Figure 1.
The curves of infection force $ g(I) $ . (a) Linear; (b) Saturated; (c) Nonmonotone
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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
School of Mathematics and Statistics, Central China Normal University, Wuhan, Hubei 430079, China |
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 and Continuous Dynamical Systems - S,
2020, 13
(11)
: 3125-3138.
doi: 10.3934/dcdss.2020115
![]()
References:
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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] |
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. |
| [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.
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| [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.
|
| [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. |
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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.
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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. |
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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. |
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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. |
| [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. |
| [11] |
W. O. Kermack and A. G. McKendrick,
A contribution to the mathematical theory of epidemics, Proc. Roal Soc. Lond., 115 (1927), 700-721.
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| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [20] |
F. Takens,
Forced oscillations and bifurcations, Applications of Global Analysis, I, Math. Inst. Rijksuniv. Utrecht, Utrecht, 3 (1974), 1-59.
|
| [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. |
| [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. |
| [23] |
D. M. Xiao and Y. G. Zhou,
Qualitative analysis of an epidemic model, Can. Appl. Math. Quart., 14 (2006), 469-492.
|
| [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. |
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] |
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. |
| [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.
|
| [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.
|
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [11] |
W. O. Kermack and A. G. McKendrick,
A contribution to the mathematical theory of epidemics, Proc. Roal Soc. Lond., 115 (1927), 700-721.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [20] |
F. Takens,
Forced oscillations and bifurcations, Applications of Global Analysis, I, Math. Inst. Rijksuniv. Utrecht, Utrecht, 3 (1974), 1-59.
|
| [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. |
| [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. |
| [23] |
D. M. Xiao and Y. G. Zhou,
Qualitative analysis of an epidemic model, Can. Appl. Math. Quart., 14 (2006), 469-492.
|
| [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. |
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 $
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
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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