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| We present recent results for the asymptotic behavior of positive solutions of delayed epidemic models with a class of nonlinear incidence rates. Here the delay denotes the length of incubation period in the vector population. By means of a threshold parameter $R_0$, known as the basic reproduction number, we establish a characterization for the incidence rate, which shows that non-monotonicity with delay in the incidence rate is necessary for destabilization of an endemic equilibrium $E_*$. This enables us to improve a stability condition obtained in Y. Yang and D. Xiao (2010). It is proven that as we increase the value of a parameter measuring saturation effect, the number of infective individuals at the endemic steady state decreases, while the equilibrium can be unstable via Hopf bifurcation. Two-parameter plane analysis together with an application of the implicit function theorem facilitates us to obtain an exact stability condition. |
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