Spatio Temporal Modelling for Government Policy the COVID-19 Pandemic in East Java

COVID-19 has cursorily spread globally. Just in four months, its status altered into a pandemic. In Indonesia, the virus epicenter is identified in Java. The first positive case was identified in West Java and later spread in all Java. The Large-scale Social Restrictions are seemingly inefficient as the SARS-CoV-2 transmission remains. As such, the government is struggling to find anticipatory policies and steps best to mitigate the transmission. In this particular article, we used a Spatiotemporal model method for the total COVID-19 cases in Java and forecasted the total cases for the next 14 days, allowing the stakeholders to make more effective policies. The data we were using was the daily data of the cumulative number of COVID-19 cases taken from www.covid19.go.id. Data modeling was conducted using a generalized Spatio-temporal autoregressive model. The model acquired to model the COVID-19 cases in Java was the GSTAR(1)(1,0,0) model.


INTRODUCTION
As stipulated by WHO on 12 March 2020, COVID-19 had become a pandemic [1]. The virus, firstly identified in Wuhan in December 2019, rapidly spread throughout China and other 190 countries [2]. No research exactly explains how the SARS-CoV-2 was initially transmitted, but, in the meantime, it is believed that humans transmit this virus to humans. Later research reveals that symptomatic patients transmit SARS-CoV-2 through droplets or sneezes [3]. Moreover, another research mentions that SARS-CoV-2 can live in gas particles, e.g., air (generated through nebulizer) for approximately three hours [4]. Due to its relatively rapid transmission and mortality rate which cannot be overlooked and no definitive therapy found, COVID-19 is one of the diseases to which we should alert [5].
The Coronavirus epicenter in Indonesia is identified in Java. The first positive case was identified in West Java and later spread in all Java. It indicates that adjacent locations closely pertain to the SARS-CoV-2 transmission. In response to the virus, China's social distancing regulation is proven effective to stabilize the virus transmission, and hence the number declines [6]. Indonesia, similar to China, issues the same regulation, namely the Large-scale Social Restrictions (PSBB). Nevertheless, the regulation is seemingly inefficient as the SARS-CoV-2 transmission remains. As such, the government is struggling to find anticipatory policies and steps best to mitigate the transmission.

Data Source
The data we were using in this research were the daily data of the cumulative number of COVID-19 cases taken from www.covid19.go.id.

Data Stationarity
According to the stationary time series data, neither a sharp decrease nor an increase in data value nor fluctuated data was found around the constant mean value [15]. Stationary data had the mean ( ) = µ and variance ( ) = σ 2 . The mean value conditioned that data had to be stationary, so neither decrease nor an increase in data from time to time was allowed [16]. Furthermore, the characteristic of a stationary time series was endlessly constant average and variance. There were two types of time series stationarity, namely stationarity to variance and the mean.

a. Stationarity to Variance
Stationarity to variance was if ( ) = ( − ) for all t and k, the variance was constant from time to time [17]. To observe whether or not the data was stationary to variance, we used a Box-Cox plot. Non-stationary data could be altered into stationary ones through transformation.

b. Stationarity to the Mean
Stationarity to the mean was if ( ) = ( − ) for all t and k, the mean function remained constant from time to time. Stationarity to the mean was observed using the ACF (Autocorrelation Function) plot or the Dickey-Fuller test. Non-stationary data could be altered into stationary ones through differencing.

Generalized Space-Time Autoregressive Integrated (GSTAR)
The AR order was determined using the MPACF plot. Correlation between Zt and Zt+k, after a dependence relationship, was linear. The variables Zt+1, Zt+2, …, and Zt+k-1 were thus negated. The formula of correlation partial matrix function is as follows: Where ϕkk = Partial correlation matrix coefficient at lag k = Observation data at the time t ̂ = Predictor for + = Observation data at the time + ̂+ = Predictor for +

