scholarly journals Global Stability of a Delayed SIRI Epidemic Model with Nonlinear Incidence

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Amine Bernoussi ◽  
Abdelilah Kaddar ◽  
Said Asserda
Keyword(s):  
Global Stability ◽  
Numerical Simulations ◽  
Epidemic Model ◽  
Sufficient Conditions ◽  
Endemic Equilibrium ◽  
Global Dynamics ◽  
Nonlinear Incidence ◽  
Lasalle Theorem ◽  
Nonlinear Incidence Function ◽  
Disease Free

In this paper we propose the global dynamics of an SIRI epidemic model with latency and a general nonlinear incidence function. The model is based on the susceptible-infective-recovered (SIR) compartmental structure with relapse (SIRI). Sufficient conditions for the global stability of equilibria (the disease-free equilibrium and the endemic equilibrium) are obtained by means of Lyapunov-LaSalle theorem. Also some numerical simulations are given to illustrate this result.

scholarly journals Global Dynamics of an Epidemic Model with a Ratio-Dependent Nonlinear Incidence Rate

2009 ◽  
Vol 2009 ◽  
pp. 1-13 ◽  
Author(s):  
Sanling Yuan ◽  
Bo Li
Keyword(s):  
Incidence Rate ◽  
Computer Simulations ◽  
Epidemic Model ◽  
Global Analysis ◽  
Endemic Equilibrium ◽  
Global Dynamics ◽  
Nonlinear Incidence Rate ◽  
Nonlinear Incidence ◽  
Set Up ◽  
Disease Free

We study an epidemic model with a nonlinear incidence rate which describes the psychological effect of certain serious diseases on the community when the ratio of the number of infectives to that of the susceptibles is getting larger. The model has set up a challenging issue regarding its dynamics near the origin since it is not well defined there. By carrying out a global analysis of the model and studying the stabilities of the disease-free equilibrium and the endemic equilibrium, it is shown that either the number of infective individuals tends to zero as time evolves or the disease persists. Computer simulations are presented to illustrate the results.

scholarly journals Hopf Bifurcation of a Delayed Epidemic Model with Information Variable and Limited Medical Resources

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Caijuan Yan ◽  
Jianwen Jia
Keyword(s):  
Hopf Bifurcation ◽  
Epidemic Model ◽  
Sufficient Conditions ◽  
Endemic Equilibrium ◽  
Logistic Growth ◽  
Sir Epidemic Model ◽  
Medical Resources ◽  
Reproduction Ratio ◽  
Disease Free ◽  
Information Variable

We consider SIR epidemic model in which population growth is subject to logistic growth in absence of disease. We get the condition for Hopf bifurcation of a delayed epidemic model with information variable and limited medical resources. By analyzing the corresponding characteristic equations, the local stability of an endemic equilibrium and a disease-free equilibrium is discussed. If the basic reproduction ratioℛ0<1, we discuss the global asymptotical stability of the disease-free equilibrium by constructing a Lyapunov functional. Ifℛ0>1, we obtain sufficient conditions under which the endemic equilibriumE*of system is locally asymptotically stable. And we also have discussed the stability and direction of Hopf bifurcations. Numerical simulations are carried out to explain the mathematical 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.

Global stability of an SEIR epidemic model with vaccination

2016 ◽  
Vol 09 (06) ◽  
pp. 1650082 ◽  
Author(s):  
Lili Wang ◽  
Rui Xu
Keyword(s):  
Global Stability ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Sufficient Conditions ◽  
Endemic Equilibrium ◽  
Globally Asymptotically Stable ◽  
Compound Matrices ◽  
Comparison Arguments ◽  
The Basic Reproduction Number

In this paper, an SEIR epidemic model with vaccination is formulated. The results of our mathematical analysis indicate that the basic reproduction number plays an important role in studying the dynamics of the system. If the basic reproduction number is less than unity, it is shown that the disease-free equilibrium is globally asymptotically stable by comparison arguments. If it is greater than unity, the system is permanent and there is a unique endemic equilibrium. In this case, sufficient conditions are established to guarantee the global stability of the endemic equilibrium by the theory of the compound matrices. Numerical simulations are presented to illustrate the main results.

Dynamic Analysis of an SIRS Model with Nonlinear Incidence Rate

2012 ◽  
Vol 155-156 ◽  
pp. 23-26
Author(s):  
Jun Hong Li ◽  
Ning Cui ◽  
Liang Cui ◽  
Cai Juan Li
Keyword(s):  
Hopf Bifurcation ◽  
Dynamic Analysis ◽  
Numerical Simulations ◽  
Epidemic Model ◽  
Global Dynamics ◽  
Nonlinear Incidence Rate ◽  
Dulac Function ◽  
Sirs Epidemic Model ◽  
Sirs Model ◽  
Disease Free

In this paper, we study the global dynamics of an SIRS epidemic model with nonlinear inci- dence rate. By means of Dulac function and Poincare-Bendixson Theorem, we proved the global asy- mptotical stable results of the disease-free equilibrium. It is then obtained the model undergoes Hopf bifurcation and existence of one limit cycle. Some numerical simulations are given to illustrate the an- alytical results.

