EFFECT OF LONG VACATION ON DAILY CASES OF COVID-19 DURING PARTIAL RESTRICTION IN JAKARTA, INDONESIA

,


INTRODUCTION
Corona Virus (COVID- 19) is an infectious disease caused by Severe Acute Respiratory Syndrome Corona virus 2 (SARS-CoV-2). Humans may be infected without any symptoms while in some cases it causes severe symptoms including pneumonia, acute respiratory syndrome, kidney failure. In Indonesia, more than ten of thousands of people have been infected and more than thousands have been died from this virus up until end of 2020 (Ministry of Health RI, 2020). Due to its fast spread, World Health Organization (WHO) officially declared the COVID-19 as a pandemic on March 11, 2020 [1]. A reasonable way to deal with this pandemic is to know the best practice to implement such as correct health policies [2]. Therefore, a theoretical prediction of the virus spreading is important to make an effective health policy. Effective reproduction number (Rt) whose value describes how many persons can be infected by one who is already infected by the disease. As reported by [3], Rt define value are determined by four main factors: the duration of infection, frequency of contacts among persons within the population, number of vaccinated populations, and the implementation of health protocols to avoid the transmission within and around the environment. The spread of COVID-19 affected almost all areas in Indonesia's provinces, including capital city of Jakarta. Therefore, many studies have been conducted and reported on various aspects related to the pandemic. For instance, many questions have been raised including "how to fulfill the demand of public health facilities i.e., hospital and drug stocks required to cure the patients during the peak period of active cases?". This question may be used to address other questions, "when the pandemic will be over and how many people will be possibly infected?" Most answers to that question raised are based on a system of differential or difference 3 EFFECT OF LONG VACATION ON DAILY CASES OF  equations for describing an epidemic condition, namely of the (susceptible (S), Infected (I), Recovered (R) = SIR) model. An elaboration of a SIR models give the output for the solutions in terms of a prediction of when and how many infected population would exist at the peak of the pandemic.
The system of differential equations, may be turn into a system of difference equations used for obtaining an effective number formulation, see [4] and [5] that solved numerically the differential equations system for simulating the sizes of population S, I and R under some predefined assumptions. Recently, [6] worked with an analytical solution to formulate an effective reproduction number under some imposed conditions. Since the solution for a differential equation system fall into the families of the growth function such as logistic functions [7], the model parameterization may be obtained using nonlinear estimation, which is available in some statistical packages, see [8]. In another work, [9] proposed ARIMA model for prediction the number of COVID-19 positive cases. The application of an optical sensor with surface plasmon resonance was shown to have high sensitivity and excellent detection limits to further reduce the spread of the disease [10].
Handling the COVID-19 pandemic in an attempt for reducing the frequencies of contacts among people in Jakarta was done through the implementation of great social restriction (PSBB).
Study in [11] shows that restriction for at least 50% from the total contacts between individual within a certain region was able to suppress the pandemic. Contact tracing for COVID-19 involves three processes namely, identifying, assessing, and managing people who already infected to prevent wider spread in order to break the chains of transmission of this infectious disease. The identified persons who may have been exposed to COVID-19 through the appropriate test will be supervised up daily for 14 days within the region where last time they found [12]. As WHO noted that one of the critical elements of the implementation of contact tracing is a real-time data analysis.
The data analysis should include some potentially factors that have significant impacts either directly or indirectly towards the correct information needed to tackle the pandemic. For the direct impact example is an evaluation testing and contact tracing, see [13]. While for the indirect impact, e.g. to do evaluation of government's interventions see [14], [15] to break the chains of human-tohuman transmission for ensuring that the number of new cases generated by each confirmed maintained below. Different to many countries which already implemented fully level of social restriction through a lock down policy to the area being isolated, Indonesia applies some partials of social restrictions due to the level of emergency. The social restriction sometimes must be rearranged due to some moments of vacation which are considered as routine traditions. As an example a long periods of vacation is during Christmas up to New Year occasion. When people usually go to visit and gather their families. The people mobility may hence increase unexpected mobility and crowds of people during the pandemic. In this occasion based on the number of cases and tests each day together with the length day of vacations, we wish to test whether the daily number of cases may be positively affected by some serial's moments of vacations along end of year time during the social restriction periods. For this purpose, a structural model for time series data is chosen as our model. By implementing a structural model, at least three benefits to gain, firstly, daily fluctuations number of positive cases in the capital city of Jakarta may be described using more than one nonlinear functional forms e.g. logarithmic function to describe the trend components according to the effects of the number of tests as well as long holiday moments. Secondly, the use transformed logarithmic form into the linear functional provides an interval confidence for the estimate. Thirdly, as already observed by [16], that in India country whose demographic characteristics are not much different to Indonesia's, the use of model formulation based on SIR nor simulation were not good enough to predict the figure of the daily cumulative COVID-19 positive cases for certain periods of time.

