key: cord-1044840-dx59l3gl
authors: Hasan, A.; Nasution, Y.; Susanto, H.; Putri, E.; Tjahjono, V.; Puspita, D.; Sukandar, K.; Nuraini, N.; Widyastuti, W.
title: Modeling COVID-19 Transmissions and Evaluation of Large Scale Social Restriction in Jakarta, Indonesia
date: 2020-11-03
journal: nan
DOI: 10.1101/2020.10.30.20222984
sha: 9a38d178590b19687bc3fe662de46254818e6014
doc_id: 1044840
cord_uid: dx59l3gl

This paper presents mathematical modeling and quantitative evaluation of Large Scale Social Restriction (LSSR) in Jakarta between 10 April and 4 June 2020. The special capital region of Jakarta is the only province among 34 provinces in Indonesia with an average Testing Positivity Rate (TPR) below 5% recommended by the World Health Organization (WHO). The transmission model is based on a discrete-time compartmental epidemiological model incorporating suspected cases. The quantitative evaluation is measured based on the estimation of the time-varying effective reproduction number (Rt). Our results show the LSSR has been successfully suppressed the spread of COVID-19 in Jakarta, which was indicated by Rt<1. However, once the LSSR was relaxed, the effective reproduction number increased significantly. The model is further used for short-term forecasting to mitigate the course of the pandemic.

The government of Indonesia has been criticized by its medical experts for its approach to handle the COVID-19 outbreak. Indonesia is among countries with the lowest testing rates, which potentially masks the scale of its outbreak.

The first COVID-19 cases in Indonesia were confirmed on March 2, 2020, with 5 two citizens tested positive. The spread of the infection in Indonesia is marked by super-spreading events [1] . At the beginning of the outbreak, the government introduced two terms for people who have symptoms but have not had a test yet: Person under Observation (ODP) and Patient under Surveillance (PDP).

The difference between ODP and PDP lies in the severity of the symptoms. 10 PDPs have more severe symptoms that need to be hospitalized. According to the Indonesian Health Ministry, a person who has made contact with people who are positive with COVID-19 or has been traveled to COVID-19 epicenter within 14 days before the onset of the symptoms will be categorized as ODP or PDP. ODPs and PDPs must be self-isolated at home or hospital and monitored Since 13 July 2020, the term ODP and PDP were no longer be used. Instead, The government of Indonesia introduced a new classification called "suspected cases" to address individuals who should be tested for COVID-19. According to the Indonesia Ministry of Health, a suspect is a patient who fulfills one of the following criteria: diagnosed with Upper Respiratory Tract Infection or URI 25 (fever more than 38 • C, and at least one symptom of respiratory illness like a cough or sore throat) and a history of travel or residence in a location reporting community transmission of COVID-19 disease during the 14 days before symptom onset, diagnosed with URI and having been in contact with a confirmed or probably COVID-19 in the last 14 days prior to symptom onset, or diagnosed 30 2 All rights reserved. No reuse allowed without permission.

(which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted November 3, 2020. with severe URI/pneumonia which requiring hospitalization and the absence of an alternative diagnosis that thoroughly explains the clinical presentation. In this paper, the number of suspected cases is obtained by combining ODP and PDP data. Figure 1 shows the number of suspected cases compared to active, recovered, and deceased cases. At the end of March, the cases soared to 1,200 cases, with 114 deaths. More than half of those cases and deaths came from Jakarta, more than any other provinces in Indonesia. Jakarta was indeed considered as the epicenter of the outbreak. On 31 March, the Indonesian government declared the outbreak as a national disaster. It announced stronger measures than social distancing to 40 suppress the virus's spread, which was called Large Scale Social Restrictions (LSSR). The LSSR allows regional governments to restrict the movement of people and goods within the regions.

