CONTACT:  

Muhammad Fajar,          mfajar@bps.go.id            Badan Pusat Statistik-Statistics Indonesia, Jakarta 10710, 

Indonesia  

 

       The article can be accessed here. https://doi.org/10.15642/mantik.2021.7.1.67-73  

Modeling of COVID-19 Epidemic Growth Curve in Indonesia 
 

 

Muhammad Fajar1, Wahyudi2 

1,2Badan Pusat Statistik-Statistics Indonesia, Indonesia 

 

 

 

Article history: 

Received Jun 26, 2020 

Revised Mar 8, 2021 

Accepted May 30, 2021 

 

 

Kata Kunci:  

Pemodelan, Logistik, 

Gompertz, Epidemi, 

COVID-19 

 

Abstrak. Tujuan studi ini adalah untuk melakukan pemodelan 

parametrik terhadap kurva pertumbuhan epidemi COVID-19 sehingga 

dapat diperoleh nilai maksimum dan waktu pada titik tersebut dari 

kumulatif kasus COVID-19. Sumber data yang digunakan adalah data 

jumlah kumulatif kasus terkonfirmasi positif COVID-19 yang berasal 

dari https://covid19.go.id/. Metode yang digunakan dalam penelitian ini 

adalah fitting data dengan model Logistik dan Gompertz. Hasil 

penelitian ini adalah (1) model Logistik dan Gompertz memiliki tingkat 

akurasi tinggi (konteks fitting data) dalam memodelkan kurva 

pertumbuhan epidemi COVID-19, diindikasikan dari nilai R2 (koefisien 

determinasi) yang mencapai lebih dari 99%; (2) Dari model Logistik 

diperoleh bahwa estimasi jumlah kasus kumulatif maksimum pada 

akhir epidemi COVID-19 adalah 7.714 kasus konfirmasi positif yang 

dicapai dalam waktu sekitar 82 hari (22 Mei 2020) dari tanggal 2 Maret 

2020 ketika kasus positif COVID-19 pertama diumumkan oleh 

pemerintah; dan (3) Dari model Gompertz diperoleh bahwa estimasi 

jumlah kasus kumulatif maksimum pada akhir epidemi COVID-19 

adalah 33.975 kasus konfirmasi positif yang dicapai dalam waktu 

sekitar 152 hari (30 Juli 2020) dari tanggal 2 Maret 2020. Temuan ini 

dapat dijadikan bahan input pembuatan langkah pemerintah dalam 

mengendalikan penyebaran COVID-19. 

 

Keywords:  

Modeling, Logistic, 

Gompertz, Epidemic, 

COVID-19 

 

 

Abstract. Aim of this study is to make parametric modeling of the 

COVID-19 epidemic growth curve so that the maximum value and time 

at that point can be obtained from the cumulative cases of COVID-19. 

The data used in this study is the cumulative number of positive 

confirmed cases of COVID-19 from https://covid19.go.id/. The method 

used in this study is fitting data with the Logistic and Gompertz models. 

Result of this study are (1) the Logistic and Gompertz models are very 

fit in modeling the COVID-19 epidemic growth curve, indicated from 

the value of R2 (coefficient of determination) which reaches more than 

99%; (2) From the Logistics model it is obtained that the estimated 

amount of the maximum cumulative case at the end of the COVID-19 

epidemic is 7,714 positive confirmed cases, achieved in about 82 days 

(May 22, 2020) from Mar 2, 2020, when the first positive COVID-19 

case was announced by the government; and (3) From the Gompertz 

model, it is obtained that the estimated maximum cumulative case at 

the end of the COVID-19 epidemic is 33,975 positive confirmed cases, 

achieved in about 152 days (Jul 30, 2020) from Mar 2, 2020. The results 

of this study can be used as input to the government to take steps in 

controlling the spread of COVID-19. 

  

How to cite: 

M. Fajar and Wahyudi, “Modelling of Covid-19 Epidemic Growth Curve in Indonesia”, J. Mat. 

Mantik, vol. 7, no. 1, pp. 67-73, May 2021. 

