Bifurcations of an sirs epidemic model with nonlinear incidence rate

Zhixing Hu, Ping Bi, Wanbiao Ma, Shigui Ruan

Research output: Contribution to journalArticle

31 Citations (Scopus)

Abstract

The main purpose of this paper is to explore the dynamics of an epidemic model with a general nonlinear incidence βSip (1 +αI q). The ex-istence and stability of multiple endemic equilibria of the epidemic model are analyzed. Local bifurcation theory is applied to explore the rich dynamical be-havior of the model. Normal forms of the model are derived for different types of bifurcations, including Hopf and Bogdanov-Takens bifurcations. Concretely speaking, the first Lyapunov coefficient is computed to determine various types of Hopf bifurcations. Next, with the help of the Bogdanov-Takens normal form, a family of homoclinic orbits is arising when a Hopf and a saddle-node bifur- cation merge. Finally, some numerical results and simulations are presented to illustrate these theoretical results.

Original languageEnglish (US)
Pages (from-to)93-112
Number of pages20
JournalDiscrete and Continuous Dynamical Systems - Series B
Volume15
Issue number1
DOIs
StatePublished - Jan 2011

Fingerprint

Nonlinear Incidence Rate
SIR Epidemic Model
Epidemic Model
Hopf Bifurcation
Normal Form
Bifurcation
Nonlinear Incidence
Bogdanov-Takens Bifurcation
Multiple Equilibria
Local Bifurcations
Saddle-node Bifurcation
Endemic Equilibrium
Hopf bifurcation
Bifurcation Theory
Homoclinic Orbit
Lyapunov
Numerical Simulation
Numerical Results
Bifurcation (mathematics)
Coefficient

Keywords

  • Bogdanov-Takens bifurcation
  • Hopf bifur-cation
  • Nonlinear incidence rate
  • SIRS epidemic model
  • Stability

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

Cite this

Bifurcations of an sirs epidemic model with nonlinear incidence rate. / Hu, Zhixing; Bi, Ping; Ma, Wanbiao; Ruan, Shigui.

In: Discrete and Continuous Dynamical Systems - Series B, Vol. 15, No. 1, 01.2011, p. 93-112.

Research output: Contribution to journalArticle

@article{964939b708fc4845bd0f6f5332836be8,
title = "Bifurcations of an sirs epidemic model with nonlinear incidence rate",
abstract = "The main purpose of this paper is to explore the dynamics of an epidemic model with a general nonlinear incidence βSip (1 +αI q). The ex-istence and stability of multiple endemic equilibria of the epidemic model are analyzed. Local bifurcation theory is applied to explore the rich dynamical be-havior of the model. Normal forms of the model are derived for different types of bifurcations, including Hopf and Bogdanov-Takens bifurcations. Concretely speaking, the first Lyapunov coefficient is computed to determine various types of Hopf bifurcations. Next, with the help of the Bogdanov-Takens normal form, a family of homoclinic orbits is arising when a Hopf and a saddle-node bifur- cation merge. Finally, some numerical results and simulations are presented to illustrate these theoretical results.",
keywords = "Bogdanov-Takens bifurcation, Hopf bifur-cation, Nonlinear incidence rate, SIRS epidemic model, Stability",
author = "Zhixing Hu and Ping Bi and Wanbiao Ma and Shigui Ruan",
year = "2011",
month = "1",
doi = "10.3934/dcdsb.2011.15.93",
language = "English (US)",
volume = "15",
pages = "93--112",
journal = "Discrete and Continuous Dynamical Systems - Series B",
issn = "1531-3492",
publisher = "Southwest Missouri State University",
number = "1",

}

TY - JOUR

T1 - Bifurcations of an sirs epidemic model with nonlinear incidence rate

AU - Hu, Zhixing

AU - Bi, Ping

AU - Ma, Wanbiao

AU - Ruan, Shigui

PY - 2011/1

Y1 - 2011/1

N2 - The main purpose of this paper is to explore the dynamics of an epidemic model with a general nonlinear incidence βSip (1 +αI q). The ex-istence and stability of multiple endemic equilibria of the epidemic model are analyzed. Local bifurcation theory is applied to explore the rich dynamical be-havior of the model. Normal forms of the model are derived for different types of bifurcations, including Hopf and Bogdanov-Takens bifurcations. Concretely speaking, the first Lyapunov coefficient is computed to determine various types of Hopf bifurcations. Next, with the help of the Bogdanov-Takens normal form, a family of homoclinic orbits is arising when a Hopf and a saddle-node bifur- cation merge. Finally, some numerical results and simulations are presented to illustrate these theoretical results.

AB - The main purpose of this paper is to explore the dynamics of an epidemic model with a general nonlinear incidence βSip (1 +αI q). The ex-istence and stability of multiple endemic equilibria of the epidemic model are analyzed. Local bifurcation theory is applied to explore the rich dynamical be-havior of the model. Normal forms of the model are derived for different types of bifurcations, including Hopf and Bogdanov-Takens bifurcations. Concretely speaking, the first Lyapunov coefficient is computed to determine various types of Hopf bifurcations. Next, with the help of the Bogdanov-Takens normal form, a family of homoclinic orbits is arising when a Hopf and a saddle-node bifur- cation merge. Finally, some numerical results and simulations are presented to illustrate these theoretical results.

KW - Bogdanov-Takens bifurcation

KW - Hopf bifur-cation

KW - Nonlinear incidence rate

KW - SIRS epidemic model

KW - Stability

UR - http://www.scopus.com/inward/record.url?scp=78651226178&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78651226178&partnerID=8YFLogxK

U2 - 10.3934/dcdsb.2011.15.93

DO - 10.3934/dcdsb.2011.15.93

M3 - Article

VL - 15

SP - 93

EP - 112

JO - Discrete and Continuous Dynamical Systems - Series B

JF - Discrete and Continuous Dynamical Systems - Series B

SN - 1531-3492

IS - 1

ER -