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November  2020, 13(11): 3125-3138. doi: 10.3934/dcdss.2020115

Bogdanov-Takens bifurcation in a SIRS epidemic model with a generalized nonmonotone incidence rate

Min Lu , Chuang Xiang and  Jicai Huang ,

School of Mathematics and Statistics, Central China Normal University, Wuhan, Hubei 430079, China

* Corresponding author

Received  November 2018 Revised  November 2018 Published  November 2020 Early access  October 2019

Fund Project: Research was partially supported by NSFC grants (No.11471133, 11871235)

In this paper, we study a SIRS epidemic model with a generalized nonmonotone incidence rate. It is shown that the model undergoes two different topological types of Bogdanov-Takens bifurcations, i.e., repelling and attracting Bogdanov-Takens bifurcations, for general parameter conditions. The approximate expressions for saddle-node, Homoclinic and Hopf bifurcation curves are calculated up to second order. Furthermore, some numerical simulations, including bifurcations diagrams and corresponding phase portraits, are given to illustrate the theoretical results.

Keywords: SIRS epidemic model, nonmonotone incidence rate, Bogdanov-Takens bifurcation, hopf bifurcation, homoclinic bifurcation.
Mathematics Subject Classification: Primary: 34C23, 34C25; Secondary: 92D25.
Citation: Min Lu, Chuang Xiang, Jicai Huang. Bogdanov-Takens bifurcation in a SIRS epidemic model with a generalized nonmonotone incidence rate. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3125-3138. doi: 10.3934/dcdss.2020115
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]

M. E. Alexander and S. M. Moghadas, Bifurcations analysis of SIRS epidemic model with generalized incidence, SIAM J. Appl. Math., 65 (2005), 1794-1816.  doi: 10.1137/040604947.  Google Scholar

[3]

R. I. Bogdanov, Bifurcations of a limit cycle for a family of vector fields on the plane, Selecta Math. Soviet., 1 (1981), 373-388.   Google Scholar

[4]

R. I. Bogdanov, Versal deformations of a singular point on the plane in the case of zero eigenvalues, Selecta Math. Soviet., 1 (1981), 389-421.   Google Scholar

[5]

V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Math. Biosci., 42 (1978), 43-61.  doi: 10.1016/0025-5564(78)90006-8.  Google Scholar

[6]

V. Capasso, E. Crosso and G. Serio, I modelli matematici nella indagine epidemiologica. I. Applicazione all'epidemia di colera verificatasi in Bari nel 1973, Annali Sclavo, 19 (1977), 193-208.   Google Scholar

[7]

W. R. Derrick and P. van den Driessche, A disease transmission model in a nonconstant population, J. Math. Biol., 31 (1993), 495-512.  doi: 10.1007/BF00173889.  Google Scholar

[8]

H. W. Hethcote and P. van den Driessche, Some epidemiological models with nonlinear incidence, J. Math. Biol., 29 (1991), 271-287.  doi: 10.1007/BF00160539.  Google Scholar

[9]

J. C. Huang, Y. J. Gong and J. Chen, Muliple bifurcations in a predator-prey system of Holling and Leslie type with constant-yield prey harvesting, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 23 (2013), 1350164, 24 pp. doi: 10.1142/S0218127413501642.  Google Scholar

[10]

J. C. Huang, Y. J. Gong and S. G. 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

[11]

W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. Roal Soc. Lond., 115 (1927), 700-721.   Google Scholar

[12]

Y. Lamontagne, C. Coutu and C. Rousseau, Bifurcation analysis of a predator-prey system with generalized Holling type Ⅲ functional response, J. Dynam. Differential Equations, 20 (2008), 535-571.  doi: 10.1007/s10884-008-9102-9.  Google Scholar

[13]

X. P. Li, J. L. Ren, S. A. Campbell, G. S. K. Wolkowicz and H. P. Zhu, How seasonal forcing influences the complexity of a predator-prey system, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 785-807.  doi: 10.3934/dcdsb.2018043.  Google Scholar

[14]

J. Li, Y. L. Zhao and H. P. Zhu, Bifurcation of an SIS model with nonlinear contact rate, J. Math. Anal. Appl., 432 (2015), 1119-1138.  doi: 10.1016/j.jmaa.2015.07.001.  Google Scholar

