American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Dynamics of the density dependent and nonautonomous predator-prey system with Beddington-DeAngelis functional response
  • DCDS-B Home
  • This Issue
  • Next Article
    A detailed balanced reaction network is sufficient but not necessary for its Markov chain to be detailed balanced
June  2015, 20(4): 1107-1116. doi: 10.3934/dcdsb.2015.20.1107

Codimension 3 B-T bifurcations in an epidemic model with a nonlinear incidence

Chengzhi Li 1, , Jianquan Li 2, and  Zhien Ma 3,

1. 

School of Mathematical Sciences and LMAM, Peking University, Beijing, 100871
2. 

Faculty of Science, Air Force Engineering University, Xi'an 710051
3. 

School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China

Received  April 2014 Revised  September 2014 Published  February 2015

It was shown in [11] that in an epidemic model with a nonlinear incidence and two compartments some complex dynamics can appear, such as the backward bifurcation, codimension 1 Hopf bifurcation and codimension 2 Bogdanov-Takens bifurcation. In this paper we prove that for the same model the codimension of Bogdanov-Takens bifurcation can be 3 and is at most 3. Hence, more complex new phenomena, such as codimension 2 Hopf bifurcation, codimension 2 homoclinic bifurcation and semi-stable limit cycle bifurcation, exhibit. Especially, the system can have and at most have 2 limit cycles near the positive singularity.
Keywords: Bogdanov-Takens bifurcation, Epidemic model, codimension three..
Mathematics Subject Classification: Primary: 34C07, 34C08; Secondary: 37G1.
Citation: Chengzhi Li, Jianquan Li, Zhien Ma. Codimension 3 B-T bifurcations in an epidemic model with a nonlinear incidence. Discrete and Continuous Dynamical Systems - B, 2015, 20 (4) : 1107-1116. doi: 10.3934/dcdsb.2015.20.1107
References:
[1]

M. E. Alexander and S. M. Moghadas, Periodicity in an epidemic model with a generalized non-linear incidence, Math. Biosci., 189 (2004), 75-96. doi: 10.1016/j.mbs.2004.01.003.

[2]

F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemics, Springer-Verlag, New York, 2000.

[3]

L. Cai, G. Chen and D. Xiao, Multiparametric bifurcations of an epidemiological model with strong Allee effect, J. Math. Biol., 67 (2013), 185-215. doi: 10.1007/s00285-012-0546-5.

[4]

S.-N. Chow, C. Li and D. Wang, Normal Forms and Bifurcations of Planar Vector Fields, Cambridge University Press, 1994. doi: 10.1017/CBO9780511665639.

[5]

C. Christopher and C. Li, Limit Cycles of Differential Equations, Birkhäuser Verlag, 2007.

[6]

J. Cui,, X. Mu and H. Wan, Saturation recovery leads to multiple endemic equilibria and backward bifurcation, J. Theoret. Biol., 254 (2008), 275-283. doi: 10.1016/j.jtbi.2008.05.015.

[7]

F. Dumortier, R. Roussarie and J. Sotomayor, Generic 3-parameter family of vector feilds on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3, Ergod. Theor. & Dyn. Sys., 7 (1987), 375-413. doi: 10.1017/S0143385700004119.

[8]

H. W. Hethcote, Mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653. doi: 10.1137/S0036144500371907.

[9]

J. Huang, Y. Gong and S. Ruan, Bifurcation analysis in a predator-prey model with constant-yield predator harvesting, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2101-2121. doi: 10.3934/dcdsb.2013.18.2101.

[10]

C. Li and H. Zhu, Canard cycles for predator-prey systems with Holling types of functional response, J. Differential Equations, 254 (2013), 879-910. doi: 10.1016/j.jde.2012.10.003.

[11]

J. Li, Y. Zhou, J. Wu and Z. Ma, Complex dynamics of a simple epidemic model with a nonlinear incidence, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 161-173. doi: 10.3934/dcdsb.2007.8.161.

[12]

W. Liu, H. W. Hethcote and S. A. Levin, Dynamical behavior of epidemiological model with nonlinear incidence rates, J. Math. Biol., 25 (1987), 359-380. doi: 10.1007/BF00277162.

[13]

W. Liu, S. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol., 23 (1986), 187-204. doi: 10.1007/BF00276956.

[14]

Z. Ma and J. Li, Dynamical Modeling and Analysis of Epidemics, World Scientific, Singapore, 2009. doi: 10.1142/9789812797506.

