A second order accurate approximation for fractional derivatives with singular and nonsingular kernel applied to a HIV model Abstract   In this manuscript we examine the CD    4       T cells model of HIV infection under the consideration of two different fractional differentiation operators namely Caputo and CaputoFabrizio (CF). Moreover, the generalized HIV model is investigated by considering Reverse Transcriptase (RT) inhibitors as a drug treatment for HIV. The threshold values for the stability of the equilibrium point belonging to noninfected case are calculated for both models with and without treatment. For the numerical solutions of the studied model, we construct trapezoidal approximation schemes having second order accuracy for the approximation of fractional operators with singular and nonsingular kernel. The stability and convergence of the proposed schemes are analyzed analytically. To illustrate the dynamics given by these two fractional operators, we perform numerical simulations of the HIV model for different biological scenarios with and without drug concentration. The studied biological cases are identified by considering different values of the parameters such as infection rate, growth rate of CD4           T cells, clearance rate of virus particles and also the order of the fractional derivative.

Positivity and global stability preserving NSFD schemes for a mixing propagation model of computer viruses Abstract   In this work, nonstandard finite difference (NSFD) schemes preserving the essential qualitative properties including positivity and stability of a mixing propagation model of computer viruses are proposed and analyzed for the first time. Especially, the model under consideration possesses the equilibrium points which are not only locally asymptotically stable but also globally asymptotically stable. Because of this reason we propose a new approach to prove theoretically that the global asymptotic stability of the original model is preserved by the proposed NSFD schemes. This approach is based on the Lyapunov stability theorem and its extension in combination with a theorem on the global stability of discretetime nonlinear cascade systems. The important result is that we obtain NSFD schemes which are dynamically consistent with the continuous model. Some numerical examples are performed to support the theoretical results. The results show that there is a good agreement between the numerical simulations and the established theoretical results. Furthermore, the numerical simulations also indicate that the proposed NSFD schemes are suitable and effective to solve the continuous model, whereas, the standard numerical schemes such as the Euler scheme, the classical fourth order Rungexe2x80x93Kutta scheme are not dynamically consistent with the continuous model and consequently, they fail to reflect the correct behavior of the continuous model.