Bifurcation of an SIRS Model with a Modified Nonlinear Incidence Rate

Yingying Zhang () and Chentong Li
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Yingying Zhang: College of Science, Northwest A&F University, Yangling 712100, China
Chentong Li: Guangdong Key Laboratory of Mordern Control Technology, Institute of Intelligent Manufacturing, Guangdong Academy of Sicence, Guangzhou 510070, China

Mathematics, 2023, vol. 11, issue 13, 1-24

Abstract: An SIRS epidemic model with a modified nonlinear incidence rate is studied, which describes that the infectivity is strong at first as the emergence of a new disease or the reemergence of an old disease, but then the psychological effect will weaken the infectivity. Lastly, the infectivity goes to a saturation state as a result of a crowding effect. The nonlinearity of the functional form of the incidence of infection is modified, which is more reasonable biologically. We analyze the stability of the associated equilibria, and the basic reproduction number and the critical value which determine the dynamics of the model are derived. The bifurcation analysis is presented, including backward bifurcation, saddle-node bifurcation, Bogdanov–Takens bifurcation of codimension two and Hopf bifurcation. To study Hopf bifurcation of codimension three of the model when some assumptions hold, the focus values are calculated. Numerical simulations are shown to verify our results.

Keywords: SIRS model; backward bifurcation; saddle-node bifurcation; Bogdanov–Takens bifurcation; Hopf bifurcation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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