In 2022, a new outbreak of the COVID-19 pandemic created considerable challenges for the Shanghai public health system. However, conventional prevention and control strategies, which only rely on formal organizations, inefficiently decrease the number of infections. Thus, a multi-organization management mode is needed for pandemic prevention. In this paper, we applied a stochastic actor-oriented model (SAOM) to analyze how these social organizations cooperate with others and further identify the mechanism that drives them to create a reliable and sustainable cooperative relationship network from the perspective of social network analysis. The model allowed us to assess the effects of the actor’s attributes, the network structure, and dynamic cooperative behavior in RSiena with longitudinal data collected from 220 participants in 19 social organizations. The results indicated that the number of cooperative relationships increased during the pandemic, from 44 to 162, which means the network between social organizations became more reliable. Furthermore, all the hypotheses set in four sub-models were significant (t-ratio < 0.1, overall max t-ratio < 0.25, and e/s > 2). Additionally, the estimated values showed that four factors played a positive role in forming the cooperative relationship network, i.e., all except the “same age group effect (−1.02)”. The results also indicated that the social organizations tend to build relationships with more active actors in the community in every time period. This paper is of great significance regarding the innovation of public health system management and the improvement of Chinese grassroots governance.