Hybrid Propagation and Control of Network Viruses on Scale-Free Networks

How to accurately model and effectively suppress the spread of network viruses has been a major concern in the field of complex networks and cybersecurity. Most existing work often considers the transmission between the infected and uninfected nodes (i.e., horizontal transmission) and assumes that all new nodes connected to the Internet are susceptible, but the nodes might have been implanted with a backdoor or virus by attackers or infected nodes before connecting to the Internet. This vertical transmission also provides an important route for virus propagation. In this paper, we investigate the propagation of network viruses under the combined influence of network topology and hybrid transmission (i.e., horizontal and vertical transmissions). Through rigorous qualitative analysis, we identify the propagation threshold R0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_0$$\end{document} which determines whether viruses in the network tend to become extinct or persist, and explore the impacts of vertical transmission on the viral spread. Furthermore, we consider the problem of how to dynamically contain the hybrid spread of network viruses with limited resources. By utilizing optimal control theory, we prove the existence of an optimal control strategy. Finally, a group of representative simulation experiments verify the validity of the theoretical findings. Specifically, the simulation results show that the optimal control strategy proposed in this paper reduces the value of the target generic function J by 67.69% compared with no control.