Delayed dynamics analysis of SEI2RS malware propagation models in cyber–Physical systems

Abstract

Cyber-physical systems facilitate seamless interaction between the physical and digital elements for improved efficiency, automation, and real-time monitoring across domains. This study analyzes a novel virus-spreading model called the delayed SEI2RS model, which is specifically designed for cyber-physical systems. This model incorporates a saturated incidence rate and treatment. An emphasis of this research is to explore the impact of time delay on the transient immunity interval of restored nodes. By using the time delay associated with the transitory immunity interval of recovered nodes as the bifurcation parameter, we derive a comprehensive set of appropriate conditions to assess the local stability of the malware-existence equilibrium and determine Hopf bifurcation. The center manifold theorem and normal form theory are employed to investigate the path and stability of Hopf bifurcation. Numerical calculations were used to validate the results, providing empirical evidence for the proposed model and its implications. © 2024 Elsevier B.V.