 

Jurnal Matematika MANTIK 

Vol. 7, No. 1, May 2021, pp. 67-73 
 

ISSN: 2527-3159 (print) 2527-3167 (online) 

mailto:mfajar@bps.go.id
https://doi.org/10.15642/mantik.2021.7.1.67-73
https://covid19.go.id/
https://covid19.go.id/
http://u.lipi.go.id/1458103791


Jurnal Matematika MANTIK 

Vol. 7, No. 1, May 2021, pp. 67-73 

68 
 

1. Introduction 

Novel Coronavirus 2019 (COVID-19) is an infectious disease caused by SARS-CoV-

2, one of the coronavirus types [5]. This virus can cause Pneumonia, which is inflammation 

of the lung tissue causing interference with oxygen exchange, resulting from breathing, 

becoming congested. Patients with COVID-19 can experience fever, dry cough, and 

breathing difficulties.  

In Indonesia, COVID-19 has spread to 32 provinces. As of Apr 12, 2020, the 

cumulative number of positive confirmed cases was recorded at 4,241 cases, with the details 

of as many as 3,509 patients in the treatment, 359 patients recovered, and 373 patients died 

[13]. A total of 80.22% of COVID-19 positive cases in Indonesia came from Java Island. 

DKI Jakarta is the province with the most COVID-19 case found, with 2,044 cases of 

positive confirmed cases, 1,707 patients in the treatment, 142 patients cured, and 195 died 

[13]. 

 

 
Figure 1. Number of cumulative cases of COVID-19 in Indonesia, as of Apr 12 2020 

 

Figure 1 visually presents the cumulative cases of COVID-19 or also called the 

epidemic curve. When we look at Figure 1, the cumulative cases of COVID-19 continue to 

increase over time, and the pattern formed is nonlinear. Generally, on the epidemic growth 

curve, the movement of the data moves nonlinearly to reach a point where the cumulative 

case number will not increase again (Stable data movement) which is visually shaped in a 

horizontal line pattern.  

To explore more detailed information on these curves, modeling is needed. Modeling 

in epidemiology [1,2,4,7,9,11,14,15,16] is commonly used to estimate parameters, such as 

how large the maximum cumulative case can be achieved by an epidemic [2,10] and the 

epidemic's growth rate and the point of time when the maximum cumulative case is 

reached. Estimations of these parameters can be used as an input for the formation of a 

mitigation plan for the Government.  

Therefore, in this study, the authors modeled the COVID-19 epidemic curve with a 

parametric model (analogy based on references [11,12,17]), namely the Logistics and 

Gompertz functions. Considerations of using these parametric models include: (1) can 

predict the time point when reaching the maximum asymptotic peak point in the cumulative 

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M, Fajar and Wahyudi 

Modelling of Covid-19 Epidemic Growth Curve in Indonesia 

69 
 

data of a disease (in this case is COVID-19), (2) can predict how large the maximum 

cumulative COVID-19 case is achieved, (3) Ease and calculation speed in parameters 

estimation process is easier and faster than Generalized Logistic model, and (4) unlike the 

SIR model (Susceptible, Infectious, Recovered) which is commonly used for studies of 

disease transmission (epidemics) [5, 14] and requires assumptions, among others [8]: (a) the 

population is constant, (b) the birth rate is equal to the death rate in the population, (c) 

changes in susceptible and infected individuals are proportional to the population, (d) 

infected individuals assumed to be able to recover with constant chance over time, (e) 

constant speed in terms of transmission and recovery, and (f) it is assumed that once the 

person has been infected and then has recovered, then that person will not be re-infected due 

to strong immunity, the Logistic and Gompertz models do not require these assumptions. 

The difference in the length of time in reaching the inflection points of the two models can 

be used as the basis for an optimistic and pessimistic scenario in controlling COVID-19 with 

the assumption that there is no intervention, meaning that the growth process of the COVID-

19 epidemic runs naturally without intervention. The implication is for stakeholders to 

manage the risk of an ongoing epidemic so that they can make the right steps. 

 

2. Methods 

2.1   Data Source 

The data used in this study is daily data on the cumulative number of COVID-19 

positive confirmed cases in Indonesia as of Apr 12 2020, sourced from www.covid19.go.id 

[13]. 

