Bifurcations of an SIRS epidemic model with a general saturated incidence rate

Fang Zhang; Wenzhe Cui; Yanfei Dai; Yulin Zhao; Fang Zhang; Wenzhe Cui; Yanfei Dai; Yulin Zhao

doi:10.3934/mbe.2022501

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 11: 10710-10730. doi: 10.3934/mbe.2022501

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Research article

Bifurcations of an SIRS epidemic model with a general saturated incidence rate

1.
School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai 519082, China
2.
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

Received: 20 March 2022 Revised: 27 May 2022 Accepted: 22 June 2022 Published: 28 July 2022

This paper is concerned with the bifurcations of a susceptible-infectious-recovered-susceptible (SIRS) epidemic model with a general saturated incidence rate $ k I^p/(1+\alpha I^p) $. For general $ p > 1 $, it is shown that the model can undergo saddle-node bifurcation, Bogdanov-Takens bifurcation of codimension two, and degenerate Hopf bifurcation of codimension two with the change of parameters. Combining with the results in ^[1] for $ 0 < p\leq 1 $, this type of SIRS model has Hopf cyclicity $ 2 $ for any $ p > 0 $. These results also improve some previous ones in ^[2] and ^[3], which are dealt with the special case of $ p = 2 $.
- SIRS model,
- generalized saturated incidence rate,
- Bogdanov-Takens bifurcation,
- Hopf bifurcation
Citation: Fang Zhang, Wenzhe Cui, Yanfei Dai, Yulin Zhao. Bifurcations of an SIRS epidemic model with a general saturated incidence rate[J]. Mathematical Biosciences and Engineering, 2022, 19(11): 10710-10730. doi: 10.3934/mbe.2022501

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Abstract

This paper is concerned with the bifurcations of a susceptible-infectious-recovered-susceptible (SIRS) epidemic model with a general saturated incidence rate $ k I^p/(1+\alpha I^p) $. For general $ p > 1 $, it is shown that the model can undergo saddle-node bifurcation, Bogdanov-Takens bifurcation of codimension two, and degenerate Hopf bifurcation of codimension two with the change of parameters. Combining with the results in ^[1] for $ 0 < p\leq 1 $, this type of SIRS model has Hopf cyclicity $ 2 $ for any $ p > 0 $. These results also improve some previous ones in ^[2] and ^[3], which are dealt with the special case of $ p = 2 $.

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