# American Institute of Mathematical Sciences

## 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

Received  November 2018 Revised  November 2018 Published  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.

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, doi: 10.3934/dcdss.2020115
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##### References:
The curves of infection force $g(I)$. (a) Linear; (b) Saturated; (c) Nonmonotone
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) $-2\sqrt{\alpha}<\beta<0$
A cusp of codimension 2 for system (9). (a) $m<m_*$; (b) $m>m_*$
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
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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