Because of the lack of reliable information on the spread parameters of COVID-19, there is an increasing demand for new approaches to efficiently predict the dynamics of new virus spread under uncertainty. The study presented in this paper is based on the Case-Based Reasoning method used in statistical analysis, forecasting and decision making in the field of public health and epidemiology. A new mathematical Case-Based Rate Reasoning model (CBRR) has been built for the short-term forecasting of coronavirus spread dynamics under uncertainty. The model allows for predicting future values of the increase in the percentage of new cases for a period of 2–3 weeks. Information on the dynamics of the total number of infected people in previous periods in Italy, Spain, France, and the United Kingdom was used. Simulation results confirmed the possibility of using the proposed approach for constructing short-term forecasts of coronavirus spread dynamics. The main finding of this study is that using the proposed approach for Russia showed that the deviation of the predicted total number of confirmed cases from the actual one was within 0.3%. For the USA, the deviation was 0.23%.