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 discrete-time 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 Runge–Kutta scheme are not dynamically consistent with the continuous model and consequently, they fail to reflect the correct behavior of the continuous model.