A new and simple condition for the global asymptotic stability of a malware spread model on WSNs

TL;DR

This paper introduces a straightforward and verifiable condition for the global asymptotic stability of a malware spread model on wireless sensor networks, improving upon previous partial results by removing complex technical hypotheses.

Contribution

It provides a simple, complete criterion for the stability of the disease-endemic equilibrium using Lyapunov functions and matrix properties, advancing the theoretical analysis of malware spread models.

Findings

Established a simple condition for GAS of the DEE point

Validated theoretical results with numerical examples

Improved upon previous partial stability results

Abstract

In a very recent work [J. D. Hern\'andez Guill\'en, A. Mart\'in del Rey, A mathematical model for malware spread on WSNs with population dynamics, Physica A: Statistical Mechanics and its Applications 545(2020) 123609], a novel theoretical model for the spread of malicious code on wireless sensor networks was introduced and analyzed. However, the global asymptotic stability (GAS) of the disease-endemic equilibrium (DEE) point was only resolved partially under technical hypotheses that are not only difficult to be verified but also restrict the space of feasible parameters for the model. In the present work, we use a simple approach to establish the complete GAS of the DEE point without the technical hypotheses proposed in the benchmark work. This approach is based on a suitable family of Lyapunov functions in combination with characteristics of Volterra-Lyapunov stable matrices. Consequently, we obtain a simple and easily-verified condition for the DEE point to be globally asymptotically stable. This result provides an important improvement for the results constructed in the benchmark work. In addition, the theoretical findings are supported by numerical and illustrative examples, which show that the numerical results are consistent with the theoretical ones.