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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 & 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.  Google Scholar

[2]

F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemics, Springer-Verlag, New York, 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), 185-215. 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), 275-283. 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), 375-413. doi: 10.1017/S0143385700004119.  Google Scholar

[8]

H. W. Hethcote, Mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653. 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), 2101-2121. 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), 879-910. 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), 161-173. 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), 359-380. 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), 187-204. doi: 10.1007/BF00276956.  Google Scholar

[14]

Z. Ma and J. Li, Dynamical Modeling and Analysis of Epidemics, World Scientific, Singapore, 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), 135-163. 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), 621-639. 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), 636-682. 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), 75-96. 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, New York, 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), 185-215. 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), 275-283. 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), 375-413. doi: 10.1017/S0143385700004119.  Google Scholar

[8]

H. W. Hethcote, Mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653. 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), 2101-2121. 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), 879-910. 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), 161-173. 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), 359-380. 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), 187-204. doi: 10.1007/BF00276956.  Google Scholar

[14]

Z. Ma and J. Li, Dynamical Modeling and Analysis of Epidemics, World Scientific, Singapore, 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), 135-163. 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), 621-639. 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), 636-682. doi: 10.1137/S0036139901397285.  Google Scholar

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