scholarly journals Global dynamics of alcoholism epidemic model with distributed delays

2021 ◽  
Vol 18 (6) ◽  
pp. 8245-8256
Author(s):  
Salih Djillali ◽  
◽  
Soufiane Bentout ◽  
Tarik Mohammed Touaoula ◽  
Abdessamad Tridane ◽  
...  
Keyword(s):  
Epidemic Model ◽  
Direct Method ◽  
Distributed Delays ◽  
Positive Equilibrium ◽  
Model Parameters ◽  
Threshold Condition ◽  
Global Dynamics ◽  
Lyapunov Direct Method ◽  
Globally Asymptotically Stable ◽  
The Social

<abstract><p>This paper aims to investigate the global dynamics of an alcoholism epidemic model with distributed delays. The main feature of this model is that it includes the effect of the social pressure as a factor of drinking. As a result, our global stability is obtained without a "basic reproduction number" nor threshold condition. Hence, we prove that the alcohol addiction will be always uniformly persistent in the population. This means that the investigated model has only one positive equilibrium, and it is globally asymptotically stable independent on the model parameters. This result is shown by proving that the unique equilibrium is locally stable, and the global attraction is shown using Lyapunov direct method.</p></abstract>

scholarly journals Global Dynamics of a Generalized SIRS Epidemic Model with Constant Immigration

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Qianqian Cui ◽  
Qinghui Du ◽  
Li Wang
Keyword(s):  
Epidemic Model ◽  
Reproduction Number ◽  
Direct Method ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Lyapunov Direct Method ◽  
Globally Asymptotically Stable ◽  
Sirs Epidemic Model ◽  
Dynamical Behaviors ◽  
Disease Case

In this paper, we discuss the global dynamics of a general susceptible-infected-recovered-susceptible (SIRS) epidemic model. By using LaSalle’s invariance principle and Lyapunov direct method, the global stability of equilibria is completely established. If there is no input of infectious individuals, the dynamical behaviors completely depend on the basic reproduction number. If there exists input of infectious individuals, the unique equilibrium of model is endemic equilibrium and is globally asymptotically stable. Once one place has imported a disease case, then it may become outbreak after that. Numerical simulations are presented to expound and complement our theoretical conclusions.

scholarly journals Global Dynamics and Sensitivity Analysis of a Vector-Host-Reservoir Model

2016 ◽  
Vol 21 (2) ◽  
pp. 120
Author(s):  
Ibrahim M. ELmojtaba ◽  
Santanu Biswas ◽  
Joydev Chattopadhyay
Keyword(s):  
Sensitivity Analysis ◽  
Positive Equilibrium ◽  
Model Parameters ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Animal Reservoir ◽  
Globally Asymptotically Stable ◽  
Numerical Result ◽  
Perform Sensitivity Analysis ◽  
Host Reservoir

The role of animal reservoir in the disease dynamics is not yet properly studied. In the present investigation a mathematical model of a vector-host-reservoir is proposed and analyzed to observe the global dynamics of the disease. We observe that the disease free equilibrium is globally asymptotically stable if the basic reproduction number ( ) is less than unity whereas unique positive equilibrium is globally asymptotically stable if and transcritical bifurcation occurs at . Our numerical result suggests that the biting rate plays an important role for the propagation of the disease and the recovery rate has not such important contribution towards eradication of the disease. We also perform sensitivity analysis of the model parameters and the results suggest that the death rate of reservoir may be used as a control parameter to eradicate the disease. 

