Best selected forecasting models for COVID-19 pandemic

2.1 ETS model

The ETS model has three parts: the first letter “E” stands for error, the second letter “T” stands for trend, and the third letter “S” is for the seasonal pattern. The readers are referred to Hyndman et al. [34,35] to understand the three-character string terminology. The first character is an error type and can be attributed as (“A”, “M,” or “Z”), the second character is a trend type, which is (“N”, “A”, “M,” or “Z”) and the third character denotes the season type (“N”, “A”, “M,” or “Z”). The characters “N” = none, “A” = additive, “M” = multiplicative, and “Z” = automatically selected in all cases. The ETS (A, N, N) model, for example, denotes basic exponential smoothing with additive error, whereas the ETS (M, A, M) model denotes the multiplicative Holt–Winters method with multiplicative error, and so forth. The model can be fitted using the “ETS” function in the R programming language, which provides the optimal parameter values automatically, using the “forecast” package. For the application and comparison of various time series models, namely, ARIMA, ETS, TBATS, and hybrid models, to anticipate COVID-19’s second wave in Italy, refer to Perone [29].

2.2 Mathematical description of the model

Let z 1 , z 2 , ⋯ ⋯ , z n denote the time series of interest, the ETS (MNM) model is given by

where H 1 , H 2 , G 1 , G 2 , F 1 and F 2 are all matrix coefficients and x t and z t are the unobserved state vectors at time t. Here ε t for each case is normally distributed with mean = 0 and variance = 1.

Here h period ahead mean is denoted by μ h and variance δ h 2

where θ 1 = μ ∼ 1 2 , θ h = μ ∼ h 2 + δ 2 ∑ j = 1 h − 1 c j 2 θ h − j , k = h − 1 m , and m = period of seasonality. When h ≤ m , the expression yields the exact solution, otherwise approximation if h > m . The prediction interval can be obtained as follows:

2.3 Accuracy measures

The following accuracy measures are used to evaluate the best model: RMSE, MAE, and mean absolute percentage error (MAPE), whose mathematical representations are reported by Naeem et al. [30].

2.4 Neural network nonlinear autoregressive (NNAR) model

The input layer, hidden layer, and output layer make up the basic structure of an artificial neural network (ANN). The NNAR mode [29,36] is a feed-forward NN with a single hidden layer and the lagged value of the time series as inputs. The inputs are lags 1 to p and lag m to mp, where m is the time series’ frequency. For example, the frequency for annual data is 12, while for daily data, the frequency is 7. Both non-seasonal and seasonal data can be used with the ANN model. The fitted model for non-seasonal data is designated as NNAR (p, k), where p is a lagged input and k is the number of nodes in the single hidden layer. The NNAR (p, 0) model is similar to the ARIMA (p, 0, 0) model, but without the stationary constraints. The fitted model for seasonal data is called an NNAR (p, P, k) [m], which is similar to ARIMA (p, 0, 0) (P, 0, 0) [m] but without stationary constraints. The model can be fitted in the R programming language using the “forecast” package, which uses the “NNETAR” function to automatically generate the best parameter values. The model uses the default setting of P = 1 for the seasonal setting, and p is chosen from the best linear model fitted to the “seasonally adjusted data.” If k is not supplied, [p + P + 1]/2 is used instead. It is difficult to determine prediction intervals for the forecasts produced by neural networks since they are not based on a clearly defined stochastic model. Despite this, we can still calculate prediction intervals through simulation, where future sample routes are produced using bootstrapped residuals. A forecast error is given by

Therefore, we can write

where y ˆ t + 1 |t a one step ahead is forecast and e t + 1 is the unknown future error term. We can substitute e t + 1 by selecting errors from the past, assuming that future errors would be similar to the past errors. We may repeat the process to get the new simulated observation by including it in our data set.

where e t + 2 represents another pull from the residuals. By continuing in this manner, we can simulate a broad range of potential values for our time series. By doing this repeatedly, we get a lot of potential futures. By determining percentiles for each forecast horizon, we can next compute the prediction intervals. Because the method only requires that we use previous data to calculate future uncertainty, it is called the “bootstrap” method, which refers to pulling ourselves up by our bootstraps. Table 1 shows the monthly average number of confirmed cases, deaths, and recovered patients.

3.1 Reported COVID-19 confirmed cases

To determine the best forecasting model for confirmed COVID-19 cases in Pakistan, we examined a set of time series models. Figure 1 depicts the presentation of the confirmed patient series from April 16 to August 15, 2021, as well as the autocorrelation function (ACF) and partial autocorrelation function (PACF).

