Dynamically consistent nonstandard numerical schemes for solving some computer virus and malware propagation models

Abstract

This work is devoted to constructing reliable numerical schemes for some computer virus and malware propagation models. We apply the Mickens' methodology to formulate nonstandard finite difference (NSFD) schemes for some epidemiological models describing the spread of computer viruses and malware. Positivity, boundedness and global asymptotic stability (GAS) of the proposed NSFD schemes are studied rigorously. It should be emphasized that the GAS of the NSFD models are established based on an extension of the classical Lyapunov's direct method. As an important consequence, we conclude that the constructed NSFD schemes are dynamically consistent with respect to the positivity, boundedness and GAS of the continuous models. Finally, a set of numerical examples is conducted to support the theoretical findings and to demonstrate advantages of the NSFD schemes over some well-known standard ones. The numerical examples show that the used standard numerical schemes fail to preserve the qualitative dynamical properties of the continuous models for all finite step sizes; consequently, they can generate numerical approximations that are completely different from the exact solutions. Conversely, the NSFD schemes can provide reliable numerical solutions regardless of chosen step sizes.