[15]

S. Ruan and W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Differential Equations, 188 (2003), 135-163. doi: 10.1016/S0022-0396(02)00089-X.

[16]

Y. Tang, D. Huang, S. Ruan and W. Zhang, Coexistence of limit cycles and homoclinic loops in an SIRS model with nonlinear incidence rate, SIAM J. Appl. Math., 69 (2008), 621-639. doi: 10.1137/070700966.

[17]

H. Zhu, S. A. Campbell and G. S. K. Wolkowicz, Bifurcation analysis of a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 63 (2002), 636-682. doi: 10.1137/S0036139901397285.

show all references

References:
[1]

M. E. Alexander and S. M. Moghadas, Periodicity in an epidemic model with a generalized non-linear incidence, Math. Biosci., 189 (2004), 75-96. doi: 10.1016/j.mbs.2004.01.003.

[2]

F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemics, Springer-Verlag, New York, 2000.

[3]

L. Cai, G. Chen and D. Xiao, Multiparametric bifurcations of an epidemiological model with strong Allee effect, J. Math. Biol., 67 (2013), 185-215. doi: 10.1007/s00285-012-0546-5.

[4]

S.-N. Chow, C. Li and D. Wang, Normal Forms and Bifurcations of Planar Vector Fields, Cambridge University Press, 1994. doi: 10.1017/CBO9780511665639.

[5]

C. Christopher and C. Li, Limit Cycles of Differential Equations, Birkhäuser Verlag, 2007.

[6]

J. Cui,, X. Mu and H. Wan, Saturation recovery leads to multiple endemic equilibria and backward bifurcation, J. Theoret. Biol., 254 (2008), 275-283. doi: 10.1016/j.jtbi.2008.05.015.

[7]

F. Dumortier, R. Roussarie and J. Sotomayor, Generic 3-parameter family of vector feilds on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3, Ergod. Theor. & Dyn. Sys., 7 (1987), 375-413. doi: 10.1017/S0143385700004119.

[8]

H. W. Hethcote, Mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653. doi: 10.1137/S0036144500371907.

[9]

J. Huang, Y. Gong and S. Ruan, Bifurcation analysis in a predator-prey model with constant-yield predator harvesting, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2101-2121. doi: 10.3934/dcdsb.2013.18.2101.

[10]

C. Li and H. Zhu, Canard cycles for predator-prey systems with Holling types of functional response, J. Differential Equations, 254 (2013), 879-910. doi: 10.1016/j.jde.2012.10.003.

[11]

J. Li, Y. Zhou, J. Wu and Z. Ma, Complex dynamics of a simple epidemic model with a nonlinear incidence, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 161-173. doi: 10.3934/dcdsb.2007.8.161.

[12]

W. Liu, H. W. Hethcote and S. A. Levin, Dynamical behavior of epidemiological model with nonlinear incidence rates, J. Math. Biol., 25 (1987), 359-380. doi: 10.1007/BF00277162.

[13]

W. Liu, S. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol., 23 (1986), 187-204. doi: 10.1007/BF00276956.

[14]

Z. Ma and J. Li, Dynamical Modeling and Analysis of Epidemics, World Scientific, Singapore, 2009. doi: 10.1142/9789812797506.

[15]

S. Ruan and W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Differential Equations, 188 (2003), 135-163. doi: 10.1016/S0022-0396(02)00089-X.

[16]

Y. Tang, D. Huang, S. Ruan and W. Zhang, Coexistence of limit cycles and homoclinic loops in an SIRS model with nonlinear incidence rate, SIAM J. Appl. Math., 69 (2008), 621-639. doi: 10.1137/070700966.

[17]

H. Zhu, S. A. Campbell and G. S. K. Wolkowicz, Bifurcation analysis of a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 63 (2002), 636-682. doi: 10.1137/S0036139901397285.

