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.

On local stability of stochastic delay nonlinear discrete systems with statedependent noise Abstract   We examine the local stability of solutions of a delay stochastic nonlinear difference equation with deterministic and statedependent Gaussian perturbations. We apply the degenerate Lyapunovxe2x80x93Krasovskii functional technique and construct a sequence of events, each term of which is defined by a bound on a normally distributed random variable. Local stability holds on the intersection of these events, which has probability at least     1  xe2x88x92  xcexb3  ,     xcexb3xe2x80xafxe2x88x88xe2x80xaf(0, 1). This probability can be made arbitrarily high by choosing the initial value sufficiently small. We also present a generalization to systems where a condition for stability is expressed in terms of the diagonal part of the unperturbed system, and computer simulations which illustrate our results.