The COVID?19 epidemic is facing the influence of a variety of complex practical factors, which makes the development of the epidemic uncertain. In order to overcome the problem of large error in epidemic forecasting results due to the limitations of many ideal assumptions based on the infectious disease compartment model, a time series forecasting model based on deep learning is adopted to predict the epidemic development, and an informer model based on transformer model is established. Attention mechanism and distillation mechanism are applied to the time series forecasting of epidemic data. The threshold autoregressive (TAR) model and a variety of mainstream recurrent neural time series prediction models are used as comparison models. Through simulation experiments, the current number of remaining infections in the epidemic data of China, America and Britain is predicted in the short term, and RMSE and MAE are used as evaluation indicators, and then the best model is selected for medium ? and long?term prediction. The experimental results show that the indicator value of the informer model is optimal in both RMSE and MAE, further indicating that the prediction accuracy of the informer model is higher than that of other comparative models in China, America and Britain. Finally, the Informer model is used for the development of the epidemic in China，America and Britain medium and long?term prediction.

For a long time, researchers mostly analyze the transmission process of infected nodes in complex networks to get the target of forecasting and arresting the extend of the infectious diseases. In this article, the SEIR propagation dynamics model was extended to the undirected and powerless large small world network, and the weights between nodes were given as infection ability. Two initial node selection methods were selected to carry out multiple simulation experiments. Based on the traditional method of judging the impact of transmission by the number of infected people and infection threshold, the specific values of infection probability, peak value and inflection point time were added to analyze the impact of initial node selection on transmission process more comprehensively. The compared experimental results show that the initial node which the degree is larger and the betweenness is larger, the larger the propagation scale, the faster the propagation speed and the shorter the equilibrium time. This study provides some reference value for guard against and control of the extend of infectious diseases.