 

2.2   Growth Model 

The cumulative growth of COVID-19 cases (by analogy as referenced [11, 12, 17]) 

indicates a phase, in which the growth rate starts from zero at t = 0, and then accelerates to 

𝜇𝑚  within a certain time interval of 𝜆. When heading to the final phase of the epidemic, 

growth rates are decreasing to reaching zero (it means the rate of addition of new cases is 

very low so that the number is approaching zero even reaches zero, the implications of the 

cumulative case data plot will be horizontally shaped), and the number of final cumulative 

cases is achieved asymptotically (maximum, a).   

 
Figure 1. Epidemic Growth Curve 

Time 



Jurnal Matematika MANTIK 

Vol. 7, No. 1, May 2021, pp. 67-73 

70 
 

In plotting1 data on the cumulative number of COVID-19 cases, the authors used the 

parametric model as follows: 

a. Logistics Model [11,12,17] 

𝑦𝑡 =
𝐴

1 + exp (
4𝜇𝑚

𝐴
(𝜆 − 𝑡) + 2)

 ,                                                       (1) 

where 𝑦𝑡  is the cumulative number of COVID-19 cases at time t, 𝐴 is the maximum 
cumulative case at the end of the epidemic, 𝜇𝑚  is the maximum growth rate, 𝜆 is the time 
point when it reaches 𝜇𝑚 . When 𝑦𝑡 = 𝐴/2, the time needed is based on equation (1): 

𝑡𝐴
2

,𝑙𝑜𝑔𝑖𝑠𝑡𝑖𝑘
= 𝜆 +

𝐴

2𝜇𝑚
                                                                           (2) 

From equation (2) it can be obtained the time needed to reach 𝐴 asymptotically is: 

𝑡𝐴,𝑙𝑜𝑔𝑖𝑠𝑡𝑖𝑘 = 2𝑡𝐴
2

,𝑙𝑜𝑔𝑖𝑠𝑡𝑖𝑘
                                                                                 (3) 

  

b. Model Gompertz [11,12,17] 

𝑦𝑡 = 𝐴 exp (− exp (
𝜇𝑚𝑒

𝐴
(𝜆 − 𝑡) + 1)) ,                                               (4) 

where 𝑒 is a natural number (2.718…). When 𝑦𝑡 = 𝐴/2, the time needed is based on 
equation (4): 

𝑡𝐴
2

,𝑔𝑜𝑚𝑝𝑒𝑟𝑡𝑧
= 𝜆 −

𝐴(ln(ln 2) − 1)

𝜇𝑚𝑒
                                                      (5) 

From equation (5) it can be obtained the time needed to reach 𝐴 asymptotically, is: 

𝑡𝐴,𝑔𝑜𝑚𝑝𝑒𝑟𝑡𝑧 = 2𝑡𝐴
2

,𝑔𝑜𝑚𝑝𝑒𝑟𝑡𝑧
                                                                    (6) 

In these two models, there are parameters 𝐴, 𝜇𝑚, and 𝜆 which must be estimated. In the 

estimation process, this study uses R software with a nonlinear square estimation algorithm 

[3].  

 

3. Results and Discussions 

The following Figure 2 is a visual presentation of the results of data plotting with the 

model proposed by the author in the previous section. 

 

 
1 Plotting is done based on the origin O. 



M, Fajar and Wahyudi 

Modelling of Covid-19 Epidemic Growth Curve in Indonesia 

71 
 

 

 

(a) (b) 
Figure 2. (a) Fitting the plot data with the Logistic model, (b) Fitting the plot data with the 

Gompertz model 

Notes: o states the actual data,        states the fitting model, and  - - -  states the slope of the 

model 

 

From Figure 2, visually, both models are accurate in plotting data patterns. This is 

indicated by the value of R2 (coefficient of determination) generated by the Logistic and 

Gompertz models which are 99.571% and 99.789%, respectively. That is, variations of 

response variables (in this case the cumulative COVID-19 case) can be explained by both 

models by more than 99%, while the remaining less than 1% are explained by other variables 

outside the model.  

