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doi: 10.3934/dcdss.2022069
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A hierarchical parametric analysis on Hopf bifurcation of an epidemic model

1. 

School of Mathematics and Statistics, Lingnan Normal University, Zhanjiang, Guandong 524048, China

2. 

Department of Mathematics, Western University, London, Ontario, N6A 5B7, Canada

* Corresponding author: Pei Yu

Received  January 2022 Revised  February 2022 Early access March 2022

A common task in studying nonlinear dynamical systems is to derive the conditions on stability and bifurcations, which becomes difficulty when the system contain multiple parameters. In particular, finding the explicit conditions under which Hopf bifurcation can occur is not easy and becomes very involved even for simple models. In this paper, an epidemic model is presented to illustrate how to use a hierarchical parametric analysis for bifurcation study, in particular to demonstrate how to choose proper parameters as bifurcation parameters, how to deal with other "control" parameters, and how to derive the conditions on stability and Hopf bifurcation, which are explicitly expressed in terms of system parameters.

Citation: Bing Zeng, Pei Yu. A hierarchical parametric analysis on Hopf bifurcation of an epidemic model. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022069
References:
[1]

M. E. Alexander and S. M. Moghadas, Periodicity in an epidmic model with a generalized nonlinear incicdence, Math. Biosci., 189 (1973), 75-96.  doi: 10.1016/j.mbs.2004.01.003.

[2]

K. L. Cooke and J. A. Yorke, Some equations modelling growth processes and gonorrhea epidemics, Math. Biosci., 16 (1973), 75-101.  doi: 10.1016/0025-5564(73)90046-1.

[3]

F. DumortierR. Roussarie and J. Sotomayor, Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3, Ergodic Theory Dynam. Systems, 7 (1987), 375-413.  doi: 10.1017/S0143385700004119.

[4]

M. Gazor and P. Yu, Spectral sequences and parametric normal forms, J. Differ. Equ., 252 (2012), 1003-1031.  doi: 10.1016/j.jde.2011.09.043.

[5]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, 42. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-1140-2.

[6]

M. Han and P. Yu, Normal Forms, Melnikov Functions, and Bifurcations of Limit Cycles, Applied Mathematical Sciences, 181. Springer, London, 2012. doi: 10.1007/978-1-4471-2918-9.

[7]

J. Jiang and P. Yu, Multistable phenomena involving equilibria and periodic motions in predator-prey systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27 (2017), 1750043, 28 pp. doi: 10.1142/S0218127417500432.

[8]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Second edition, Applied Mathematical Sciences, 112. Springer-Verlag, New York, 1998.

[9]

C. LiJ. Li and Z. Ma, Codimension 3 B-T bifurcations in an epidemic model with a nonlinear incidence, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1107-1116.  doi: 10.3934/dcdsb.2015.20.1107.

[10]

J. LiY. ZhouJ. 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.

[11]

P. van den Driessche and J. Watmough, A simple SIS epidemic model with a backward bifurcation, J. Math. Biol., 40 (2000), 525-540.  doi: 10.1007/s002850000032.

[12]

P. Yu, Computation of normal forms via a perturbation technique, J. Sound Vib., 211 (1998), 19-38.  doi: 10.1006/jsvi.1997.1347.

[13]

P. Yu and A. Y. L. Leung, The simplest normal form of Hopf bifurcation, Nonlinearity, 16 (2003), 277-300.  doi: 10.1088/0951-7715/16/1/317.

[14]

P. Yu and W. Zhang, Complex dynamics in a unified SIR and HIV disease model: A bifurcation theory approach, J. Nonlinear Sci., 29 (2019), 2447-2500.  doi: 10.1007/s00332-019-09550-7.

[15]

B. ZengS. Deng and P. Yu, Bogdanov-Takens bifurcation in predator-prey systems, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 3253-3269.  doi: 10.3934/dcdss.2020130.

show all references

References:
[1]

M. E. Alexander and S. M. Moghadas, Periodicity in an epidmic model with a generalized nonlinear incicdence, Math. Biosci., 189 (1973), 75-96.  doi: 10.1016/j.mbs.2004.01.003.

[2]

K. L. Cooke and J. A. Yorke, Some equations modelling growth processes and gonorrhea epidemics, Math. Biosci., 16 (1973), 75-101.  doi: 10.1016/0025-5564(73)90046-1.

[3]

F. DumortierR. Roussarie and J. Sotomayor, Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3, Ergodic Theory Dynam. Systems, 7 (1987), 375-413.  doi: 10.1017/S0143385700004119.

[4]

M. Gazor and P. Yu, Spectral sequences and parametric normal forms, J. Differ. Equ., 252 (2012), 1003-1031.  doi: 10.1016/j.jde.2011.09.043.

[5]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, 42. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-1140-2.

[6]

M. Han and P. Yu, Normal Forms, Melnikov Functions, and Bifurcations of Limit Cycles, Applied Mathematical Sciences, 181. Springer, London, 2012. doi: 10.1007/978-1-4471-2918-9.

[7]

J. Jiang and P. Yu, Multistable phenomena involving equilibria and periodic motions in predator-prey systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27 (2017), 1750043, 28 pp. doi: 10.1142/S0218127417500432.

[8]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Second edition, Applied Mathematical Sciences, 112. Springer-Verlag, New York, 1998.

[9]

C. LiJ. Li and Z. Ma, Codimension 3 B-T bifurcations in an epidemic model with a nonlinear incidence, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1107-1116.  doi: 10.3934/dcdsb.2015.20.1107.

[10]

J. LiY. ZhouJ. 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.

[11]

P. van den Driessche and J. Watmough, A simple SIS epidemic model with a backward bifurcation, J. Math. Biol., 40 (2000), 525-540.  doi: 10.1007/s002850000032.