[15]

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

[16]

W. M. 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

[17]

J. L. Ren and L. P. Yu, Codimension-two bifurcation, chaos and control in a discrete-time information diffusion model, Journal of Nonlinear Science, 26 (2016), 1895-1931.  doi: 10.1007/s00332-016-9323-8.  Google Scholar

[18]

J. L. Ren, L. P. Yu and S. Siegmind, Bifurcations and chaos in a discrete predator-prey model with Crowley-Martin functional response, Nonlinear Dynamics, 90 (2017), 19-41.  doi: 10.1007/s11071-017-3643-6.  Google Scholar

[19]

S. G. Ruan and W. D. 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

[20]

F. Takens, Forced oscillations and bifurcations, Applications of Global Analysis, I, Math. Inst. Rijksuniv. Utrecht, Utrecht, 3 (1974), 1-59.   Google Scholar

[21]

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

[22]

D. M. Xiao and S. G. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Math. Biosci., 208 (2007), 419-429.  doi: 10.1016/j.mbs.2006.09.025.  Google Scholar

[23]

D. M. Xiao and Y. G. Zhou, Qualitative analysis of an epidemic model, Can. Appl. Math. Quart., 14 (2006), 469-492.   Google Scholar

[24]

Y. G. Zhou, D. M. Xiao and Y. L. Li, Bifurcations of an epidemic model with non-monotonic incidence rate of saturated mass action, Chaos Solitons Fractals, 32 (2007), 1903-1915.  doi: 10.1016/j.chaos.2006.01.002.  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]

M. E. Alexander and S. M. Moghadas, Bifurcations analysis of SIRS epidemic model with generalized incidence, SIAM J. Appl. Math., 65 (2005), 1794-1816.  doi: 10.1137/040604947.  Google Scholar

[3]

R. I. Bogdanov, Bifurcations of a limit cycle for a family of vector fields on the plane, Selecta Math. Soviet., 1 (1981), 373-388.   Google Scholar

[4]

R. I. Bogdanov, Versal deformations of a singular point on the plane in the case of zero eigenvalues, Selecta Math. Soviet., 1 (1981), 389-421.   Google Scholar

[5]

V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Math. Biosci., 42 (1978), 43-61.  doi: 10.1016/0025-5564(78)90006-8.  Google Scholar

[6]

V. Capasso, E. Crosso and G. Serio, I modelli matematici nella indagine epidemiologica. I. Applicazione all'epidemia di colera verificatasi in Bari nel 1973, Annali Sclavo, 19 (1977), 193-208.   Google Scholar

[7]

W. R. Derrick and P. van den Driessche, A disease transmission model in a nonconstant population, J. Math. Biol., 31 (1993), 495-512.  doi: 10.1007/BF00173889.  Google Scholar

[8]

H. W. Hethcote and P. van den Driessche, Some epidemiological models with nonlinear incidence, J. Math. Biol., 29 (1991), 271-287.  doi: 10.1007/BF00160539.  Google Scholar

[9]

J. C. Huang, Y. J. Gong and J. Chen, Muliple bifurcations in a predator-prey system of Holling and Leslie type with constant-yield prey harvesting, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 23 (2013), 1350164, 24 pp. doi: 10.1142/S0218127413501642.  Google Scholar

[10]

J. C. Huang, Y. J. Gong and S. G. 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

[11]

W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. Roal Soc. Lond., 115 (1927), 700-721.   Google Scholar

[12]

Y. Lamontagne, C. Coutu and C. Rousseau, Bifurcation analysis of a predator-prey system with generalized Holling type Ⅲ functional response, J. Dynam. Differential Equations, 20 (2008), 535-571.  doi: 10.1007/s10884-008-9102-9.  Google Scholar

[13]

X. P. Li, J. L. Ren, S. A. Campbell, G. S. K. Wolkowicz and H. P. Zhu, How seasonal forcing influences the complexity of a predator-prey system, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 785-807.  doi: 10.3934/dcdsb.2018043.  Google Scholar

[14]

