Global threshold dynamics of an age structured alcoholism model

2021 ◽  
pp. 2150013
Author(s):  
Soufiane Bentout ◽  
Salih Djilali ◽  
Abdenasser Chekroun
Keyword(s):  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Threshold Dynamics ◽  
Global Behavior ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Global Threshold ◽  
Age Structured ◽  
The Basic Reproduction Number ◽  
Proper Strategy

We consider in this research an age-structured alcoholism model. The global behavior of the model is investigated. It is proved that the system has a threshold dynamics in terms of the basic reproduction number (BRN), where we obtained that alcohol-free equilibrium (AFE) is globally asymptotically stable (GAS) in the case [Formula: see text], but for [Formula: see text] we found that the system persists and the nontrivial equilibrium (EE) is GAS. Furthermore, the effects of the susceptible drinkers rate and the repulse rate of the recovers to alcoholics are investigated, which allow us to provide a proper strategy for reducing the spread of alcohol use in the studied populations. The obtained mathematical results are tested numerically next to its biological relevance.

An age-structured epidemic model with waning immunity and general nonlinear incidence rate

2018 ◽  
Vol 11 (05) ◽  
pp. 1850069 ◽  
Author(s):  
Xia Wang ◽  
Ying Zhang ◽  
Xinyu Song
Keyword(s):  
Steady State ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Threshold Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Waning Immunity ◽  
Age Structured ◽  
The Basic Reproduction Number

In this paper, a susceptible-vaccinated-exposed-infectious-recovered epidemic model with waning immunity and continuous age structures in vaccinated, exposed and infectious classes has been formulated. By using the Fluctuation lemma and the approach of Lyapunov functionals, we establish a threshold dynamics completely determined by the basic reproduction number. When the basic reproduction number is less than one, the disease-free steady state is globally asymptotically stable, and otherwise the endemic steady state is globally asymptotically stable.

ANALYSIS OF AN AGE-STRUCTURED HIV-1 INFECTION MODEL WITH LOGISTIC TARGET CELL GROWTH

2020 ◽  
Vol 28 (04) ◽  
pp. 927-944
Author(s):  
HUIJUAN LIU ◽  
FEI XU ◽  
JIA-FANG ZHANG
Keyword(s):  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Logistic Growth ◽  
Infection Model ◽  
Target Cells ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Age Structured ◽  
Hiv 1 ◽  
The Basic Reproduction Number

In this work, we construct an age-structured HIV-1 infection model to investigate the interplay between [Formula: see text] cells and viruses. In our model, we assume that the variations in the death rate of productively infected [Formula: see text] cells and the production rate of virus in infected cells are all age-dependent, and the target cells follow logistic growth. We perform mathematical analysis and prove the persistence of the semi-flow of the system. We calculate the basic reproduction number and prove the local and global stability of the steady states. We show that if the basic reproduction number is less than one, the disease-free equilibrium is globally asymptotically stable, and if the basic reproduction number is greater than one, the infected steady state is locally asymptotically stable.

Dynamics of a dengue epidemic model with class-age structure

2017 ◽  
Vol 10 (08) ◽  
pp. 1750109 ◽  
Author(s):  
Wen-Jing Feng ◽  
Li-Ming Cai ◽  
Kaihui Liu
Keyword(s):  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Interaction Model ◽  
Threshold Value ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Dengue Disease ◽  
Age Structured ◽  
The Basic Reproduction Number

We introduce the class-age-dependent rates of the infected and vaccinated class in the compartmental model of dengue transmission. An age-structured host-vector interaction model incorporating vaccination effects is formulated and analyzed for the spread of dengue. Moreover, the basic reproduction number is derived, which serves as a threshold value determining the stability of the equilibrium points. By constructing suitable Lyapunov functional, the global asymptotic stability of the equilibria of the model is established in terms of the basic reproduction number. In particular, the disease-free equilibrium of the model is globally asymptotically stable if the basic reproduction number is less than one, while the disease persists and the unique endemic equilibrium is globally asymptotically stable if the basic reproduction number is greater than one. The analysis of our model indicates that our model is realistic to give a hint to control the transmission of dengue. Furthermore, it follows from the formulation of the infection-free equilibrium of susceptible humans [Formula: see text] and the basic reproduction number [Formula: see text] that both of them are decreasing with respect to the vaccination parameter [Formula: see text], which indicates that appropriate vaccinating program may contribute to prevent the transmission of Dengue disease.

