A simple approach to the study of global asymptotic stability of some modified continuous-time epidemiological models for distributed denial of service attacks

Abstract

In this work, we first revisit two recognized continuous-time epidemiological models for distributed denial of service (DDoS) attacks on targeted sources in computer networks, which are described by systems of nonlinear ordinary differential equations (ODEs) with complex dynamics. These models were formulated and analyzed in existing literature but the global asymptotic stability (GAS) of disease-free equilibrium (DFE) points has not been established.  Our main objective is to perform a rigorous mathematical analysis for the complete GAS of the two mathematical models under consideration. We use a simple approach, which is based on utilizing the cascade structure of the ODE systems, to study the GAS problem. More clearly, by taking advantage of the cascade structure, the GAS analysis of the original nonlinear systems is reduced to the GAS analysis of simple linear systems. After that, the GAS analysis of the reduced linear systems is completed in a straightforward manner. As an important consequence, the GAS is confirmed not only for the DEE points but also for possible disease-endemic equilibrium (DEE) points. The theoretical findings improve the resuls presented in the benchmark works. Furthermore, the present approach can be applied to a broad range of mathematical models arising in real-world applications, with a specific focus on DDoS attacks. To show advantages of the proposed approach, we consider some other mathematical models of DDoS attacks constructed previously. It is proved that the used approach is not only simple but also useful in investigating the GAS of the mathematical models being considered. Finally, the theoretical insights are illustrated by a set of illustrative numerical experiments, in which the validity of the theoretical findings is supported.