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

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    * Corresponding author

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

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  • 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.

    Mathematics Subject Classification: Primary: 34C23, 34C25; Secondary: 92D25.

    Citation:

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  • Figure 1.  The curves of infection force $ g(I) $. (a) Linear; (b) Saturated; (c) Nonmonotone

    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 3.  A cusp of codimension 2 for system (9). (a) $ m<m_* $; (b) $ m>m_* $

    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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