By using the nonlinear least square estimation mechanism, the estimation results 

obtained are presented in table 1, as follows, 

Table 1. The estimated results of the Logistics Model and Gompertz  

which are visualized in Figure 2. 

No Model R2 𝑨 𝝁𝒎 𝝀 
𝒕𝑨

𝟐
 𝒕𝑨 

1 Logistic 99.571% 

7,713.719 

[6125.587, 

9301.852] 

253.860 

[220.735, 

286.984] 

25.961 

[24.213, 

27.709] 

41.154 82.307 

2 Gompertz 99.789% 

33,975.144  

[16247.870, 

51702.420] 

409.535 

[269.645, 

549.426] 

34.356 

[28.378, 

40.335] 

76.062 152.123 

Note: 95% confidence interval [lower limit, upper limit] 

 

Based on Table 1, there are few information that can be obtained, namely: (1) From 

the Logistics model it is found that the estimated maximum cumulative cases at the end of 

the COVID-19 epidemic is 7,714 positive confirmed cases achieved in about 82 days (May 

22, 2020) from Mar 2, 2020, when the first positive COVID-19 case was announced by the 

Government and the maximum growth rate achieved was the slope of the model, which was 

253.860% (it means when the λ = 25.961 days from the start of the COVID-19 epidemic 

began, then the maximum growth rate reached by the epidemic was 253.860%). (2) From 

the Gompertz model, it is found that the estimated maximum cumulative cases at the end of 

the COVID-19 epidemic is 33,975 positive confirmed cases achieved in about 152 days (Jul 

30, 2020) from Mar 2, 2020 when the first positive COVID-19 case was announced by the 

government and the maximum growth rate achieved is the slope of the model, which is 

409.535% (that is, when the time λ = 34.356 days from the start of the COVID-19 epidemic 

began, the maximum growth rate achieved by the epidemic is 409.535%).  



Jurnal Matematika MANTIK 

Vol. 7, No. 1, May 2021, pp. 67-73 

72 
 

Based on the results of the estimated maximum cumulative number and time2 needed 

to reach the maximum cumulative case from Table 1, it can be considered that the Logistic 

model is an optimistic scenario. In contrast, the Gompertz model is a pessimistic scenario, 

because: (1) the estimated maximum cumulative case achieved by COVID-19 from the 

Logistic model is lower than the results of the Gompertz model, and (2) the estimated time 

from the Logistic model to reach the maximum cumulative case is almost twice as fast as 

the time obtained from the Gompertz model.  

Therefore, this information can be used as input for the COVID-19 control policy. For 

example, according to an optimistic scenario (Logistics model), it is predicted that the peak 

of the COVID-19 epidemic occurred on May 22, 2020 (about 1 week before Idul Fitri) with 

a cumulative number of 7,714 cases, so the Government has to do a lockdown policy in 

major cities throughout provinces in Java (not only in DKI Jakarta) by prohibiting people 

returning home to celebrate Lebaran and encouraging people to stay at home to reduce the 

number of positive cases does not reach the maximum point. If government authorities 

assume that the growth of the COVID-19 epidemic follows the Gompertz model, then the 

Government only needs to implement social distancing instead of lockdown or quarantine. 

 

4. Conclusions 

Based on the results of the discussion, it can be concluded that: (1) Logistics and 

Gompertz models are very fit in modeling the COVID-19 epidemic growth curve, indicated 

from the value of R2 (coefficient of determination) which reaches more than 99%, (2) From 

the Logistic model it is obtained that the estimation the maximum cumulative cases at the 

end of the COVID-19 epidemic is 7,714 positive confirmed cases which are achieved in 

about 82 days (May 22, 2020) from Mar 2, 2020, when the Government announced the first 

positive COVID-19 case. (3) From the Gompertz model, it is obtained that the estimated 

maximum cumulative cases at the end of the COVID-19 epidemic is 33,975 positive 

confirmed cases which are achieved in about 152 days (Jul 30, 2020) from Mar 2, 2020, 

when Government announced the first positive COVID-19 case. From this finding, the 

results of this study can be used as input to the Government to take steps in controlling the 

spread of COVID-19. 

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