J. Li, Y. L. Zhao and H. P. Zhu, Bifurcation of an SIS model with nonlinear contact rate, J. Math. Anal. Appl., 432 (2015), 1119-1138.  doi: 10.1016/j.jmaa.2015.07.001.  Google Scholar

[15]

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

[16]

W. M. 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

[17]

J. L. Ren and L. P. Yu, Codimension-two bifurcation, chaos and control in a discrete-time information diffusion model, Journal of Nonlinear Science, 26 (2016), 1895-1931.  doi: 10.1007/s00332-016-9323-8.  Google Scholar

[18]

J. L. Ren, L. P. Yu and S. Siegmind, Bifurcations and chaos in a discrete predator-prey model with Crowley-Martin functional response, Nonlinear Dynamics, 90 (2017), 19-41.  doi: 10.1007/s11071-017-3643-6.  Google Scholar

[19]

S. G. Ruan and W. D. 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

[20]

F. Takens, Forced oscillations and bifurcations, Applications of Global Analysis, I, Math. Inst. Rijksuniv. Utrecht, Utrecht, 3 (1974), 1-59.   Google Scholar

[21]

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

[22]

D. M. Xiao and S. G. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Math. Biosci., 208 (2007), 419-429.  doi: 10.1016/j.mbs.2006.09.025.  Google Scholar

[23]

D. M. Xiao and Y. G. Zhou, Qualitative analysis of an epidemic model, Can. Appl. Math. Quart., 14 (2006), 469-492.   Google Scholar

[24]

Y. G. Zhou, D. M. Xiao and Y. L. Li, Bifurcations of an epidemic model with non-monotonic incidence rate of saturated mass action, Chaos Solitons Fractals, 32 (2007), 1903-1915.  doi: 10.1016/j.chaos.2006.01.002.  Google Scholar

Figure 1.  The curves of infection force $ g(I) $. (a) Linear; (b) Saturated; (c) Nonmonotone
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Figure 2.  The curves of infection force $ g(I) = \frac{kI}{1+\beta I+\alpha I^2} $ with $ \alpha = \frac{9}{2} $ and $ k = 1 $. (a) $ \beta\geq 0 $; (b) $ -2\sqrt{\alpha}<\beta<0 $
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Figure 3.  A cusp of codimension 2 for system (9). (a) $ m<m_* $; (b) $ m>m_* $
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Figure 4.  The repelling Bogdanov-Takens bifurcation diagram and phase portraits of system (11) with $ p = 3 $, $ q = 2 $, $ m = -3 $, $ A = \frac{9}{4} $, $ n = 4 $. (a) Bifurcation diagram; (b) No equilibria when $ (\lambda_1, \lambda_2) = (0.03, 0.25) $ lies in the region Ⅰ; (c) An unstable focus when $ (\lambda_1, \lambda_2) = (0.03, 0.12097) $ lies in the region Ⅱ; (d) An unstable limit cycle when $ (\lambda_1, \lambda_2) = (0.03, 0.1197) $ lies in the region Ⅲ; (e) An unstable homoclinic loop when $ (\lambda_1, \lambda_2) = (0.03, 0.119045) $ lies on the curve HL; (f) A stable focus when $ (\lambda_1, \lambda_2) = (0.03, 0.115) $ lies in the region Ⅳ
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Figure 5.  The attracting Bogdanov-Takens bifurcation diagram and phase portraits of system (9) with $ p = 3 $, $ q = 2 $, $ m = -\frac{3}{2} $, $ A = \frac{36}{13} $, $ n = \frac{13}{16} $. (a) Bifurcation diagram; (b) No equilibria when $ (\lambda_1, \lambda_2) = (0.03, 0.11) $ lies in the region Ⅰ; (c) A stable focus when $ (\lambda_1, \lambda_2) = (0.03, 0.098) $ lies in the region Ⅱ; (d) A stable limit cycle when $ (\lambda_1, \lambda_2) = (0.03, 0.096) $ lies in the region Ⅲ; (e) A stable homoclinic loop when $ (\lambda_1, \lambda_2) = (0.03, 0.09342) $ lies on the curve HL; (f) An unstable focus when $ (\lambda_1, \lambda_2) = (0.03, 0.09) $ lies in the region Ⅳ
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