[1]

Jicai Huang, Sanhong Liu, Shigui Ruan, Xinan Zhang. Bogdanov-Takens bifurcation of codimension 3 in a predator-prey model with constant-yield predator harvesting. Communications on Pure and Applied Analysis, 2016, 15 (3) : 1041-1055. doi: 10.3934/cpaa.2016.15.1041
[2]

Min Lu, Chuang Xiang, Jicai Huang. Bogdanov-Takens bifurcation in a SIRS epidemic model with a generalized nonmonotone incidence rate. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3125-3138. doi: 10.3934/dcdss.2020115
[3]

Bing Zeng, Shengfu Deng, Pei Yu. Bogdanov-Takens bifurcation in predator-prey systems. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3253-3269. doi: 10.3934/dcdss.2020130
[4]

Xun Cao, Xianyong Chen, Weihua Jiang. Bogdanov-Takens bifurcation with $ Z_2 $ symmetry and spatiotemporal dynamics in diffusive Rosenzweig-MacArthur model involving nonlocal prey competition. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022031
[5]

Hebai Chen, Xingwu Chen, Jianhua Xie. Global phase portrait of a degenerate Bogdanov-Takens system with symmetry. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1273-1293. doi: 10.3934/dcdsb.2017062
[6]

Hebai Chen, Xingwu Chen. Global phase portraits of a degenerate Bogdanov-Takens system with symmetry (Ⅱ). Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4141-4170. doi: 10.3934/dcdsb.2018130
[7]

Qiuyan Zhang, Lingling Liu, Weinian Zhang. Bogdanov-Takens bifurcations in the enzyme-catalyzed reaction comprising a branched network. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1499-1514. doi: 10.3934/mbe.2017078
[8]

Zhihua Liu, Rong Yuan. Takens–Bogdanov singularity for age structured models. Discrete and Continuous Dynamical Systems - B, 2020, 25 (6) : 2041-2056. doi: 10.3934/dcdsb.2019201
[9]

Fátima Drubi, Santiago Ibáñez, David Rivela. Chaotic behavior in the unfolding of Hopf-Bogdanov-Takens singularities. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 599-615. doi: 10.3934/dcdsb.2019256
[10]

Bing Zeng, Pei Yu. A hierarchical parametric analysis on Hopf bifurcation of an epidemic model. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022069
[11]

Majid Gazor, Mojtaba Moazeni. Parametric normal forms for Bogdanov--Takens singularity; the generalized saddle-node case. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 205-224. doi: 10.3934/dcds.2015.35.205
[12]

Xiaomei Feng, Zhidong Teng, Kai Wang, Fengqin Zhang. Backward bifurcation and global stability in an epidemic model with treatment and vaccination. Discrete and Continuous Dynamical Systems - B, 2014, 19 (4) : 999-1025. doi: 10.3934/dcdsb.2014.19.999
[13]

Shangzhi Li, Shangjiang Guo. Permanence and extinction of a stochastic SIS epidemic model with three independent Brownian motions. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2693-2719. doi: 10.3934/dcdsb.2020201
[14]

Benjamin H. Singer, Denise E. Kirschner. Influence of backward bifurcation on interpretation of $R_0$ in a model of epidemic tuberculosis with reinfection. Mathematical Biosciences & Engineering, 2004, 1 (1) : 81-93. doi: 10.3934/mbe.2004.1.81
[15]

Linda J. S. Allen, P. van den Driessche. Stochastic epidemic models with a backward bifurcation. Mathematical Biosciences & Engineering, 2006, 3 (3) : 445-458. doi: 10.3934/mbe.2006.3.445
[16]

Hongyong Zhao, Peng Wu, Shigui Ruan. Dynamic analysis and optimal control of a three-age-class HIV/AIDS epidemic model in China. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3491-3521. doi: 10.3934/dcdsb.2020070
[17]

Zhouchao Wei, Wei Zhang, Irene Moroz, Nikolay V. Kuznetsov. Codimension one and two bifurcations in Cattaneo-Christov heat flux model. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5305-5319. doi: 10.3934/dcdsb.2020344
[18]

Xiao-Qiang Zhao, Wendi Wang. Fisher waves in an epidemic model. Discrete and Continuous Dynamical Systems - B, 2004, 4 (4) : 1117-1128. doi: 10.3934/dcdsb.2004.4.1117
[19]

Hongying Shu, Xiang-Sheng Wang. Global dynamics of a coupled epidemic model. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1575-1585. doi: 10.3934/dcdsb.2017076
[20]

F. Berezovskaya, G. Karev, Baojun Song, Carlos Castillo-Chavez. A Simple Epidemic Model with Surprising Dynamics. Mathematical Biosciences & Engineering, 2005, 2 (1) : 133-152. doi: 10.3934/mbe.2005.2.133

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (250)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy