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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

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 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.

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