Numerical treatments for the optimal control of two types variable-order COVID-19 model

In this paper, a novel variable-order COVID-19 model with modified parameters is presented. The variable-order fractional derivatives are defined in the Caputo sense. Two types of variable order Caputo definitions are presented here. The basic reproduction number of the model is derived. Properties of the proposed model are studied analytically and numerically. The suggested optimal control model is studied using two numerical methods. These methods are non-standard generalized fourth-order Runge–Kutta method and the non-standard generalized fifth-order Runge–Kutta technique. Furthermore, the stability of the proposed methods are studied. To demonstrate the methodologies’ simplicity and effectiveness, numerical test examples and comparisons with real data for Egypt and Italy are shown.