## 3 Results and Discussion
### Discussion
The GRU DL network has an intermediately complex structure, situated between a simple structure (RNN) and a complex structure (LSTM). Therefore, this may result in a moderate training time. In conclusion, the GRU can be recommended for the case study of Italy. The relatively bad performance of the three DL techniques can be attributed to the fluctuations in daily new infections noticed particularly starting from Day 95 (Figure 3c). Note that, among recent studies concerning the prediction of the number COVID-19 infections in Italy, the authors of [36] focused on predicting the reproduction number, which is complementary to the number of infections predicted in our study. Future directions can be built around a combination of the two topics. * Case of India: The best MAPE was provided by the RNN (17.58%) and the best \(R^{2}\) and RMSE (0.8759 and 3902, respectively) were yielded by the GRU DL algorithm. In this case study, the GRU can be adopted as the best forecasting technique. From Figure 3d, it can be observed that LSTM fails in capturing the strong fluctuation in daily new infections occurring around Day 120. However, the average value of the MAPE of the three techniques is relatively high, which reflects the relatively bad predictability of the daily new infections using DL. As an overall observation, the values of MAPE yielded by the three DL techniques can be considered moderate (ranging from 8% to 25%). The values of \(R^{2}\) can be considered to range from good to very good (ranging from 0.8 to more than 0.9). The RNN technique is recommended for three case studies (Saudi Arabia, Egypt, and India), and GRU is recommended for Italy. From this, it can be concluded that there is no need to use complex-structure DL networks to model the daily new infections dynamics. Simpler DL algorithms can yield better forecasts within reasonable training time. Moreover, DL techniques are known to require large amounts of data, which is not true in the case of COVID-19, since health authorities are releasing data on a daily basis. The 80% used (540 observations out of 675 collected) can be considered a relatively small dataset to train DL algorithms. It should also be noted that, when compared with the results of model-based techniques, the forecasts yielded by the three DL techniques can be considered good, since the coefficient of determination (\(R^{2}\)) ranges between 0.8 and 0.9, which was the case in many studies using classical models. However, a fair comparison should be conducted between methods from the same family, under the same conditions, and using the same datasets in the same country/location. Forecasting COVID-19 dynamics was shown through several studies (and confirmed in this study) to be case-sensitive. Overfitting is the situation where a DL algorithm performance on the training dataset is good (in-sample) and \"bad\" for new dataset or testing (out-of-sample). To investigate this problem, we performed the following:  
* We conducted various experiments (for the four case studies) while varying the number of iterations (epochs). The quality of the DL models was measured by calculating three performance metrics (the RMSE, the coefficient of determination (\(R^{2}\)), and the mean absolute percentage error (MAPE)) for the out-of-sample datasets. * As seen from Tables 4 and 5, the coefficient of determination was found to be more than 0.8 and reached 0.97 in some cases (this value varies between 0 and 1 for the ideal fit). In addition, the MAPE and RMSE were in good ranges, which indicates the \"good\" quality of our investigated models and no overfitting. Predictions and modeling in the case of pandemics are very useful for supporting decision-making. In fact, in the case of the unprecedented COVID-19 pandemic, several models have been developed under different circumstances in different locations. DLmodels including those developed in this study, were found to be very useful since they were able to take into consideration hidden factors implicitly.