• Previous Article
    A detailed balanced reaction network is sufficient but not necessary for its Markov chain to be detailed balanced
  • DCDS-B Home
  • This Issue
  • Next Article
    Dynamics of the density dependent and nonautonomous predator-prey system with Beddington-DeAngelis functional response
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

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.
Citation: Chengzhi Li, Jianquan Li, Zhien Ma. Codimension 3 B-T bifurcations in an epidemic model with a nonlinear incidence. Discrete & 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. doi: 10.1016/j.mbs.2004.01.003. Google Scholar

[2]

F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemics,, Springer-Verlag, (2000). Google Scholar

[3]

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

[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. Google Scholar

[5]

C. Christopher and C. Li, Limit Cycles of Differential Equations,, Birkhäuser Verlag, (2007). Google Scholar

[6]

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

[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. doi: 10.1017/S0143385700004119. Google Scholar

[8]

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

[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. doi: 10.3934/dcdsb.2013.18.2101. Google Scholar

[10]

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

[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. doi: 10.3934/dcdsb.2007.8.161. Google Scholar

[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. doi: 10.1007/BF00277162. Google Scholar

[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. doi: 10.1007/BF00276956. Google Scholar

[14]

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

[15]

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

[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. doi: 10.1137/070700966. Google Scholar

[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. doi: 10.1137/S0036139901397285. Google Scholar

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. doi: 10.1016/j.mbs.2004.01.003. Google Scholar

[2]

F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemics,, Springer-Verlag, (2000). Google Scholar

[3]

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

[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. Google Scholar

[5]

C. Christopher and C. Li, Limit Cycles of Differential Equations,, Birkhäuser Verlag, (2007). Google Scholar

[6]

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

[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. doi: 10.1017/S0143385700004119. Google Scholar

[8]

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

[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. doi: 10.3934/dcdsb.2013.18.2101. Google Scholar

[10]

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

[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. doi: 10.3934/dcdsb.2007.8.161. Google Scholar

[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. doi: 10.1007/BF00277162. Google Scholar

[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. doi: 10.1007/BF00276956. Google Scholar

[14]

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

[15]

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

[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. doi: 10.1137/070700966. Google Scholar

[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. doi: 10.1137/S0036139901397285. Google Scholar

[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 & Applied Analysis, 2016, 15 (3) : 1041-1055. doi: 10.3934/cpaa.2016.15.1041

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

Elisabeth Logak, Isabelle Passat. An epidemic model with nonlocal diffusion on networks. Networks & Heterogeneous Media, 2016, 11 (4) : 693-719. doi: 10.3934/nhm.2016014

[13]

Ming Zhao, Cuiping Li, Jinliang Wang, Zhaosheng Feng. Bifurcation analysis of the three-dimensional Hénon map. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 625-645. doi: 10.3934/dcdss.2017031

[14]

Jianquan Li, Xiaoqin Wang, Xiaolin Lin. Impact of behavioral change on the epidemic characteristics of an epidemic model without vital dynamics. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1425-1434. doi: 10.3934/mbe.2018065

[15]

Yuan Lou, Daniel Munther. Dynamics of a three species competition model. Discrete & Continuous Dynamical Systems - A, 2012, 32 (9) : 3099-3131. doi: 10.3934/dcds.2012.32.3099

[16]

Weigu Li, Kening Lu. Takens theorem for random dynamical systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3191-3207. doi: 10.3934/dcdsb.2016093

[17]

Kai Wang, Zhidong Teng, Xueliang Zhang. Dynamical behaviors of an Echinococcosis epidemic model with distributed delays. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1425-1445. doi: 10.3934/mbe.2017074

[18]

Yukihiko Nakata, Yoichi Enatsu, Yoshiaki Muroya. On the global stability of an SIRS epidemic model with distributed delays. Conference Publications, 2011, 2011 (Special) : 1119-1128. doi: 10.3934/proc.2011.2011.1119

[19]

Meihong Qiao, Anping Liu, Qing Tang. The dynamics of an HBV epidemic model on complex heterogeneous networks. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1393-1404. doi: 10.3934/dcdsb.2015.20.1393

[20]

Aili Wang, Yanni Xiao, Huaiping Zhu. Dynamics of a Filippov epidemic model with limited hospital beds. Mathematical Biosciences & Engineering, 2018, 15 (3) : 739-764. doi: 10.3934/mbe.2018033

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (17)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]