A stochastic vaccinated epidemic model incorporating Lévy processes with a general awareness-induced incidence

2021 ◽  
pp. 2150044
Author(s):  
Khadija Akdim ◽  
Adil Ez-Zetouni ◽  
Mehdi Zahid
Keyword(s):  
Numerical Simulations ◽  
Positive Solution ◽  
Dynamic Behavior ◽  
Epidemic Model ◽  
Equilibrium Points ◽  
Sufficient Conditions ◽  
Lévy Noise ◽  
Global Positive Solution ◽  
Levy Noise ◽  
Disease Free

In this paper, we investigate a stochastic vaccinated epidemic model with a general awareness-induced incidence perturbed by Lévy noise. First, we show that this model has a unique global positive solution. Therefore, we establish the dynamic behavior of the solution around both disease-free and endemic equilibrium points. Furthermore, when [Formula: see text], we give sufficient conditions for the existence of an ergodic stationary distribution to the model when the jump part in the Lévy noise is null. Finally, we present some examples to illustrate the analytical results by numerical simulations.

Global stability and vaccination of an SEIVR epidemic model with saturated incidence rate

2016 ◽  
Vol 09 (05) ◽  
pp. 1650068 ◽  
Author(s):  
Muhammad Altaf Khan ◽  
Yasir Khan ◽  
Sehra Khan ◽  
Saeed Islam
Keyword(s):  
Global Stability ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Endemic Equilibrium ◽  
Asymptotically Stable ◽  
Saturated Incidence Rate ◽  
Saturated Incidence ◽  
Disease Free ◽  
The Basic Reproduction Number

This study considers SEIVR epidemic model with generalized nonlinear saturated incidence rate in the host population horizontally to estimate local and global equilibriums. By using the Routh–Hurwitz criteria, it is shown that if the basic reproduction number [Formula: see text], the disease-free equilibrium is locally asymptotically stable. When the basic reproduction number exceeds the unity, then the endemic equilibrium exists and is stable locally asymptotically. The system is globally asymptotically stable about the disease-free equilibrium if [Formula: see text]. The geometric approach is used to present the global stability of the endemic equilibrium. For [Formula: see text], the endemic equilibrium is stable globally asymptotically. Finally, the numerical results are presented to justify the mathematical 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.

Analysis of a delayed epidemic model with non-monotonic incidence rate and vertical transmission

2014 ◽  
Vol 07 (04) ◽  
pp. 1450041
Author(s):  
Jinhu Xu ◽  
Wenxiong Xu ◽  
Yicang Zhou
Keyword(s):  
Vertical Transmission ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Sufficient Conditions ◽  
Endemic Equilibrium ◽  
Globally Asymptotically Stable ◽  
Comparison Arguments ◽  
Disease Free ◽  
The Basic Reproduction Number

A delayed SEIR epidemic model with vertical transmission and non-monotonic incidence is formulated. The equilibria and the threshold of the model have been determined on the bases of the basic reproduction number. The local stability of disease-free equilibrium and endemic equilibrium is established by analyzing the corresponding characteristic equations. By comparison arguments, it is proved that, if R0 < 1, the disease-free equilibrium is globally asymptotically stable. Whereas, the disease-free equilibrium is unstable if R0 > 1. Moreover, we show that the disease is permanent if the basic reproduction number is greater than one. Furthermore, the sufficient conditions are obtained for the global asymptotic stability of the endemic equilibrium when R0 > 1.

scholarly journals Rich dynamics of an SIR epidemic model

2010 ◽  
Vol 15 (1) ◽  
pp. 71-81 ◽  
Author(s):  
S. Pathak ◽  
A. Maiti ◽  
G. P. Samanta
Keyword(s):  
Numerical Simulations ◽  
Epidemic Model ◽  
Transmission Function ◽  
Endemic Equilibrium ◽  
Sir Epidemic Model ◽  
Sir Epidemic ◽  
The Stability ◽  
Disease Free

This paper aims to study an SIR epidemic model with an asymptotically homogeneous transmission function. The stability of the disease-free and the endemic equilibrium is addressed. Numerical simulations are carried out. Implications of our analytical and numerical findings are discussed critically.

Sign in / Sign up

Export Citation Format

Share Document