METHODS
This study used secondary data from the website of the capital city of Jakarta (DKI Jakarta) Provincial Government's Covid-19 Task Force Provide Citation. The time series data was taken starting from 10 th April 2020 to 20 th December 2021 after the implementation of the first large- scale social restrictions. The data is arranged in the form of a table containing data of daily date, number of people tested per day, number of positive cases per day. At the early stage, the collected purposive data sampling was processed with MS Excel to yield the required the data modeling.
The processed data then were imported to a statistical program to perform model parameters estimation. The estimation of the structural model parameters consists of two steps followed by statistical testing for the feasibility of the model to be used as a predictor for the number of daily new positive cases.
The training data considered was obtained from April 10 to December 21, 2020 while the sample of predicted data was calculated starting from December 22, 2020. The level of accuracy was evaluated using Mean Absolute Percent Error (MAPE). The structural model in the study is illustrated by Figure 1.
Since DL value must represents a risk degree while maintaining the consistency of its dimensions with the number of tested persons per day so the dimension must be in unit day. The unit time is taken to ease the interpretation of the analysis results. Besides it seems to be a common sense, the longer the exposure time is the bigger infectious chance to happen. And also, the number of people tested per day is a kind of a controlled variable as its value may be adjusted or determined by the regulator according to a certain degree of emergency in handling the pandemic. In the literature of SIR model, [7] stated that number of positive cases per unit time and the infectious period changes exponentially with a mean one over the recovery rate.
Applying the logarithmic of based 1.

RESULTS AND DISCUSSION
Parameter estimation for the structural model is obtained by exchanging the random error t with the estimated error et as given in equations (5) below:  As parametric inferential statistical method, regression analysis requires normality assumption of residuals, i.e., the difference between the value of the dependent variable and the estimated results.
The Least Squares method used to obtain the estimates of the model parameters was based on sample data obtained from the URL address https://public.tableau.com/profile/jsc.data#!/vizhome/ LandingPageCovid-19Jakarta2/TablePositivityRate. Meanwhile, data on the start and end of the PSBB in DKI Jakarta were obtained from the URL https://ppid.jakarta.go.id/siaran-pers. Table 3, according to [17] the Kolmogorov Smirnov's statistics test on the residual model showed normality assumption because all values in the sig column indicating the z statistic values were less than 1.97 with significant level of 5%.  The SPSS outputs presented in Table 3  The structural model equations for prediction purposes presented by Equations (6)  SPSS software showed all coefficient estimate were significant at a significant level  < 1%.

