Methodology

This research is based on the Prophet model, which is generally used for business forecasting. As mentioned, this model includes parameters like holidays, trend, and seasonality, which would help mold the prediction results. The Prophet model is usually not used in the scope of predicting a pandemic, but we have lined the parameters to analyze predictions based on the Prophet model, such as using lockdown as holidays. This research is conducted in 2 phases. The first phase is conducted during the lockdown and the second phase, after the lockdown. During the lockdown phase, strict rules were imposed by the government and no relaxation was there. This phase enlightens the fact of how controlled the situation can be, keeping the complete lockdown rules in sight. This was one of the primary periods of concern for this research. The next circumstance, which needs to be understood, is the amount of change in the number of positive cases of COVID-19 that comes into account when laws are relaxed for the lockdown and people start returning to the new normal. This period is termed as period after the lockdown. Since the number of deaths is highly dependent on the total number of cases, it’s why the prediction made on deaths is done taking the whole period of lockdown into consideration.

COVID-19 miserably affected everyone’s lives, so there is a need for a prediction model that can forecast the situation and alert us to be better prepared for the future. This research has been performed using the data science website, www.kaggle.com (accessed May 24, 2020), using the data set named “COVID-19 in India.” The data set gets updated daily for cases for all states and union territories since January 30, 2020. For this research, the interval from March 24 to May 24 has been considered. As of May 24, 2020, the dimensions of the data are 2306 rows and 7 columns. The data set has been pre-processed by removing the missing values where no case for a state was recorded and total cases have been added as shown in Table 1. For the prediction purpose, a pivot table was obtained from the data set where all the total cases and deaths were summed cumulatively according to the dates. Python is used as the programming language.

Table 1.

Data set of COVID-19 cases in India as of May 24, 2020

Figure 1 demonstrates the procedure adopted for conducting this research.

Figure 1.

Sequence diagram for methodology.

Commonly used approaches for predicting the number of cases in epidemics are either time series analysis or the SIR model, where S stands for the number of people who are still at risk that the disease might infect them. The number of infected people at time t is given by I(t). We assume that I(0) > 0 because, otherwise, the epidemic cannot start. R(t) symbolizes the number of removed people at time t. In other words, the number of deaths that have been seen at time t. ^{Reference Taylor and Letham14} There are some limitations to this model. This model assumes that every person is moving and has an equal chance of getting in contact with every other person among the population, irrespective of the space or distance between them. It is assumed that the transmission rate remains constant throughout the pandemic. Moreover, this model does not cater to those infected who have been diagnosed or are in quarantine. It treats the same as those who have not been quarantined. Therefore, both are considered to have the same transmission rate. A time series is the sequence of one or more values per time step. This analysis can take the trend, seasonality, and other factors into account for forecasting purposes. There are some other methods that can be used for a time series analysis like the Prophet, LSTM, and ARIMA models.

In this research, the Prophet model has been used for the analysis and prediction as it has some merits over other models such as the ARIMA model; in the Prophet model, the interpolation of missing values is not required. Measurements need not be regularly spaced. Analysts can also add seasonality and holidays to improve the model performance and perhaps obtain a better prediction and forecasting. As in the case of COVID-19, the lockdown was imposed, which can be considered as an influencing factor toward the prediction. Analysts tend to have experience with regression and therefore can easily extend the model to include new components. Regression refers to a set of statistical processes for the estimation of the relationship between the target variable and 1 or more features. The Prophet model has interpretable parameters, which can be changed by the analyst to provide some assumptions for the prediction. The analyst can make assumptions on different trends. While using the ARIMA model, trend and seasonality need to be removed and later added while making the forecasts. There are some reasons in which Prophet performs better than the LSTM model. LSTM requires more data as compared with Prophet for a time series analysis in order to compute more accurate results. The reason behind LSTM requiring more data is that it needs to understand the non-linearity of the data provided. The Prophet model can add the seasonality and holiday parameters, which produce assumptions on the predictions. The Prophet model is easy to implement as compared with LSTM networks. LSTM networks do not learn the trend or seasonality of the data. The time complexity of the neural network that consists of LSTM is ( $O\left( {{n^5}} \right)$ ), which is quite high as compared with the complexity of the Prophet model that is $O\left( {{n^2}} \right)$ .

The Prophet model was developed by Facebook in 2017. ^{Reference Taylor and Letham14} The forecasting has been done for the next 30 days starting from May 25, 2020. This model is based on the regression technique (Equation 1). This uses only time as a regressor, but several linear and non-linear functions of time can be used as components.

(1) $$y\left( t \right) = g\left( t \right) + h\left( t \right) + s\left( t \right)$$

Here, $g\left( t \right)$ is the trend function that models non-periodic changes in the value of the time series, $s\left( t \right)$ represents periodic changes (eg, weekly and yearly seasonality), and $h\left( t \right)$ represents the effects of holidays that occur on potentially irregular schedules over 1 or more days.

Trend $g\left( t \right)$ can be classified into 2 types: saturation and piecewise linear models.

