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

Min Lu; Chuang Xiang; Jicai Huang

doi:10.3934/dcdss.2020115

Article Contents

2020, Volume 13, Issue 11: 3125-3138. Doi: 10.3934/dcdss.2020115

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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

^* Corresponding author
^* Corresponding author

Received: October 31, 2018

Revised: October 31, 2018

Early access: October 2019

Published: July 2020

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

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Abstract

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.

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

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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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