this model are analyzed by both theoretical and numerical means. Firstly, the basic reproduction number R 0 , which determines whether the disease is extinct or not, is derived. Secondly, by LaSalles invariance principle, it is proved that the disease-free equilibrium is globally asymptotically stable when R 0 < 1 , and the disease dies out. By Routh-Hurwitz criterion theory, we also prove that the disease-free equilibrium is unstable and the unique endemic equilibrium is locally asymptotically stable when R 0 > 1 . Thirdly, by constructing a suitable Lyapunov function, we obtain that the unique endemic equilibrium is globally asymptotically stable and the disease persists at this endemic equilibrium if it initially exists when R 0 > 1 . Finally, some numerical simulations are presented to illustrate the analysis results.

Global dynamics for a live-virus-blood model with distributed time delay

2017 ◽  
Vol 10 (05) ◽  
pp. 1750067 ◽  
Author(s):  
Ding-Yu Zou ◽  
Shi-Fei Wang ◽  
Xue-Zhi Li
Keyword(s):  
Steady State ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Live Virus ◽  
Number Formula ◽  
Distributed Time Delay ◽  
The Basic Reproduction Number

In this paper, the global properties of a mathematical modeling of hepatitis C virus (HCV) with distributed time delays is studied. Lyapunov functionals are constructed to establish the global asymptotic stability of the uninfected and infected steady states. It is shown that if the basic reproduction number [Formula: see text] is less than unity, then the uninfected steady state is globally asymptotically stable. If the basic reproduction number [Formula: see text] is larger than unity, then the infected steady state is globally asymptotically stable.

scholarly journals On the Global Dynamics of a Vector-Borne Disease Model with Age of Vaccination

2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Stanislas Ouaro ◽  
Ali Traoré
Keyword(s):  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Mass Action ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Vector Borne Disease ◽  
Special Cases ◽  
Vector Borne ◽  
The Basic Reproduction Number

We study a vector-borne disease with age of vaccination. A nonlinear incidence rate including mass action and saturating incidence as special cases is considered. The global dynamics of the equilibria are investigated and we show that if the basic reproduction number is less than 1, then the disease-free equilibrium is globally asymptotically stable; that is, the disease dies out, while if the basic reproduction number is larger than 1, then the endemic equilibrium is globally asymptotically stable, which means that the disease persists in the population. Using the basic reproduction number, we derive a vaccination coverage rate that is required for disease control and elimination.

scholarly journals Stability Analysis of a Reaction-Diffusion Heroin Epidemic Model

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-16
Author(s):  
Liang Zhang ◽  
Yifan Xing
Keyword(s):  
Epidemic Model ◽  
Reproduction Number ◽  
Reaction Diffusion ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Ode Model ◽  
Globally Asymptotically Stable ◽  
D Model ◽  
Drug Free ◽  
The Basic Reproduction Number

A reaction-diffusion (R-D) heroin epidemic model with relapse and permanent immunization is formulated. We use the basic reproduction number R0 to determine the global dynamics of the models. For both the ordinary differential equation (ODE) model and the R-D model, it is shown that the drug-free equilibrium is globally asymptotically stable if R0≤1, and if R0>1, the drug-addiction equilibrium is globally asymptotically stable. Some numerical simulations are also carried out to illustrate our analytical results.

scholarly journals Global Stability of an Epidemic Model with Incomplete Treatment and Vaccination

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Hai-Feng Huo ◽  
Li-Xiang Feng
Keyword(s):  
Global Stability ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Incomplete Treatment ◽  
Disease Free ◽  
The Basic Reproduction Number

An epidemic model with incomplete treatment and vaccination for the newborns and susceptibles is constructed. We establish that the global dynamics are completely determined by the basic reproduction numberR0. IfR0≤1, then the disease-free equilibrium is globally asymptotically stable. IfR0>1, the endemic equilibrium is globally asymptotically stable. Some numerical simulations are also given to explain our conclusions.

