Modeling and Simulation of Cyber Epidemics Using a Delayed Variable-Order Fractional Framework

In today’s digital era, cyber epidemics are evolving rapidly. They pose a growing global threat. A cyber epidemic involves malware that spreads across connected networks. In this paper, we propose a delayed variable-order fractional model to study the spread of malware. The model incorporates time-varying memory effects through Caputo’s variable-order fractional derivative. It also accounts for delayed antivirus response via a time-delay term. The system is solved numerically using a predictor–corrector scheme. To the best of our knowledge, this work presents the first delayed variable-order fractional cyber epidemic model that integrates both time-dependent memory and time-delay effects. The results indicate that as the fractional order decreases, the system dynamics become slower and more stable, with lower infection levels. In addition, the variable-order model provides a more flexible description of evolving memory effects in cyber epidemic systems.