Based on
Hence each unit changes in the value of the independent variables significantly contributes to the change for the dependent variable value yt. The SPSS output in Table 3 showed both the independent variables simultaneously contribute in the variability of daily number of new positive COVID-19 for about cases by 84% for level of significant 1%, and about 16% supplied by other factors. By recalling the independent variables mentioned above, hence the daily predicting new cases given by Equation (7): = 1.2 (11.392+0.0535 +.307 1 ) for 1 = 35.640 + 2.385 .
Based on the plot of rescaled time corresponding to the daily cases, it was noticeable that a structural change present mainly due to the implementation of large-scale social restrictions, confirmed with a decrease in daily new positive cases within some passages of time. However, due to several factors, such as discipline in implementing social restrictions, public obedience in carrying out health procedures, and also the 11-day long holiday during Christmas up to the New Year, an exponential spike was seen with new parameters greater than the original parameter values.
This similar pattern also met in the study by [14] who found nearly 44.06% increase in number of cases due to untimely end of lockdown scenarios in a certain region in India. As agreed with it, [11] reported in their simulation that of at least 50% social restriction were quite effective to decrease the cases. Moreover, as long as the social restrictions were carried out using appropriate procedures, [15] reported that the number of COVID-19 cases in Kuwait can be reduced. They use a modified ARIMA model to evaluate the impact of government's interventions through the partial and total lockdown interventions in Kuwait. The study showed that both interventions were very effective in lowering the number COVID-19 cases in Kuwait. The similar study was also conducted by [18] which integrate some aspects relating with the entities of demographic in the Boston metropolitan. It was found that a combination of a period of strict social distancing and a robust level of testing, contact-tracing and household quarantine could keep the disease within the capacity of the healthcare system while enabling the reopening of economic activities.
Based on the SPSS output in Table 4, where the statistical value of F is significant at the < 1% level of significance, it is possible to determine the predictive model for the periodic data structural model as follows. By applying the base 1.2 logarithm gives the result:  16, 2021 is 2778 persons. Table 4 shows the predicted value starting on January 8, 2021 to January 19, 2021 along with the prediction error using MAPE, which measures forecast accuracy by concentrating on a rescaled version of a measure.  Table 4 provides prediction results using the value of the DL variable, namely the moment of long holiday with a value range of 8 -8.5, considering that according to epidemiologists, the severity of outbreak transmission follows a normal distribution, in this case the data of the moment of long vacation in the period from 8 to 19 January 2021 were used. The total difference in the prediction value and the actual values MAPE is relatively less than 1.3%. As stated by [19], a forecast can be categorized accurate if MAPE is less than 5%.
The COVID-19 pandemic has resulted in many adverse outcomes and challenges including diagnosis can result in significant consequences such as delaying surgeries, unnecessary quarantine and treatments, transplant lists omission, and unnecessary sick leaves as stated by [20], [21].
Despite of these issues which may lead to substantial impact, there is a scarcity in the literature of its prevalence or impact, as admitted by [21], hence those issues must be considered to our result and then based on the gathered information relevant to our research papers we make some comparisons towards our findings as follows.
Firstly, [14] found doubling testing levels may results in identifying more cases about 13.84% in part of India. Also stated by [13], good and proper contact tracing system prior to the testing procedures were associated with a reduction in subsequent new infections of 63%. Those findings are critical elements toward the implementation of contact tracing in order to be able to work with a real-time data analysis which supplies valid information for monitoring the pandemic.
Secondly, according to [20], false positive results in the low prevalence level can have several adverse effects as already mentioned above, a particular one is a risk of subsequent increased exposure due to the believing that having been infected and being placed with other patients with COVID-19 and to more exposed to the virus, which may bring to more severe health condition.
Beside this negative consequently individually, the reported daily positive case results in bias information caused by unnecessarily addition of positive cases. Thirdly, [22], stated that 58% of COVID-19 patients may have initial false-negative RT-PCR, this figure was obtained based on their 95% CI finding for the overall false-negative rate 0.12 (0.10 to 0.14). To overcome this problem, [23] suggested a screening strategy in order to lower false-negative rate using group Mix with less than six members. In which it was claimed to be up to 63% saving in the number of tests compared to individual testing. By combining the two above mentioned methods, it is worth to expect the amount of tests can be increased more to reduce false-negative rate up to 58%. Fourthly, for the question of what will happen if large numbers of metropolitan people visit a less populated province during the long vacation? As qualitatively argued by [24] that minimizing events gathering both in the province and metropolitan groups and/or reducing the number of short-term visitors could substantially decrease spreading as could measure to lower the fraction initially infectious upon arrival. So based it is understandable during the long vacation moments the 15 EFFECT OF LONG VACATION ON DAILY CASES OF COVID-19 government control towards the citizen mobility still being relevant.
In the absence of quantitative measure toward the social restrictions at the time of the collected data being analyzed, a prediction function for the daily cases which measures the effect of long vacation moments through the number of daily testing done in the end of year of 2020 through the early of year 2021 was obtained. If we bring the result of [14] that doubling testing levels may results in identifying more cases about 13.84%, in terms of our predicted model increment as stated in Equation (7), doubling the number of testing may increase the positivity rate for about 12.9%, which is almost less than 2 times greater apart from the possibility of both negative and or positive false in the reported data as well as the mathematical functional being used.

CONCLUSION
The results of secondary data analysis using a two-step structural time series model with a