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 \(R_0\) 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.

如何准确建模并有效抑制网络病毒的传播一直是复杂网络和网络安全领域的一大关注点。大多数现有工作通常考虑受感染和未受感染节点之间的传播（即水平传播），并假设所有连接到互联网的新节点都容易受到影响，但这些节点之前可能已被攻击者或受感染节点植入后门或病毒。连接到互联网。这种垂直传播也为病毒传播提供了重要途径。在本文中，我们研究了网络拓扑和混合传播（即水平和垂直传播）的综合影响下网络病毒的传播。通过严格的定性分析，我们确定了决定网络中病毒是否趋于灭绝或持续存在的传播阈值\(R_0\) ，并探讨了垂直传播对病毒传播的影响。此外，我们还考虑如何在有限的资源下动态遏制网络病毒的混合传播问题。利用最优控制理论，证明了最优控制策略的存在性。最后，一组有代表性的仿真实验验证了理论研究结果的有效性。具体来说，仿真结果表明，本文提出的最优控制策略与无控制相比， 使目标泛函数J的值降低了67.69%。