The COVID-19 disease has spread worldwide since 2020, causing a high number of deaths as well as infections, and impacting economic, social and health systems. Understanding its dynamics may facilitate a better understanding of its behavior, reducing the impact of similar diseases in the future. Classical modeling techniques have failed in predicting the behavior of this disease, since they have been unable to capture hidden features in the data collected about the disease. The present research benefits from the high capacity of modern computers and new trends in artificial intelligence (AI), specifically three deep learning (DL) neural networks: recurrent neural network (RNN), gated recurrent unit (GRU), and long short-term memory (LSTM). We thus modelled daily new infections of COVID-19 in four countries (Saudi Arabia, Egypt, Italy, and India) that vary in their climates, cultures, populations, and health systems. The results show that a simple-structure RNN algorithm is better at predicting daily new infections and that DL techniques have promising potential in disease modeling and can be used efficiently even in the case of limited datasets.