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

Abstract

In this paper, we extend the classical vaccination epidemic model from a deterministic framework to a model with historical eects by formulating it as a system of fractional-order dierential equations (FDEs). The basic reproduction number R0 of the resulting fractional model is computed and it is shown that if R0 is less than one, the disease-free equilibrium is locally asymptotically stable. Particularly, we analytically calculate a certain threshold-value for R0 and present the existence conditions of endemic equilibrium. By using stability analysis, we prove stability and α-stability of the endemic equilibrium points. The proposed model is applied on Pertussis disease and the fractional nonlinear system of the model is solved by applying multi-step generalized dierential transform method (MSGDTM). Our results show that historical eects play an important role on the disease spreading.