The novel coronavirus pneumonia (COVID-19) has created great demands for medical resources. Determining these demands timely and accurately is critically important for the prevention and control of the pandemic. However, even if the infection rate has been estimated, the demands of many medical materials are still difficult to estimate due to their complex relationships with the infection rate and insufficient historical data. To alleviate the difficulties, we propose a co-evolutionary transfer learning (CETL) method for predicting the demands of a set of medical materials, which is important in COVID-19 prevention and control. CETL reuses material demand knowledge not only from other epidemics, such as severe acute respiratory syndrome (SARS) and bird flu but also from natural and manmade disasters. The knowledge or data of these related tasks can also be relatively few and imbalanced. In CETL, each prediction task is implemented by a fuzzy deep contractive autoencoder (CAE), and all prediction networks are cooperatively evolved, simultaneously using intrapopulation evolution to learn task-specific knowledge in each domain and using interpopulation evolution to learn common knowledge shared across the domains. Experimental results show that CETL achieves high prediction accuracies compared to selected state-of-the-art transfer learning and multitask learning models on datasets during two stages of COVID-19 spreading in China.