Abstract
In this paper, we introduce a new fractional-order epidemic model with vaccination, using fractional derivative in the sense of the Caputo derivative of order \(\alpha \in [0,1)\). We analyze the forward and backward bifurcation by determining the basic reproduction number \(R_{0}^{\alpha }\) and also a certain threshold-value of \(R_{0}^{\alpha }\). Furthermore, we present some theorems about the stability of the endemic equilibrium points and show that the stability region of the model is also related to value of the fractional-order \(\alpha\). Finally, we present some numerical simulations for real data of pertussis disease to show the benefits of our results.
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