An SIRS model with a nonlinear incidence rate. (English) Zbl 1152.34339

Summary: The global dynamics of an SIRS model with a nonlinear incidence rate is investigated. We establish a threshold for a disease to be extinct or endemic, analyze the existence and asymptotic stability of equilibria, and verify the existence of bistable states, i.e., a stable disease free equilibrium and a stable endemic equilibrium or a stable limit cycle. In particular, we find that the model admits stability switches as a parameter changes. We also investigate the backward bifurcation, the Hopf bifurcation and Bogdanov-Takens bifurcation and obtain the Hopf bifurcation criteria and Bogdanov-Takens bifurcation curves, which are important for making strategies for controlling a disease.


34C23 Bifurcation theory for ordinary differential equations
92D30 Epidemiology
37N25 Dynamical systems in biology
Full Text: DOI


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