Riccati equation as topology-based model of computer worms and discrete SIR model with constant infectious period

Abstract

We propose discrete and continuous infection models of computer worms via e-mail or social networking site (SNS) messengers that were previously classified as worms spreading through topological neighbors. The discrete model is made on the basis of a new classification of worms as “permanently” or “temporarily” infectious. A temporary infection means that only the most recently infected nodes are infectious according to a difference equation. The discrete model is reduced to a Riccati differential equation (the continuous model) at the limit of a zero difference interval for the difference equation. The discrete and continuous models well describe actual data and are superior to a linear model in terms of the Akaike information criterion (AIC). Both models overcome the overestimation that is generated by applying a scan-based model to topology-based infection, especially in the early stages. The discrete model gives a condition in which all nodes are infected because the vulnerable nodes of the Ricatti difference equation are finite and the solution of the Riccati difference equation plots discrete values on the exact solution of the Riccati differential equation. Also, the discrete model can also be understood as a model for the spread of infections of an epidemic virus with a constant infectious period and is described with a discrete susceptible–infected–recovered (SIR) model. The discrete SIR model has an exact solution. A control to reduce the infection is considered through the discrete SIR model.