  In this paper, we propose a numerical scheme of the predictor-corrector type
for solving nonlinear fractional initial value problems, the chosen fractional
derivative is called the Atangana-Baleanu derivative defined in Caputo sense
(ABC). This proposed method is based on Lagrangian quadratic polynomials to
approximate the nonlinearity implied in the Volterra integral which is obtained
by reducing the given fractional differential equation via the properties of
the ABC-fractional derivative. Through this technique, we get corrector formula
with high accuracy which is implicit as well as predictor formula which is
explicit and has the same precision order as the corrective formula. On the
other hand, the so-called memory term is computed only once for both prediction
and 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 new
method on two differential equations. Moreover, a case study is considered
where the proposed predictor-corrector scheme is used to obtained approximate
solutions of ABC-fractional generalized SI (susceptible-infectious) epidemic
model for the purpose of analyzing dynamics of the suggested system as well as
demonstrating the effectiveness of the new method to solve systems dealing with
real-world problems.