Six distinct time series models were used: ARIMA, neural networks, ETS models, and exponential smoothing methods (Brown, Holt, and Winters).

Table 3 shows a full description of fitted models with accuracy measurements, such as the RMSE and MAE. The ETS (M, N, M) forecasting model was found to be the best of all the competing models, with the lowest RMSE (439.55) and MAE (316.44). Table 4 shows the 10-point COVID-19 instances anticipated in Pakistan, with lower and upper confidence intervals (CIs) of 80 and 95%, respectively.

Figure 2 shows a graphical representation of the actual and fitted data, while Figure 3 shows a forecast with an 80 and 95% confidence band. The graphical representation of the real and confirmed cases indicates that the ETS has a high level of agreement (M, N, and M).

Figure 2

A plot of actual and predicted ETS (M, N, M) model.

Figure 3

A plot of forecasted ETS (M, N, M) model, with CI 80 and 95%.

3.2 Reported COVID-19-related deaths

Table 5 provides a detailed description of the fitted time series modes for the reported COVID-related deaths in Pakistan. Figure 4 depicts the ACF and PACF, as well as a series of deaths from April 16 to August 15, 2021. Table 6 shows the 10-point reported COVID-related deaths in Pakistan, with lower and upper CIs of 80 and 95%, respectively.

Figure 4

Reported COVID-19-related deaths, Pakistan.

When comparing the various fitted models, the NNAR (3, 1, 2) model is shown to be the most optimal. Based on the least forecasting error, the best forecasting model for predicting death due to COVID-19 was chosen. The RMSE and MAE errors of the NNAR (3, 1, 2) model are 17.25 and 12.22, respectively, as shown in Table 5.

The graphical representation of actual and fitted models are demonstrated in Figure 5, whereas Figure 6 shows the forecasted with an 80 and 95% confidence band. The real and fitted confirmed cases are represented graphically, which are in good agreement under the NNAR (3, 1, 2) model.

Figure 5

A plot of actual and predicted NNAR (3, 1, 2) model.

Figure 6

A plot of the forecasted NNAR (3, 1, 2) model, with CI 80 and 95%.

3.3 Recovered or discharged patients

Figure 7 shows the display of recovered patient series from April 16, 2021 to August 15, 2021 along with ACF and PACF.

Figure 7

Reported COVID-19 recovered patients, Pakistan.

The NNAR (8, 1, 4) model is shown to be the most optimal among the numerous fitted models when compared to its counterpart models. Based on the least forecasting error, the best forecasting model for predicting daily recovered or discharged patients from the hospital was chosen. Table 7 shows the NNAR (8, 1, 4) model’s RMSE (551.805) and MAE (325.68). Table 8 shows the 10-point COVID-19 instances anticipated in Pakistan, with lower and upper CIs of 80 and 95%, respectively.

The graphical representation of the actual and fitted models are demonstrated in Figure 8, whereas Figure 9 shows the forecast with an 80% and 95% confidence band. The graphical representation of the actual and fitted cases shows good agreement under the NNAR (8, 1, 4) model.

Figure 8

A plot of the actual and predicted NNAR (8, 1, 4) model.

Figure 9

A plot of the forecasted NNAR (8, 1, 4) model, with CI 80 and 95%.

Pakistan cannot afford a state-wide total lockdown due to the country’s weak economic status, as people will die of famine rather than COVID-19. As a result, Pakistan adopted a smart lockdown, and the concept was widely accepted around the world. In Pakistan, it was proven to be effective in controlling COVID-19 instances. Despite its lack of healthcare facilities, Pakistan’s government successfully executed the lockdown. On the other hand, the situation in India is not as bad as it is in many other developing countries. However, the fight is far from over. COVID-19-related policies that are wise, effective, and proactive are still needed, as is the maximal administration of COVID-19 vaccination doses to the elderly population. Simultaneously, the government and other non-governmental organizations must conduct an effective public awareness campaign to encourage people to get vaccinated, while simultaneously exposing the COVID-19 vaccine’s uncertainties and misinformation. Because the selected forecasting models ETS (M, N, M), NNAR (3, 1, 2), and NNAR (8, 1, 4) predicted that COVID-19 confirmed outbreaks, deaths, and recovered or discharged patients are expected to rise by 0.75, 5.08, and 19.11% daily in Pakistan, to prevent the spread of the delta form, Pakistan’s fourth spike, the national command and operation center should strictly enforce the social vaccination, which includes masks, social distance, and handwashing.