The frequent outbreak of severe foodborne diseases warns of a potential threat that the global trade networks could spread fatal pathogens. The global trade network is a typical overlay network, which compounds multiple standalone trade networks representing the transmission of a single product and connecting the same set of countries and territories through their own set of trade interactions. Although the epidemic dynamic implications of overlay networks have been debated in recent studies, some general answers for the overlay of multiple and diverse standalone networks remain elusive, especially the relationship between the heterogeneity and diversity of a set of standalone networks and the behavior of the overlay network. In this paper, we establish a general analysis framework for multiple overlay networks based on diversity theory. The framework could reveal the critical epidemic mechanisms beyond overlay processes. Applying the framework to global trade networks, we found that, although the distribution of connectivity of standalone trade networks was highly heterogeneous, epidemic behavior on overlay networks is more dependent on cooperation among standalone trade networks rather than on a few high-connectivity networks as the general property of complex systems with heterogeneous distribution. Moreover, the analysis of overlay trade networks related to 7 real pathogens also suggested that epidemic behavior is not controlled by high-connectivity goods but that the actual compound mode of overlay trade networks plays a critical role in spreading pathogens. Finally, we study the influence of cooperation mechanisms on the stability of overlay networks and on the control of global epidemics. The framework provides a general tool to study different problems on overlay networks.