Spatio-temporal Modelling for Government Policy the COVID-19 Pandemic in Java
Atiek Iriany 220 The partial autoregression matrix at lag s became the last matrix coefficient when the data were leveraged for the vector autoregression process of the order s.
The best model was selected among some models considered feasible for MPACF testing. Model selection was conducted using AIC. The less the AIC value in a model, the better the model. The quantification of the AIC value was as follows: = the number of predicted parameters in the model T = the number of observations S(p) = residual sum of squares p = VAR model order The GSTAR model was introduced by Borovkova, Lopuha, and Ruchjana in 2020 in Wutsqa et al. [18]. It was more flexible and generalized than the STAR model and did not require the same parameter values at all locations. The GSTAR model ( , 1 , … . , ) is written as follows [19]: , diagonal matrix of the parameter space-time lag spatial 0 and the parameter autoregressive lag at the time k th Φ 1 = diag ( 1 1 , … , 1 ), diagonal matrix of the parameter space-time lag spatial 1 and the parameter autoregressive lag at the time k th W = weighing matrix (N×N) selected as such that ( ) = 0 dan ∑ ( ) = 1 ≠ e(t) = the white-nose vector in size of (N × 1) Z(t) = the random vector in size of (N × 1) at the time t Suhartono and Subanar [20] introduced a new method for determining weight using the result of cross-correlation normalization between locations at a congruent time lag.
The determination of location weight for the GSTAR model (1;p) is as follows: with i ≠ j and the weight had fulfilled ∑ ≠ = 1. The weight of cross-correlation normalization represented the variance of correlation between locations occurring in the data.

Spatio-temporal Modelling for Government Policy the COVID-19 Pandemic in Java
Atiek Iriany 221

RESULTS AND DISCUSSION
The COVID-19 cases in Indonesia were ever-increasing, and Java was regarded as the transmission epicenter. The increase in the COVID-19 cases is depicted in Figure 1.  Figure 1 indicates that as of 2 March-18 May 2020, the COVID-19 cases increased in all provinces in Java. On 18 May 2020, the highest number of cases, 5,555, was reportedly in Jakarta, whereas the lowest one, 185, was in Yogyakarta. Using the data of the total COVID-19 cases in six provinces in Java, we identified the correlation between provinces and the COVID-19 transmission in Java. Correlation between locations was identified using Pearson's correlation between provinces. The result of Pearson correlation quantification is presented in Table 1. In Table 1, we can see that the data of the number of the COVID-19 cases in six provinces in Java had a high Pearson's correlation value which was higher than 0.9. It implies that the correlation of the COVID-19 cases between provinces in Java was strong.

Data Stationarity Test
Data stationarity testing was performed in two stages which were stationarity to variance and stationarity to the mean. Stationarity to variance was tested using the boxcox transformation. Data were regarded stationary if the lambda value was 1, signifying that Var(Zt) = Var(Zt-k). The result of the stationarity test to variance is shown in Table 2.  Table 2, the initial data had not fulfilled the stationarity to variance yet. Several transformations were thus called for. After conducting the data stationarity test, we did the stationarity test to the mean. The test was conducted using an augmented Dickey-Fuller test. The result of the stationarity test to the mean is indicated in Table 3. From the augmented Dickey-Fuller test, we acquired predicted values less than the real ones (0.05). It indicates that the data had fulfilled the stationarity to variance.

Interpretation of the GSTAR Model Parameter
Model identification was aimed to find the autoregressive GSTAR model order. The order was elicited by identification using AIC. The lag with the smallest AIC value was regarded as the autoregressive GSTAR model order. Table 4 lists the AIC values.  Table 4 shows the smallest AIC value at the lag AR(1) and MA(0), hence the GSTAR(1)(1,0,0) model.

Interpretation of the GSTAR Model Parameter
The GSTAR model was a particular form of VAR engaging spatial elements. Estimating the GSTAR(1) (1,0,0) spatial parameters with the Ordinary Least Square method using cross-correlation normalization weight generated the following parameters. 0.088 ∅ 11 (6) -0.013 Referring to Table 5, we generated the matrix equation of the GSTAR(1)(1,0,0) model, which is as follows: The following matrix equation was derived from the above equation.  From the model generated, we made a comparison between the actual and predicted data, in which we acquired an RMSE and MAPE value of 0.005 and 1.43, respectively. The two gave us a hint that the model generated was good.

Prediction Result
From the equation, we forecasted the total cases for the next 14 days, namely 19 May-1 June 2020, the result of which is presented in Table 6.