SIR epidemic model of childhood diseases through fractional operators with Mittag-Leffler and exponential kernels

Abstract

Vaccination programs for infants have significantly affected childhood morbidity and mortality. The primary goal of health administrators is to protect children against diseases that can be prevented by vaccination. In this manuscript, we have applied the homotopy perturbation Elzaki transform method to obtain the solutions of the epidemic model of childhood diseases involving time-fractional order Atangana–Baleanu and Caputo–Fabrizio derivatives. The present method is the combination of the classical homotopy perturbation method and the Elzaki transform. Although Elzaki transform is an effective method for solving fractional differential equations, this method sometimes fails to handle nonlinear terms from the fractional differential equations. These difficulties may be overcome by coupling this transform with that of HPM. This method offers a rapidly convergent series solutions. Validation and usefulness of the technique are incorporated with new fractional-order derivatives with exponential decay law and with general Mittag-Leffler law. Obtained results are compared with the established solution defined in the Caputo sense. Further, a comparative study among Caputo, Atangana–Baleanu, and Caputo–Fabrizio derivatives is discussed.