The city of Jakarta became the first region to implement LSSR. The first stage of LSSR in Jakarta was implemented for two weeks, from 10 to 24 April would be required to wear face masks. LSSR restricted motorcycle taxis from carrying passengers and made it obligatory for hotels to accept people who are self-isolating. The LSSR has been extended twice. The first extension (LSSR II) This paper is organized as follows. In Section 2, we present a mathematical model incorporating suspected cases to describe COVID-19 transmission in Jakarta. A method to estimate the time-varying effective reproduction number (R t ) based on the Extended Kalman filter (EKF) is discussed in Section 3. In 60 Section 4, we discuss the evaluation of LSSR in Jakarta from 10 April until 4 June. A short-term forecast using the model is presented in Section 5. Finally, conclusions and recommendations are presented in Section 6.

In this paper, we propose a new compartmental epidemic model for COVID-65 19 transmissions in Jakarta. The motivation behind this new model is due to the fact that there are significant numbers of PDP and ODP (referred to as suspected cases) who show COVID-19 symptoms but have not been tested.

Our idea is to include this data when estimating the time-varying reproduction number R t . Our approach to estimate R t is based on Extended Kalman Filter 70 (EKF) implemented on a discrete-time stochastic augmented compartmental model [2] . Figure 2 shows the model diagram. Let us denote S, X, I, R, and D respectively as susceptible, suspected, active/confirmed, recovered, and deceased compartment. The force of infection from compartment S will enter compart-75 ment I and X equally. The only difference is that when the testing result 4 in compartment X is negative, the individual will go back to compartment S, whilst positive they will go to compartment I. Thus, we denote as the negative testing rate, while κ is the positive testing rate. Individuals from compartment X who died will go to compartment D at the rate of ω. An infected individual 80 from compartment I has two outcomes, either to R or to D. Thus, γ and δ denote the recovery and death rate, respectively. 

The SXIRD model consists of five ordinary differential equations (ODEs) and is given by:

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The system satisfies

notes the total number of population. It can be shown using Picard-Lindelöf theorem [3] that (1)-(5) has a unique solution, i.e., for any initial condition

Let us assume that S(0), X(0), I(0), R(0), D(0) ≥ 0. The solution for (2) is given by:

Thus, for X(0) ≥ 0 we have X(t) ≥ 0. Furthermore, the solution for (1) is given by:

Since X(t) ≥ 0, then S(t) ≥ 0. Using similar approach, we can prove that

Thus the SXIRD is well-posed mathematically and biologically.

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Following Lemma 1 in [4], and [5] , the basic reproduction number for the 100 SXIRD model (1)-(5) is defined as the spectral radius of the next generation matrix and is given by:

where Λ denotes the eigenvalue of the next generation matrix. Since in practice β = β(t) due to intervention, to take into account the decline number of susceptible individuals, the time-varying effective reproduction number is calculated 105 using the following formula [6] :

In this section, we use EKF algorithm to estimate the time-varying effective reproduction number R t (t). The EKF is an algorithm that provides estimates of unknown variables or parameters given measurements observed over time.

To estimate R t (t), the EKF is implemented to a discrete-time augmented compartmental SXIRD model. 7 All rights reserved. No reuse allowed without permission.

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Applying forward Euler method and augmented the infectious rate β(t) as a new variable, the discrete-time augmented SXIRD model is given by:

β(k + 1) = β(k).

where ∆t denotes the time step. The last equation is obtained assuming the infection rate as a piece-wise continuous function with rare jumps. Indeed, the jumps are assumed to happen every one day when a new data is obtained.

To simplify the presentation, we define an augmented state vector 120

x(k) = S(k) X(k) I(k) R(k) D(k) β(k) .

Thus, the discrete-time augmented SXIRD model (10)-(15) can be written as follows:

In order to use the EKF, we need to linearize the nonlinear model (17) 

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The copyright holder for this preprint this version posted November 3, 2020. where J f (x(k)) is the Jacobian matrix of f , given by:

where

After we obtain the Jacobian matrix, we can use a standard EKF algorithm to estimate the state and the parameter [2] . The idea of EKF algorithm is to 130 perform real-time data fitting on the suspected, active, recovered, and deceased cases. While performing the fitting, the algorithm produces the estimate of R t from (9). The EKF consists of two main steps: predict and update. In the prediction step, the method uses the model (17) to predict the state variable.

