Numerical solution and stability analysis of a nonlinear vaccination model with historical effects

Abstract

In this paper, we extend the classical vaccination epidemic model from a deterministic framework to a model with historical effects by formulating it as a system of fractional-order differential equations (FDEs). The basic reproduction number $R_0$ of the resulting fractional model is computed and it is shown that if $R_0$ is less than one, the disease-free equilibrium is locally asymptotically stable. Particularly, we analytically calculate a certain threshold-value for $R_0$ and present the existence conditions of endemic equilibrium. By using stability analysis, we prove stability and $\alpha$-stability of the endemic equilibrium points. The proposed model is applied on \emph{Pertussis} disease and the fractional nonlinear system of the model is solved by applying multi-step generalized differential transform method (MSGDTM). Our results show that historical effects play an important role on the disease spreading.

Keywords

• epidemic model,fractional-order differential equations,equilibrium points,stability,differential transform method

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