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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 & Continuous Dynamical Systems - B, 2015, 20 (4) : 1107-1116. doi: 10.3934/dcdsb.2015.20.1107
##### References:
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##### 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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