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Stochastic and fractional differentiation have been developed independently to depicting processes following randomness and power, a declining memory and passage from one process to another respectively. Very recently, fractional stochastic differential equations were suggested with the aim to capture processes following at the same time randomness and memory nonlocality. In this paper to further explore the applicability of this type of differential equations, a SIR model was considered and analyzed analytically and numerically. Some numerical simulations are presented for different values of fractional orders and densities of randomness.

Very recently, the concept of instantaneous change was extended with the aim to accommodate prediction of more complex real world problems that could not be predicted or depicted by the existing rate of change. The extension gave birth to a more general differential operator that to be a derivative associate to the well-known Riemann-Stieltjes integral. In addition to this, using specific functions, one is able to recover all existing local differential operators defined as rate of change. This extended concept is still at its genesis and more works need to be done to establish a Riemann-Stieltjes calculus. In this paper, we aim to present a detailed analysis of an important class of differential equations called stochastic equations with the new classes of differential operators with the global derivative with integer and non-integer orders. We considered many classes as nonlinear Cauchy problems, then we presented existence and the uniqueness of their solutions using the linear growth and the Lipchitz conditions. We derived numerical solutions for each class and presented the error analysis. To show the applicability of these operators, we considered three epidemiological problems, including the zombie virus spread model, the zika virus spread model and Ebola model. We solved each model using the suggested numerical scheme and presented the numerical solutions for different values of fractional order and the global function gt. Our results showed that, more complex real world problems could be depicted using these classes of differential equations.

Very recently an extension of the concept of rate of change was suggested, with the aim to depict more real world problems that could not be depicted due to the limitation of the existing rate of change definition. One of the particular discovery in this is the existence of a derivative associate to Riemann-Stieltjes integral, a connection that has never been presented before. The global differential operator with its associat integral have now constructed a new calculus that is an extension of former calculus based on existing rate of change. To further see the application of this new extension, we investigated in this paper, a lassa fever model using the global differentiation in local and nonlocal sense. Existence and uniqueness conditions were verified using the theorem suggesting the linear growth and Lipschitz conditions. A numerical scheme based on the Newton polynomial was used to solve numerically the obtained models and some simulations presented.