scholarly journals Global Dynamics of a Discretized Heroin Epidemic Model with Time Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Xamxinur Abdurahman ◽  
Ling Zhang ◽  
Zhidong Teng
Keyword(s):  
Time Delay ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Nonstandard Finite Difference Scheme ◽  
Nonstandard Finite Difference ◽  
The Basic Reproduction Number

We derive a discretized heroin epidemic model with delay by applying a nonstandard finite difference scheme. We obtain positivity of the solution and existence of the unique endemic equilibrium. We show that heroin-using free equilibrium is globally asymptotically stable when the basic reproduction numberR0<1, and the heroin-using is permanent when the basic reproduction numberR0>1.

Global dynamics of a heroin epidemic model with age structure and nonlinear incidence

2016 ◽  
Vol 09 (03) ◽  
pp. 1650033 ◽  
Author(s):  
Junyuan Yang ◽  
Xiaoxia Li ◽  
Fengqin Zhang
Keyword(s):  
Age Structure ◽  
Drug Users ◽  
Reproduction Number ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Nonlinear Incidence Rate ◽  
Globally Asymptotically Stable ◽  
Nonlinear Incidence ◽  
Drug Free ◽  
Heroin Model

A heroin model with nonlinear incidence rate and age structure is investigated. The basic reproduction number is determined whether or not a heroin epidemic breaks out. By employing the Lyapunov functionals, the drug-free equilibrium is globally asymptotically stable if [Formula: see text]; while the drug spread equilibrium is also globally asymptotically stable if [Formula: see text]. Our results imply that improving detected rates and drawing up the efficient prevention play more important role than increasing the treatment for drug users.

GLOBAL DYNAMICS OF AN IN-HOST HIV-1 INFECTION MODEL WITH THE LONG-LIVED INFECTED CELLS AND FOUR INTRACELLULAR DELAYS

2012 ◽  
Vol 05 (06) ◽  
pp. 1250058 ◽  
Author(s):  
SHIFEI WANG ◽  
YICANG ZHOU
Keyword(s):  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Infection Model ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Infected Cells ◽  
Hiv 1 ◽  
Intracellular Delays ◽  
The Basic Reproduction Number

In this paper, we investigate global dynamics for an in-host HIV-1 infection model with the long-lived infected cells and four intracellular delays. Our model admits two possible equilibria, an uninfected equilibrium and infected equilibrium depending on the basic reproduction number. We derive that the global dynamics are completely determined by the values of the basic reproduction number: if the basic reproduction number is less than one, the uninfected equilibrium is globally asymptotically stable, and the virus is cleared; if the basic reproduction number is larger than one, then the infection persists, and the infected equilibrium is globally asymptotically stable.

GLOBAL DYNAMICS OF AN SVIR EPIDEMIOLOGICAL MODEL WITH INFECTION AGE AND NONLINEAR INCIDENCE

2017 ◽  
Vol 25 (03) ◽  
pp. 419-440
Author(s):  
ZHIPING WANG ◽  
RUI XU
Keyword(s):  
Steady State ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Epidemiological Model ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Nonlinear Incidence ◽  
Infection Age ◽  
The Basic Reproduction Number

In this paper, an SVIR epidemiological model with infection age (time elapsed since the infection) and nonlinear incidence is studied. In the model, in order to reflect the dependence of disease progress on the infection age, the infected individual is structured by the infection age, and transmission and removal rates are assumed to depend on the infection age. By analyzing corresponding characteristic equations, the local stability of each of steady states of the model is established. It is proved that the semi-flow generated by this system is asymptotically smooth, and if the basic reproduction number is greater than unity, the system is uniformly persistent. By using Lyapunov functional and LaSalle’s invariance principle, the global dynamics of the model is investigated. It is shown that if the basic reproduction number is less than unity, the disease-free steady state is globally asymptotically stable and hence the disease dies out; and if the basic reproduction number is greater than unity, the endemic steady state is globally asymptotically stable and the disease persists. Numerical simulations are carried out to illustrate the main analytic results.

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