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In this paper, we propose a numerical scheme of the predictor-corrector typefor solving nonlinear fractional initial value problems, the chosen fractionalderivative is called the Atangana-Baleanu derivative defined in Caputo sense(ABC). This proposed method is based on Lagrangian quadratic polynomials toapproximate the nonlinearity implied in the Volterra integral which is obtainedby reducing the given fractional differential equation via the properties ofthe ABC-fractional derivative. Through this technique, we get corrector formulawith high accuracy which is implicit as well as predictor formula which isexplicit and has the same precision order as the corrective formula. On theother hand, the so-called memory term is computed only once for both predictionand correction phases, which indicates the low cost of the proposed method.Also, the error bound of the proposed numerical scheme is offered. Furthermore,numerical experiments are presented in order to assess the accuracy of the newmethod on two differential equations. Moreover, a case study is consideredwhere the proposed predictor-corrector scheme is used to obtained approximatesolutions of ABC-fractional generalized SI (susceptible-infectious) epidemicmodel for the purpose of analyzing dynamics of the suggested system as well asdemonstrating the effectiveness of the new method to solve systems dealing withreal-world problems.

A high-order predictor-corrector method for initial value problems with  fractional derivative involving Mittag-Leffler kernel: epidemic model case  study