[12]

P. Yu, Computation of normal forms via a perturbation technique, J. Sound Vib., 211 (1998), 19-38.  doi: 10.1006/jsvi.1997.1347.

[13]

P. Yu and A. Y. L. Leung, The simplest normal form of Hopf bifurcation, Nonlinearity, 16 (2003), 277-300.  doi: 10.1088/0951-7715/16/1/317.

[14]

P. Yu and W. Zhang, Complex dynamics in a unified SIR and HIV disease model: A bifurcation theory approach, J. Nonlinear Sci., 29 (2019), 2447-2500.  doi: 10.1007/s00332-019-09550-7.

[15]

B. ZengS. Deng and P. Yu, Bogdanov-Takens bifurcation in predator-prey systems, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 3253-3269.  doi: 10.3934/dcdss.2020130.

Figure 1.  Simulation of the epidemic model (2) for $ m = 2 $, $ n = \frac{1}{6} $, $ \varepsilon = \frac{1}{2} $ and $ k = \frac{13}{100} $, showing non-well-posedness solution preperty
Figure 2.  (a) bifurcation diagram for the epidemic model (2) projected on the $ k $-$ Y_2 $ plane with $ m = 2, \, n = \frac{1}{6}, \, \varepsilon = \frac{1}{2} $, corresponding to Case (1b) in Theorem 3.1 having two Hopf critical points, with $ k_{\rm T} = 0.097222 $, $ k_{\rm H_-} = 0.110329 $ and $ k_{\rm H_+} = 0.149040 $; and (b) simulations of three stable limit cycles, starting from the initial point $ (X, Y) = (0.25, 5) $, for three values of $ k $ (marked by the red circles on the $ k $-axis): $ k = 0.112 $ (blue color), $ k = 0.13 $ (red color), $ k = 0.147 $ (green color)
Figure 3.  (a) bifurcation diagram for the epidemic model (2) projected on the $ k $-$ Y_2 $ plane with $ m \! = \! 2, \, n \! = \! \frac{2}{5}, \, \varepsilon \! = \! 2 $, corresponding to the Case (2c)(i) in Theorem 3.1 having one Hopf critical point, with $ k_{\rm SN} \! = \!0.215089 $, $ k^* \! = \!0.228381 $, $ k_{\rm H_+} \! = \! 0.235367 $ and $ k_{\rm T} \! = \! 0.28 $; (b) simulated phase portrait with $ k \! = \! 0.222 $, showing the bistable phenomenon with two stable equilibria $ {\rm E_1} $ and $ {\rm E_{2-}} $; (c) simulated phase portrait with $ k \! = \! 0.232 $, showing the bistable phenomenon with the stable equilibrium $ {\rm E_1} $ and a stable limit cycle; and (d) simulated phase portrait with $ k \! = \! 0.25 $, showing the bistable phenomenon with two stable equilibria $ {\rm E_1} $ and $ {\rm E_{2-}} $. The three values of $ k $ are marked by the circles on the $ k $-axis in Figure 3(a)
Figure 4.  (a) bifurcation diagram for the epidemic model (2) projected on the $ k $-$ Y_2 $ plane with $ m = 2, \, n = \frac{3}{4}, \, \varepsilon = 8 $, corresponding to the Case (2c)(ii) in Theorem 3.1 having one Hopf critical point, with $ k_{\rm SN} = 0.233287 $, $ k_{\rm H_+} = 0.233834 $, $ k^* \! = \! 0.240547 $, and $ k_{\rm T} = 0.656250 $; (b) simulated phase portrait with $ k \! = \! 0.2336 $, showing the stable node $ {\rm E_1} $ and the unstable focus $ {\rm E_{2-}} $; (c) simulated phase portrait with $ k \! = \! 0.234 $, showing the stable node $ {\rm E_1} $ and an unstable limit cycle; and (d) simulated phase portrait with $ k \! = \! 0.24 $, showing the bistable phenomenon with two stable equilibria $ {\rm E_1} $ and $ {\rm E_{2-}} $. The three values of $ k $ are marked by the circles on the $ k $-axis in Figure 4(a)
Figure 5.  (a) bifurcation diagram for the epidemic model (2) projected on the $ k $-$ Y_2 $ plane with $ m = 2, \, n = \frac{1}{3}, \, \varepsilon = \frac{5}{4} $, corresponding to the Case (2d) in Theorem 3.1 having two Hopf critical points, with $ k^* \! = \! 0.197348 $, $ k_{\rm SN} = 0.201638 $, $ k_{\rm H_-} = 0.202731 $, $ k_{\rm H_+} = 0.221025 $ and $ k_{\rm T} = 0.222222 $; (b) simulated phase portrait with $ k \! = \! 0.202 $, showing the bistable phenomenon with two stable equilibria $ {\rm E_1} $ and $ {\rm E_{2-}} $; (c) simulated phase portrait with $ k \! = \! 0.205 $, showing the bistable phenomenon with the stable equilibrium $ {\rm E_1} $ and a stable limit cycle; and (d) simulated phase portrait with $ k \! = \! 0.22 $, showing the bistable phenomenon with the stable equilibrium $ {\rm E_1} $ and a stable limit cycle. The three values of $ k $ are marked by the circles on the $ k $-axis in Figure 5(a)
Figure 6.  Bifurcation diagram for the epidemic model (2) on the $ k $-$ \varepsilon $ parameter plane with $ m = 2, \, n = \frac{5}{11} $, where SN, T, H, BT and GH denote the saddle-node, transcritical, Hopf, Bogdanov-Takens and generalized Hopf bifurcations, respectively
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