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>

Global dynamics in a heroin epidemic model with different conscious stages and two distributed delays

2019 ◽  
Vol 12 (03) ◽  
pp. 1950038 ◽  
Author(s):  
Xamxinur Abdurahman ◽  
Zhidong Teng ◽  
Ling Zhang
Keyword(s):  
Epidemic Model ◽  
Drug Users ◽  
Distributed Delays ◽  
Threshold Dynamics ◽  
Uniform Persistence ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Number Formula ◽  
Drug Free

A heroin epidemic model with different conscious stages and distributed delays is constructed. The model allows for conscious drug users and unconscious drug users. The threshold dynamics of the model is established. It is shown that drug-free equilibrium is globally asymptotically stable when basic reproduction number [Formula: see text]; when [Formula: see text], the uniform persistence of the model is proved, and it is proved that the endemic equilibrium is globally asymptotically stable.

scholarly journals Stability and Hopf Bifurcation for a Delayed SIR Epidemic Model with Logistic Growth

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Yakui Xue ◽  
Tiantian Li
Keyword(s):  
Incidence Rate ◽  
Epidemic Model ◽  
Threshold Value ◽  
Endemic Equilibrium ◽  
Logistic Growth ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Sir Epidemic Model ◽  
Globally Asymptotically Stable ◽  
Sir Epidemic

We study a delayed SIR epidemic model and get the threshold value which determines the global dynamics and outcome of the disease. First of all, for anyτ, we show that the disease-free equilibrium is globally asymptotically stable; whenR0<1, the disease will die out. Directly afterwards, we prove that the endemic equilibrium is locally asymptotically stable for anyτ=0; whenR0>1, the disease will persist. However, for anyτ≠0, the existence conditions for Hopf bifurcations at the endemic equilibrium are obtained. Besides, we compare the delayed SIR epidemic model with nonlinear incidence rate to the one with bilinear incidence rate. At last, numerical simulations are performed to illustrate and verify the conclusions.

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.

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 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 Mathematical analysis and global dynamics for a time-delayed Chronic Myeloid Leukemia model with treatment

2020 ◽  
Vol 15 ◽  
pp. 68
Author(s):  
Nawal Kherbouche ◽  
Mohamed Helal ◽  
Abdennasser Chekroun ◽  
Abdelkader Lakmeche
Keyword(s):  
Myeloid Leukemia ◽  
Stem Cell Population ◽  
Positive Equilibrium ◽  
Uniform Persistence ◽  
Global Dynamics ◽  
Globally Asymptotically Stable ◽  
Hematopoietic Stem ◽  
Local Asymptotic Stability ◽  
Necessary And Sufficient ◽  
Delayed Model

In this paper, we investigate a time-delayed model describing the dynamics of the hematopoietic stem cell population with treatment. First, we give some property results of the solutions. Second, we analyze the asymptotic behavior of the model, and study the local asymptotic stability of each equilibrium: trivial and positive ones. Next, a necessary and sufficient condition is given for the trivial steady state to be globally asymptotically stable. Moreover, the uniform persistence is obtained in the case of instability. Finally, we prove that this system can exhibits a periodic solutions around the positive equilibrium through a Hopf bifurcation.

scholarly journals Global Dynamics Analysis of Homogeneous New Products Diffusion Model

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Shuping Li ◽  
Zhen Jin
Keyword(s):  
Decision Making ◽  
Interpersonal Communication ◽  
Product Information ◽  
New Products ◽  
Threshold Value ◽  
Positive Equilibrium ◽  
Global Dynamics ◽  
Globally Asymptotically Stable ◽  
Exchange Product ◽  
Parameter Condition

A mathematical model with stage structures is presented that incorporates the awareness stage and the decision-making stage; individuals exchange product information by two channels: mass media and interpersonal communication. When the persuasive advertisement is neglected in the decision-making stage, we find a threshold value about whether new products diffusion is successful or not. When the persuasive advertisement is considered, there must exist a positive equilibrium under some parameter condition; moreover, it must be globally asymptotically stable as long as it exists. Results show that the persuasive advertisement in the decision-making stage does not influence new products' successful diffusion, but the critical value that new products diffuse successfully decreases when the persuasive advertisement is considered. Some numerical simulations confirm our theoretical analysis and demonstrate the influence of parameters on the system. Corresponding to the analysis results, some diffusion strategies are provided.

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