Stability analysis of integer-order and fractional-order malware spread models on WSNs

Abstract

In a recent work [J. D. Hernández Guillén, A. Martín del Rey, A mathematical model for malware spread on WSNs with population dynamics, Physica A: Statistical Mechanics and its Applications 545(2020) 123609], an integer-order model for the spread of malicious code on wireless sensor networks was introduced and analyzed. The global asymptotic stability (GAS) of the disease-endemic equilibrium (DEE) point was only resolved partially under technical hypotheses. In the present work, we use a simple approach, which is based on a suitable family of Lyapunov functions in combination with characteristics of Volterra-Lyapunov stable matrices, to establish the GAS of the DEE point. Consequently, we obtain a simple and easily-verified condition for the DEE point of the integer-order model to be globally asymptotically stable. In addition, we generalize the inter-order model by considering it in the context of the Caputo fractional derivative. After that, the proposed approach is utilized to analyze the GAS of the fractional-order model. The result is that the GAS of the DEE point of the fractional-order model is also established. As an important consequence, the advantage of the present approach is shown. Finally, the theoretical findings are supported by numerical and illustrative examples, which indicate that the numerical results are consistent with the theoretical ones. 2010 MSC: 34C60, 34D05, 33C75, 37N99