Stability and Bogdanov-Takens Bifurcation of an SIS Epidemic Model with Saturated Treatment Function

Stability and Bogdanov-Takens Bifurcation of an SIS Epidemic Model with Saturated Treatment Function

This paper introduces the global dynamics of an SIS model with bilinear incidence rate and saturated treatment function. The treatment function is a continuous and differential function which shows the effect of delayed treatment when the rate of treatment is lower and the number of infected individ...

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Main Authors: Yanju Xiao, Weipeng Zhang, Guifeng Deng, Zhehua Liu
Format: Article
Language:English
Published: Hindawi Limited 2015-01-01
Series:Mathematical Problems in Engineering
Online Access:http://dx.doi.org/10.1155/2015/745732
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spelling doaj-d7b3d384670c40b7b8231bce67b38b562020-11-24T22:28:18ZengHindawi LimitedMathematical Problems in Engineering1024-123X1563-51472015-01-01201510.1155/2015/745732745732Stability and Bogdanov-Takens Bifurcation of an SIS Epidemic Model with Saturated Treatment FunctionYanju Xiao0Weipeng Zhang1Guifeng Deng2Zhehua Liu3School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, ChinaSchool of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, ChinaSchool of Mathematics and Information, Shanghai Lixin University of Commerce, Shanghai 201620, ChinaSchool of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, ChinaThis paper introduces the global dynamics of an SIS model with bilinear incidence rate and saturated treatment function. The treatment function is a continuous and differential function which shows the effect of delayed treatment when the rate of treatment is lower and the number of infected individuals is getting larger. Sufficient conditions for the existence and global asymptotic stability of the disease-free and endemic equilibria are given in this paper. The first Lyapunov coefficient is computed to determine various types of Hopf bifurcation, such as subcritical or supercritical. By some complex algebra, the Bogdanov-Takens normal form and the three types of bifurcation curves are derived. Finally, mathematical analysis and numerical simulations are given to support our theoretical results.http://dx.doi.org/10.1155/2015/745732
collection DOAJ
language English
format Article
sources DOAJ
author Yanju Xiao
Weipeng Zhang
Guifeng Deng
Zhehua Liu
spellingShingle Yanju Xiao
Weipeng Zhang
Guifeng Deng
Zhehua Liu
Stability and Bogdanov-Takens Bifurcation of an SIS Epidemic Model with Saturated Treatment Function
Mathematical Problems in Engineering
author_facet Yanju Xiao
Weipeng Zhang
Guifeng Deng
Zhehua Liu
author_sort Yanju Xiao
title Stability and Bogdanov-Takens Bifurcation of an SIS Epidemic Model with Saturated Treatment Function
title_short Stability and Bogdanov-Takens Bifurcation of an SIS Epidemic Model with Saturated Treatment Function
title_full Stability and Bogdanov-Takens Bifurcation of an SIS Epidemic Model with Saturated Treatment Function
title_fullStr Stability and Bogdanov-Takens Bifurcation of an SIS Epidemic Model with Saturated Treatment Function
title_full_unstemmed Stability and Bogdanov-Takens Bifurcation of an SIS Epidemic Model with Saturated Treatment Function
title_sort stability and bogdanov-takens bifurcation of an sis epidemic model with saturated treatment function
publisher Hindawi Limited
series Mathematical Problems in Engineering
issn 1024-123X
1563-5147
publishDate 2015-01-01
description This paper introduces the global dynamics of an SIS model with bilinear incidence rate and saturated treatment function. The treatment function is a continuous and differential function which shows the effect of delayed treatment when the rate of treatment is lower and the number of infected individuals is getting larger. Sufficient conditions for the existence and global asymptotic stability of the disease-free and endemic equilibria are given in this paper. The first Lyapunov coefficient is computed to determine various types of Hopf bifurcation, such as subcritical or supercritical. By some complex algebra, the Bogdanov-Takens normal form and the three types of bifurcation curves are derived. Finally, mathematical analysis and numerical simulations are given to support our theoretical results.
url http://dx.doi.org/10.1155/2015/745732
work_keys_str_mv AT yanjuxiao stabilityandbogdanovtakensbifurcationofansisepidemicmodelwithsaturatedtreatmentfunction
AT weipengzhang stabilityandbogdanovtakensbifurcationofansisepidemicmodelwithsaturatedtreatmentfunction
AT guifengdeng stabilityandbogdanovtakensbifurcationofansisepidemicmodelwithsaturatedtreatmentfunction
AT zhehualiu stabilityandbogdanovtakensbifurcationofansisepidemicmodelwithsaturatedtreatmentfunction
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