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The propagation of the state error covariance P is calculated based on the Ja-135 cobian matrix (19) and the error covariance of the model Q F (k). In this paper, Q F (k) is considered as a tuning parameter that minimize the error between the reported data and the estimated data. Let us denotex(n|m) as the estimate ofx at time n given observations up to and including at time m ≤ n. The prediction step is given below:

In the update step, first we compute the residualỹ(k + 1). The residual is the difference between the reported data and the estimated data. The filter estimates the current data by multiplying the predicted state and the measurement/data matrix C. Since R t is not measured, the data matrix is given by: 

The residual is then multiplied by the Kalman gain K(k + 1|k) to provide the 145 estimation of the state variablex(k + 1|k + 1). The update step is given below:

x(k + 1|k + 1) =x(k + 1|k) + K(k + 1)ỹ(k + 1) (35)

Here, R F (k) is also considered as a tuning parameter that minimize the error between the reported data and the estimated data. Figure 3 shows the 10 All rights reserved. No reuse allowed without permission.

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The copyright holder for this preprint this version posted November 3, 2020. ; https://doi.org/10.1101/2020.10.30.20222984 doi: medRxiv preprint results of EKF for real-time data fitting of COVID-19 cases in Jakarta. We can observe that the EKF estimates the number of cases accurately. 

The local government of Jakarta used the time-varying effective reproduction number R t as an indicator to assess the duration of LSSR. The estimated R t was obtained from daily confirmed cases and without considering the number of suspected cases. In this section, we use the estimated value of R t incorporating 155 the number of suspected cases to evaluate the LSSR in Jakarta. We first evaluate the difference between the estimated R t involving suspected cases and the one without suspected cases. In our calculation, we assume the infectious time T i is 12 days with standard deviation of 3 days. The recovery rate γ and the death In Figure 5 , we plotted the normalized R t and the period of LSSR in Jakarta.

The effect of LSSR can be shown in the next two weeks after the policy is applied because the symptoms of COVID-19 may appear 2-14 days after exposure [8] .

During the first period of LSSR, starting from 10 April 2020, the estimated value of R t for both models are decreasing. It may happen as the effect of several policies of local government, such as social distancing, wearing mask, and work from home [9] . During the second period of LSSR until early LSSR transition period, the estimate value of R t continues decreasing and subsequently the value 180 is less than 1. Consecutive decreasing of R t during LSSR I-III, indicates that 12 All rights reserved. No reuse allowed without permission.

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The copyright holder for this preprint this version posted November 3, 2020. ; https://doi.org/10.1101/2020.10.30.20222984 doi: medRxiv preprint the policy was applied successfully. As R t < 1 for several weeks during the LSSR transition, the restrictions were relaxed gradually for economic reasons [10] . Unfortunately, easing the policy restrictions seems to result in the increase of value of R t > 1 in just a short 185 time period. As a consequence, a more tightened policy should be considered by the local government of Jakarta to avoid the second wave of the outbreak.

In order to apply a proper LSSR policy based on the estimated value of R t , forecasts on the number of active, recovered, and deceased cases are required and are given in the next section. 190 

In this section, we provide short-term forecasts (60-day) on the number of COVID-19 cases. The forecasts are conducted in two presentations. First, by assuming the current measures are continued, a short-term forecast for the total number of cases with a 95% confidence interval is shown in blue-dashed line in 195 Figure 6 . The forecast is done by calculating the total case from (10)-(14). The actual total number of cases is shown in red-dashed line. From this figure, we can 13 All rights reserved. No reuse allowed without permission.

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The copyright holder for this preprint this version posted November 3, 2020. ; https://doi.org/10.1101/2020.10.30.20222984 doi: medRxiv preprint observe that our forecast provide a reasonable result. The longer the forecast, the wider the confidence interval. Furthermore, the forecast is conducted using (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted November 3, 2020. ; https://doi.org/10.1101/2020.10.30.20222984 doi: medRxiv preprint

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