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• A new stochastic fractional Coronavirus (2019-nCov) mathematical model with modified parameters are presented. • Milstein's higher order method is constructed to study the model problem • Mean square stability of Milstein algorithm is proved. • Comparative studies and numerical simulations are implemented In this paper, a new stochastic fractional Coronavirus (2019-nCov) model with modified parameters is presented. The proposed stochastic COVID-19 model describes well the real data of daily confirmed cases in Wuhan. Moreover, a novel fractional order operator is introduced, it is a linear combination of Caputo's fractional derivative and Riemann-Liouville integral. Milstein's higher order method is constructed with the new fractional order operator to study the model problem. The mean square stability of Milstein algorithm is proved. Numerical results and comparative studies are introduced. [ABSTRACT FROM AUTHOR]

• Chaotic systems such as: the fractal-fractional hyperchaotic finance system and the fractal-fractional Bloch system with time delay are studied. • Two Grüijnwald-Letnikov nonstandard finite difference schemes are presented to study the proposed chaotic systems. • The stability analysis of the used methods are proved. • Comparative studies. The purpose of this paper is to study numerically the behavior of two novel different classes of fractional order chaotic systems. These systems are; the fractal-fractional hyperchaotic finance system and the fractal-fractional Bloch system with time delay. The fractal-fractional derivatives are defined in the Caputo and Riemann-Liouville senses. Two Grünwald-Letnikov nonstandard finite difference schemes are presented to study the proposed chaotic systems. Moreover the stability analysis of the used methods are proved. In order to show the simplicity and effectively of the proposed methods, numerical simulations and comparative studies are given. [ABSTRACT FROM AUTHOR]

This article presents a variable order nonlinear mathematical model and its optimal control for a Tumor under immune suppression. The formulation generalizes the one proposed by Kim et. al. consisting of eleven integer order differential equations. The new approach adopts a variable-order fractional model with the derivatives defined in the Caputo sense. Two control variables, one for immunotherapy and one for Chemotherapy, are proposed to eliminate or reduce the Tumor cells. A simple numerical technique called the nonstandard generalized Euler method is developed to solve the proposed optimal control problem. Moreover, the stability analysis and the truncation error are studied. Numerical simulations and comparative studies are implemented. Our findings disclose that the proposed scheme used here has two main advantages: it is faster than the generalized Euler scheme and it can reduce the number of Tumor cells in a proper process better than the second scheme. [ABSTRACT FROM AUTHOR]