scholarly journals Threshold Dynamics of a Huanglongbing Model with Logistic Growth in Periodic Environments

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Jianping Wang ◽  
Shujing Gao ◽  
Yueli Luo ◽  
Dehui Xie
Keyword(s):  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Logistic Growth ◽  
Threshold Dynamics ◽  
Global Dynamics ◽  
Globally Asymptotically Stable ◽  
Linear Integral ◽  
The Impact ◽  
Disease Free ◽  
The Basic Reproduction Number

We analyze the impact of seasonal activity of psyllid on the dynamics of Huanglongbing (HLB) infection. A new model about HLB transmission with Logistic growth in psyllid insect vectors and periodic coefficients has been investigated. It is shown that the global dynamics are determined by the basic reproduction numberR0which is defined through the spectral radius of a linear integral operator. IfR0< 1, then the disease-free periodic solution is globally asymptotically stable and ifR0> 1, then the disease persists. Numerical values of parameters of the model are evaluated taken from the literatures. Furthermore, numerical simulations support our analytical conclusions and the sensitive analysis on the basic reproduction number to the changes of average and amplitude values of the recruitment function of citrus are shown. Finally, some useful comments on controlling the transmission of HLB are given.

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 Dynamics Analysis of a Viral Infection Model with a General Standard Incidence Rate

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Yu Ji ◽  
Muxuan Zheng
Keyword(s):  
Viral Infection ◽  
Incidence Rate ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Infection Model ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Reproduction Numbers ◽  
The Basic Reproduction Number ◽  
General Standard

The basic viral infection models, proposed by Nowak et al. and Perelson et al., respectively, have been widely used to describe viral infection such as HBV and HIV infection. However, the basic reproduction numbers of the two models are proportional to the number of total cells of the host's organ prior to the infection, which seems not to be reasonable. In this paper, we formulate an amended model with a general standard incidence rate. The basic reproduction number of the amended model is independent of total cells of the host’s organ. When the basic reproduction numberR0<1, the infection-free equilibrium is globally asymptotically stable and the virus is cleared. Moreover, ifR0>1, then the endemic equilibrium is globally asymptotically stable and the virus persists in the host.

Dynamics of a delayed epidemic model with varying immunity period and nonlinear transmission

2015 ◽  
Vol 08 (02) ◽  
pp. 1550027 ◽  
Author(s):  
Aadil Lahrouz
Keyword(s):  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Sufficient Conditions ◽  
Distributed Delay ◽  
Incidence Rates ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Nonlinear Incidence ◽  
The Basic Reproduction Number

An epidemic model with a class of nonlinear incidence rates and distributed delay is analyzed. The nonlinear incidence is used to describe the saturated or the psychological effect of certain serious epidemics on the community when the number of infectives is getting larger. The distributed delay is derived to describe the dynamics of infectious diseases with varying immunity. Lyapunov functionals are used to show that the disease-free equilibrium state is globally asymptotically stable when the basic reproduction number is less than or equal to one. Moreover, it is shown that the disease is permanent if the basic reproduction number is greater than one. Furthermore, the sufficient conditions under which the endemic equilibrium is locally and globally asymptotically stable are obtained.

The impact of media coverage with a piecewise transmission rate in the spread of diseases

2016 ◽  
Vol 10 (01) ◽  
pp. 1750003
Author(s):  
Maoxing Liu ◽  
Lixia Zuo
Keyword(s):  
Basic Reproduction Number ◽  
Media Coverage ◽  
Reproduction Number ◽  
Transmission Rate ◽  
Three Dimensional ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Contact Transmission ◽  
The Impact ◽  
The Basic Reproduction Number

A three-dimensional compartmental model with media coverage is proposed to describe the real characteristics of its impact in the spread of infectious diseases in a given region. A piecewise continuous transmission rate is introduced to describe that media coverage exhibits its effect only when the number of the infected exceeds a certain critical level. Further, it is assumed that the impact of media coverage on the contact transmission is described by an exponential decreasing factor. Stability analysis of the model shows that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number is less than unity. On the other hand, when the basic reproduction number is greater than unity and media coverage impact is sufficiently small, a unique endemic equilibrium exists, which is globally asymptotically stable.

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