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.

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.

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.

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.

scholarly journals Dynamical analysis of a generalized hepatitis B epidemic model and its dynamically consistent discrete model

2021 ◽  
Author(s):  
Manh Tuan Hoang
Keyword(s):  
Hepatitis B ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Discrete Model ◽  
Reproduction Number ◽  
Continuous Model ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Nsfd Scheme ◽  
The Basic Reproduction Number

The aim of this work is to study qualitative dynamical properties of a generalized hepatitis B epidemic model and its dynamically consistent discrete model. Positivity, boundedness, the basic reproduction number and asymptotic stability properties of the model are analyzed rigorously. By the Lyapunov stability theory and the Poincare-Bendixson theorem in combination with the Bendixson-Dulac criterion, we show that a disease-free equilibrium point is globally asymptotically stable if the basic reproduction number $\mathcal{R}_0 \leq 1$ and a disease-endemic equilibrium point is globally asymptotically stable whenever $\mathcal{R}_0 > 1$. Next, we apply the Mickens’ methodology to propose a dynamically consistent nonstandard finite difference (NSFD) scheme for the continuous model. By rigorously mathematical analyses, it is proved that the constructed NSFD scheme preserves essential mathematical features of the continuous model for all finite step sizes. Finally, numerical experiments are conducted to illustrate the theoretical findings and to demonstrate advantages of the NSFD scheme over standard ones. The obtained results in this work not only improve but also generalize some existing recognized works.

scholarly journals Global proprieties of a delayed epidemic model with partial susceptible protection

2021 ◽  
Vol 19 (1) ◽  
pp. 209-224
Author(s):  
Abdelheq Mezouaghi ◽  
◽  
Salih Djillali ◽  
Anwar Zeb ◽  
Kottakkaran Sooppy Nisar ◽  
...  
Keyword(s):  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Threshold Dynamics ◽  
Positive Equilibrium ◽  
Global Dynamics ◽  
Isolation Rate ◽  
Globally Asymptotically Stable ◽  
The Government ◽  
The Basic Reproduction Number

<abstract><p>In the case of an epidemic, the government (or population itself) can use protection for reducing the epidemic. This research investigates the global dynamics of a delayed epidemic model with partial susceptible protection. A threshold dynamics is obtained in terms of the basic reproduction number, where for $ R_0 &lt; 1 $ the infection will extinct from the population. But, for $ R_0 &gt; 1 $ it has been shown that the disease will persist, and the unique positive equilibrium is globally asymptotically stable. The principal purpose of this research is to determine a relation between the isolation rate and the basic reproduction number in such a way we can eliminate the infection from the population. Moreover, we will determine the minimal protection force to eliminate the infection for the population. A comparative analysis with the classical SIR model is provided. The results are supported by some numerical illustrations with their epidemiological relevance.</p></abstract>

scholarly journals On Oscillatory Pattern of Malaria Dynamics in a Population with Temporary Immunity

2007 ◽  
Vol 8 (3) ◽  
pp. 191-203 ◽  
Author(s):  
J. Tumwiine ◽  
J. Y. T. Mugisha ◽  
L. S. Luboobi
Keyword(s):  
Steady State ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Oscillatory Pattern ◽  
The Stability ◽  
Disease Free ◽  
The Basic Reproduction Number ◽  
Temporary Immunity

We use a model to study the dynamics of malaria in the human and mosquito population to explain the stability patterns of malaria. The model results show that the disease-free equilibrium is globally asymptotically stable and occurs whenever the basic reproduction number,R0is less than unity. We also note that whenR0>1, the disease-free equilibrium is unstable and the endemic equilibrium is stable. Numerical simulations show that recoveries and temporary immunity keep the populations at oscillation patterns and eventually converge to a steady state.

THE DYNAMICS AND THERAPEUTIC STRATEGIES OF A SEIS EPIDEMIC MODEL

2013 ◽  
Vol 06 (05) ◽  
pp. 1350029 ◽  
Author(s):  
XINZHU MENG ◽  
ZHITAO WU ◽  
TONGQIAN ZHANG
Keyword(s):  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Therapeutic Strategy ◽  
Reproduction Number ◽  
Asymptotically Stable ◽  
Epidemic Disease ◽  
Globally Asymptotically Stable ◽  
Effective Therapeutic Strategy ◽  
The Impact ◽  
The Basic Reproduction Number

Based on an epidemic model which Manvendra and Vinay [Mathematical model to simulate infections disease, VSRD-TNTJ3(2) (2012) 60–68] have proposed, we consider the dynamics and therapeutic strategy of a SEIS epidemic model with latent patients and active patients. First, the basic reproduction number is established by applying the method of the next generation matrix. By means of appropriate Lyapunov functions, it is proven that while the basic reproduction number 0 < R0 < 1, the disease-free equilibrium is globally asymptotically stable and the disease eliminates; and if the basic reproduction number R0 > 1, the endemic equilibrium is globally asymptotically stable and therefore the disease becomes endemic. Numerical investigations of their basin of attraction indicate that the locally stable equilibria are global attractors. Second, we consider the impact of treatment on epidemic disease and analytically determine the most effective therapeutic strategy. We conclude that the most effective therapeutic strategy consists of treating both the exposed and the infectious, while treating only the exposed is the least effective therapeutic strategy. Finally, numerical simulations are given to illustrate the effectiveness of the proposed results.

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.

STABILITY OF AN AGE-STRUCTURED SEIS EPIDEMIC MODEL WITH INFECTIVITY IN INCUBATIVE PERIOD

2010 ◽  
Vol 03 (03) ◽  
pp. 299-312 ◽  
Author(s):  
SHU-MIN GUO ◽  
XUE-ZHI LI ◽  
XIN-YU SONG
Keyword(s):  
Epidemic Model ◽  
Reproduction Number ◽  
Endemic Equilibrium ◽  
Stability Conditions ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Age Structured ◽  
The Stability ◽  
Disease Free ◽  
The Basic Reproduction Number

In this paper, an age-structured SEIS epidemic model with infectivity in incubative period is formulated and studied. The explicit expression of the basic reproduction number R0 is obtained. It is shown that the disease-free equilibrium is globally asymptotically stable if R0 < 1, at least one endemic equilibrium exists if R0 > 1. The stability conditions of endemic equilibrium are also given.

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