Epidemic spreading dynamic is usually used to describe the virus spread in the biology system and information diffusion in the social system. Previous studies indicated that multiple epidemics rarely evolve dependently, but constantly interact and coevolve. In this paper, a novel mathematical model is proposed to study the link cooperation effect of two epidemics cooperatively spreading on complex networks. The link cooperation effect means that the first epidemic is transmitted through this link, and the second epidemic will have a higher transmission probability when it is transmitted through the same link. We develop an edge-based compartmental method and can well predict the numerical simulations. We find the link cooperation effect promote the epidemic outbreak size. The phase transition is closely relates to the strength of the link cooperation effect and network topology. For the cooperate epidemics on ER networks, the phase transition is discontinuous for a stronger link cooperation effect, otherwise the phase transition is continuous. When cooperative epidemics spread on SF networks, the phase transition is always continuous because of the existence of a few hubs. Our developed theoretical method can reasonably predict the above phenomena that are obtained from the numerical simulations based on the proposed mathematical model.

流行病传播动态通常用于描述病毒在生物系统中的传播和信息在社会系统中的传播。先前的研究表明，多种流行病很少独立发展，而是不断相互作用和共同发展。本文提出了一种新的数学模型来研究两种流行病在复杂网络上协同传播的链接协同效应。链路协同效应是指第一次疫情通过这个环节传播，第二次疫情通过同一环节传播的概率会更高。我们开发了一种基于边缘的分区方法，并且可以很好地预测数值模拟。我们发现链接合作效应促进了流行病的爆发规模。相变与链路协作效果的强弱和网络拓扑结构密切相关。对于ER网络上的合作流行病，相变是不连续的，以获得更强的链接合作效果，否则相变是连续的。当合作流行病在 SF 网络上传播时，由于存在几个枢纽，相变总是连续的。我们开发的理论方法可以根据所提出的数学模型合理地预测从数值模拟中获得的上述现象。由于存在几个枢纽，相变总是连续的。我们开发的理论方法可以根据所提出的数学模型合理地预测从数值模拟中获得的上述现象。由于存在几个枢纽，相变总是连续的。我们开发的理论方法可以根据所提出的数学模型合理地预测从数值模拟中获得的上述现象。