Equation 2 represents the non-linear and saturation growth trend. $C$ is the carrying capacity, $\kappa $ is the growth rate, and $m$ is an offset parameter. Carrying capacity is some maximum achievable point like the total market size, total population, and so on. An offset can be described as the location of a data point with respect to another location of a data point. The carrying capacity in this equation is replaced by a time-varying capacity C(t) because the capacity of any application is usually not constant, and it could change depending upon the time. The growth rate is also not constant.

(2) $$g\left( t \right)\; = {C \over {1 + \;exp( - \kappa \left( {t - m} \right)}}$$

Piecewise Function (Equation 3) is the function that is broken down into parts where each part is based on change points provided. ^{Reference Taylor and Letham14}

(3) $$a( t ) = \cases{1, t \ge s_j \cr 0, Otherwise}$$

(4) $$g\left( t \right)\; = \;\left( {\kappa + a\left( t \right)T\;\delta } \right)t\; + \left( {m + a\left( t \right)T\;\gamma } \right)$$

Equation 4 represents the Linear Trend with change points. Change points are defined as the points where the growth rate is allowed to change. In Equation 4, κ is the growth rate, δ is the rate of adjustment, $m$ is the offset parameter, and ${\gamma _j}$ is set to $ - {s_j} \times {\delta _j}$ to make the function continuous; ${s_j}$ is a specific change point where $j\; = 1, \ldots ,S$ . S is the total number of change points; ${\delta _j}$ is the change in the rate that takes place at ${s_j}$ .

Seasonality models are periodic functions of t (Equation 5). The Fourier series provide a flexible model of periodic effects. Let P be the regular period we expect the time series to have. For example, P = 365:25 for yearly data and P = 7 for weekly data, if we scale our time variable in days.

(5) $$s\left( t \right) = \;\sum\nolimits_{n = 1}^N {\left( {{a_n}cos\left( {{{2\pi nt} \over P}} \right) + {b_n}sin\left( {{{2\pi nt} \over P}} \right)} \right)} $$

Holidays, h(t), can be defined as time, t, during the holiday (Equations 6 and 7). Assign each holiday a parameter, ${\kappa _{i{\rm{\;}}}}$ , which is the corresponding change in the forecast.

(6) $$Z( t) = [ {\bf 1}(t \; \varepsilon \;D1) \ldots .{\bf 1}( {t\;\varepsilon \;DL)} ]$$

and taking

(7) $$h\left( t \right)\; = \;Z\left( t \right) \times \kappa$$

The parameters such as holidays, growth, and seasonality are responsible for the working of the Prophet model, which is evident from Equation 1. The Prophet procedure adopts an additive regressive model using 4 main components, that is, growth function, a yearly seasonal component modeled using Fourier series, a weekly seasonal component modeled using dummy variables, and analyst provided holidays list.

The horizon (H) is the number of days that the model is forecasting (Equation 8). This is typically 30, 60, 90, and so on, depending on the application that the user is trying to build. For any forecast that takes place to H, the forecast states will be produced with some error. Let $\hat y(t|T)$ represent a forecast at time t made with historical information up to time T. Let $y\left( t \right)$ represent the actual value at time t.

(8) $$\emptyset (T,h) = d(\hat y(T + hT),y(T + h))$$

The reliability and validity of the prediction observed through the Prophet model is based on its performance metrics, which are as follows.

The absolute error is computed by the absolute difference of the predicted result ( $\hat y$ ) and the actual result (y) for each example in the data set. Mean absolute error (MAE) is the average of absolute errors computed over the data set. Squared error is the square of the difference of the predicted result and actual result for each example in the data set. Mean squared error (MSE) is the average of squared errors computed over the data set. Root mean squared error (RMSE) is the square root of MSE. The mean absolute percentage error (MAPE) is a measure of the prediction accuracy of a forecasting method in statistics, for example, in trend estimation, it is also used as a loss function for regression problems in machine learning. MAPE is preferred for its interpretability as an error metric for forecasts. Equations 9, 10, 11, and 12 show equations for MAE, MSE, RMSE, and MAPE, respectively.

(9) $$MAE = {1 \over n}\sum\nolimits_{j = 1}^n {| {{{\hat y}}_{j\;}} - {y_j}} | $$

(10) $$MSE = \;{1 \over n}\sum\nolimits_{j = 1}^n {{{\left| {{{\hat y}_{j\;}} - {y_j}} \right|}^2}} $$

(11) $$RMSE = \;\sqrt {{1 \over n}\sum\nolimits_{j = 1}^n {{{\left| {{{\hat y}_{j\;}} - {y_j}} \right|}^2}} } $$

(12) $$MAPE = {1 \over n}\sum\nolimits_{t = 1}^n {\left| {{{{A_t} - {F_t}} \over {{A_t}}}} \right|} $$

where A_t  is the actual value, F_t  is the forecast value, and $n$ is the total number of observations. The absolute value in this calculation is summed for every forecasted point in time and divided by the number of fitted points, n. Multiplying by 100% makes it a percentage error. ¹⁶