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The emergence of infectious disease COVID-19 has challenged and changed the world in an unprecedented manner. The integration of wireless networks with edge computing (namely wireless edge networks) brings opportunities to address this crisis. In this paper, we aim to investigate the prediction of the infectious probability and propose precautionary measures against COVID-19 with the assistance of wireless edge networks. Due to the availability of the recorded detention time and the density of individuals within a wireless edge network, we propose a stochastic geometry-based method to analyze the infectious probability of individuals. The proposed method can well keep the privacy of individuals in the system since it does not require to know the location or trajectory of each individual. Moreover, we also consider three types of mobility models and the static model of individuals. Numerical results show that analytical results well match with simulation results, thereby validating the accuracy of the proposed model. Moreover, numerical results also offer many insightful implications. Thereafter, we also offer a number of countermeasures against the spread of COVID-19 based on wireless edge networks. This study lays the foundation toward predicting the infectious risk in realistic environment and points out directions in mitigating the spread of infectious diseases with the aid of wireless edge networks.