**AUTHORS:**Wirawan Chinviriyasit, Sutawas Janreung, Settapat Chinviriyasit

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**ABSTRACT:**
An SEIRS epidemic model with a nonlinear incidence rate is investigated. Mathematical analysis
reveals that the model has a locally asymptotically stable disease–free equilibrium (DFE) whenever a certain epidemiological
threshold, known as the basic reproduction number R0, is less than unity. Using the theory of centre
manifold, the model exhibits the phenomenon of backward bifurcation, where the stable DFE coexists with a stable
endemic equilibrium when R0 < 1. The epidemiological consequence of this phenomenon is that the classical
epidemiological requirement of the reproduction number being less than unity becomes only a necessary, but not
sufficient, for disease elimination (hence, the presence of this phenomenon in the transmission dynamics of a disease
makes its effective control in the community difficult).

**KEYWORDS:**
SEIRS epidemic model, Nonlinear incidence rate, Backward bifurcation

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