The prediction of the spread of coronavirus disease 2019 (COVID-19) is vital in taking preventive and control measures to reduce human health damage. The Grey Modeling (1,1) is a popular approach used to construct a predictive model with a small-sized data set. In this study, a hybrid model based on grey prediction and rolling mechanism optimized by particle swarm optimization algorithm (PSO) was applied to create short-term estimates of the total number of confirmed COVID-19 cases for three countries, Germany, Turkey, and the USA. A rolling mechanism that updates data in equal dimensions was applied to improve the forecasting accuracy of the models. The PSO algorithm was used to optimize the Grey Modeling parameters (1,1) to provide more robust and efficient solutions with minimum errors. To compare the accuracy of the predictive models, a nonlinear autoregressive neural network (NARNN) was also developed. According to the analysis results, Grey Rolling Modeling (1,1) optimized by PSO algorithm performs better than the classical Grey Modeling (1,1), Grey Rolling Modelling (1,1), and NARNN models for predicting the total number of confirmed COVID-19 cases. The present study can provide an important basis for countries to allocate health resources and formulate epidemic prevention policies effectively.