An accurate and efficient forecasting system is imperative to the prevention of emerging infectious diseases such as COVID-19 in public health. This system requires accurate transient modeling, lower computation cost, and fewer observation data. To tackle these three challenges, we propose a novel deep learning approach using black-box knowledge distillation for both accurate and efficient transmission dynamics prediction in a practical manner. First, we leverage mixture models to develop an accurate, comprehensive, yet impractical simulation system. Next, we use simulated observation sequences to query the simulation system to retrieve simulated projection sequences as knowledge. Then, with the obtained query data, sequence mixup is proposed to improve query efficiency, increase knowledge diversity, and boost distillation model accuracy. Finally, we train a student deep neural network with the retrieved and mixed observation-projection sequences for practical use. The case study on COVID-19 justifies that our approach accurately projects infections with much lower computation cost when observation data are limited.