STABILITY ANALYSIS OF A DELAYED SIRS EPIDEMIC MODEL WITH VACCINATION AND NONLINEAR INCIDENCE

2012 ◽  
Vol 05 (06) ◽  
pp. 1250050 ◽  
Author(s):  
XIAOHONG TIAN
Keyword(s):  
Epidemic Model ◽  
Local Stability ◽  
Lyapunov Functional ◽  
Sufficient Conditions ◽  
Threshold Value ◽  
Iteration Scheme ◽  
Endemic Equilibrium ◽  
Globally Asymptotically Stable ◽  
Sirs Epidemic Model ◽  
Disease Free

In this paper, an SIRS epidemic model with time delay and vaccination is investigated. By analyzing the corresponding characteristic equation, the local stability of disease-free equilibrium of the model is established. By constructing Lyapunov functional, sufficient conditions are established for the local stability of an endemic equilibrium of the model. Further, a threshold value is obtained. By using comparison arguments, it is proved when the threshold value is less than unity, the disease-free equilibrium is globally asymptotically stable. When the threshold value is greater than unity, by using an iteration scheme and by constructing appropriate Lyapunov functional, respectively, sufficient conditions are derived for the global stability of the endemic equilibrium of the model. Numerical simulations are carried out to illustrate the theoretical results.

scholarly journals An SIRS Epidemic Model Supervised by a Control System for Vaccination and Treatment Actions Which Involve First-Order Dynamics and Vaccination of Newborns

Mathematics ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 36
Author(s):  
Santiago Alonso-Quesada ◽  
Manuel De la Sen ◽  
Raúl Nistal
Keyword(s):  
Control System ◽  
Equilibrium Point ◽  
Epidemic Model ◽  
Equilibrium Points ◽  
Threshold Value ◽  
Design Parameters ◽  
Globally Asymptotically Stable ◽  
Sirs Epidemic Model ◽  
Existence And Stability ◽  
Disease Free

This paper analyses an SIRS epidemic model with the vaccination of susceptible individuals and treatment of infectious ones. Both actions are governed by a designed control system whose inputs are the subpopulations of the epidemic model. In addition, the vaccination of a proportion of newborns is considered. The control reproduction number Rc of the controlled epidemic model is calculated, and its influence in the existence and stability of equilibrium points is studied. If such a number is smaller than a threshold value Rc, then the model has a unique equilibrium point: the so-called disease-free equilibrium point at which there are not infectious individuals. Furthermore, such an equilibrium point is locally and globally asymptotically stable. On the contrary, if Rc>Rc, then the model has two equilibrium points: the referred disease-free one, which is unstable, and an endemic one at which there are infectious individuals. The proposed control strategy provides several free-design parameters that influence both values Rc and Rc. Then, such parameters can be appropriately adjusted for guaranteeing the non-existence of the endemic equilibrium point and, in this way, eradicating the persistence of the infectious disease.

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 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 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.

A VECTOR-HOST EPIDEMIC MODEL WITH HOST MIGRATION

2011 ◽  
Vol 04 (01) ◽  
pp. 93-108
Author(s):  
QINGKAI KONG ◽  
ZHIPENG QIU ◽  
YUN ZOU
Keyword(s):  
Epidemic Model ◽  
Reproduction Number ◽  
Threshold Value ◽  
Endemic Equilibrium ◽  
Uniform Persistence ◽  
Asymptotically Stable ◽  
Host Disease ◽  
Disease Free

The host migration is one of the important elements that cause the worldwide diffusion and outbreak of many vector-host diseases. In this paper, we formulate a patchy model to investigate the effect of host migration between two patches on the spread of a vector-host disease. The results of the paper show that the reproduction number R0 is a threshold value that determines the uniform persistence and extinction of the disease. If the reproduction number R0 < 1 the disease free equilibrium (DFE) is locally asymptotically stable. If the reproduction number R0 > 1 then the DFE is unstable and the system is uniformly persistent. It is also shown that a unique endemic equilibrium, which exists when R0 > 1, is locally stable if both regions are identical.

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.

SVIR epidemic model with age structure in susceptibility, vaccination effects and relapse

2017 ◽  
Vol 82 (5) ◽  
pp. 945-970 ◽  
Author(s):  
Jinliang Wang ◽  
Min Guo ◽  
Shengqiang Liu
Keyword(s):  
Age Structure ◽  
Epidemic Model ◽  
Lyapunov Functional ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Combined Effects ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
The Stability ◽  
Disease Free

Abstract An SVIR epidemic model with continuous age structure in the susceptibility, vaccination effects and relapse is proposed. The asymptotic smoothness, existence of a global attractor, the stability of equilibria and persistence are addressed. It is shown that if the basic reproductive number $\Re_0&lt;1$, then the disease-free equilibrium is globally asymptotically stable. If $\Re_0&gt;1$, the disease is uniformly persistent, and a Lyapunov functional is used to show that the unique endemic equilibrium is globally asymptotically stable. Combined effects of susceptibility age, vaccination age and relapse age on the basic reproductive number are discussed.

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.

Backward Bifurcation in a Fractional-Order SIRS Epidemic Model with a Nonlinear Incidence Rate

2016 ◽  
Vol 17 (7-8) ◽  
pp. 401-412 ◽  
Author(s):  
A. M. Yousef ◽  
S. M. Salman
Keyword(s):  
Incidence Rate ◽  
Fractional Order ◽  
Epidemic Model ◽  
Local Stability ◽  
Host Population ◽  
Backward Bifurcation ◽  
Nonlinear Incidence Rate ◽  
Nonlinear Incidence ◽  
Sirs Epidemic Model ◽  
Disease Free

Abstract:In this work we study a fractional-order susceptible-infective-recovered-susceptible (SIRS) epidemic model with a nonlinear incidence rate. The incidence is assumed to be a convex function with respect to the infective class of a host population. Local and uniform stability analysis of the disease-free equilibrium is investigated. The conditions for the existence of endemic equilibria (EE) are given. Local stability of the EE is discussed. Conditions for the existence of Hopf bifurcation at the EE are given. Most importantly, conditions ensuring that the system exhibits backward bifurcation are provided. Numerical simulations are performed to verify the correctness of results obtained analytically.

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