Biology, Medicine, & Natural Product Chemistry  ISSN 2089-6514 (paper) 
Volume 9, Number 1, April 2020 | Pages: 15-19 | DOI: 10.14421/biomedich.2020.91.15-19 ISSN 2540-9328 (online) 
 

 

 

 

A Mathematical Model of the Covid-19 Cases in Indonesia 

 (Under and Without Lockdown Enforcement)  

 
Sugiyanto*, Muchammad Abrori 

Department of Mathematics, Universitas Islam Negeri Sunan Kalijaga Yogyakarta 

Jl. Marsda Adisucipto No 1 Yogyakarta 55281, Indonesia. Tel. +62-274-540971, Fax. +62-274-519739 

 

Corresponding author* 
sugimath@yahoo.co.id 

 

 
Manuscript received: 29 March 2020. Revision accepted: 10 April, 2020. Published: 15 April, 2020. 

 

 

Abstract 

 

COVID-19 stands for Corona (CO), Virus (VI), Disease (D) and year 2019 (19), which is COVID-19 first appeared in 2019. 

Mathematical model of covid deployment in Indonesia under and without lockdown case uses the SIRV model, such as Susceptible, 

Infected, Recovery, and Virus. The results of this model state that under lockdown the spread of COVID-19 could be stopped. If it were 

not under lockdown it can multiply 1,276 times higher over two months. 

 

Keywords: Corona virus novel 2019 (COVID-19); mathematics model; lockdown; Indonesia 

 

 

INTRODUCTION 
 

Novel corona virus can pneumonia (Pan et al., 2020). 
Pneumonia is an inflammatory lung disease which is 

characterized by coughing, chest pain, fever, and 

difficulty breathing. COVID-19 is a very contagious 

disease and declared by WHO as a pandemic on March, 

12, 2020 (Chen et al., 2020). Pandemic is an infectious 
disease that is widely spread and it can reach all 

countries in the world. COVID-19 was first identified in 

Wuhan, Hubei Province, China in December 2019 (Liu 

et al., 2020). The understanding of lockdown in a 

pandemic context of COVID-19 is an emergency state 

where someone is not allowed to enter or out of a certain 

area in a certain period of time. This kind of action is 

carried out during the emergency situation (Ku et al., 
2020).  

Mathematical model of COVID-19 deployment is 

using susceptible sub populations, infected, recovery 

and virus or abbreviated as SIRV. This population is 

divided into four subpopulations. Susceptible 

subpopulation (S) is a subpopulation that is vulnerable 

to COVID-19. Infected subpopulation (I) is 

subpopulation that is infected by COVID-19. Recovery 

subpopulation (R) is subpopulation that recover from the 

COVID-19 virus. The population of COVID-19 (V) is 

the population number of COVID-19 outside the human 

body. 
 
 
 
 
 

MODEL FORMULATIONS 
 

The COVID-19 deployment model uses several 

parameters. Parameter 𝑑1 is the rate of natural death and 
other diseases besides COVID-19. Parameter 𝑎1 

is the 

level of susceptibility to COVID-19 disease. parameter 

𝑎2 is the level of COVID-19 virus infection. Parameter 
𝑎3 is the cure rate of COVID-19 disease. Parameter 𝑑2 
is the death rate due to COVID-19 virus. 

This models are defined into: 

a. The susceptible subpopulation are affected by: (1) 
the increasing in birth rate; (2) the increasing in 

susceptible individuals from other regions; (3) the 

increasing number of individuals recovering from the 

diseases that is caused by COVID-19; (4) the reduce 

susceptibility of COVID-19 which infected 

individuals; (5) the reduce susceptible individuals 

going to other areas; (6) the loss of individuals due to 

natural deaths, other than diseases caused by 

COVID-19. 

b. The infected subpopulation are affected by: (1) the 
increasing number of individual births from a mother 

that infected by COVID-19; (2) the increasing in 

individuals due to arrivals from other regions; (3) the 

increasing in individuals from susceptible and 

infected by COVID-19; (4) the reduce individual 

recovery from COVID-19; (5) the reduction in 

infected individuals who went to other areas; (6) the 

 

 

https://doi.org/10.14421/biomedich.2020.91.15-19


 

 

 

16 Biology, Medicine, & Natural Product Chemistry 9 (1), 2020: 15-19 
 

 

reduction in individuals who die from diseases that 

caused by COVID-19; (7) the loss of individuals due 

to deaths other than COVID-19. 

c. The recovery subpopulation are affected by: (1) the 
increase in births at recovery time; (2) the increase in 

individual recovery from other regions; (3) the 

increase in infected; (4) the reduce susceptible 

individuals; (5) the reduction in recovery individuals 

that went to other areas; (6) the loss of individuals 

due to deaths other than COVID-19. 

d. The population of COVID-19 (population of virus in 
the environment spread by infected people) are 

affected by: (1) the COVID-19 deployment because 

individuals are infected; (2) the unsurvive of 

COVID-19 in a free environment due to physical and 

chemical factors.  

 
 

 

 

Table 1. The population, subpopulations, and parameters used in the COVID-19 spread by mathematics model. 

 

No. Symbol Explanation Unit 

1. S Susceptible subpopulation person/km2 

2. I Infected subpopulation person/km2 

3. R Recovery subpopulation person/km2 

4. V The population of COVID-19 virus/km2 

5. 1a  Birth rate of susceptible subpopulation day
-1 

6. 2a  Birth rate of infected subpopulation day
-1 

7. 3a  Birth rate of recovery subpopulation day
-1 

8. 1b  Arrival rates from other countries susceptible subpopulation day
-1 

9. 2b  Arrival rates from other countries infected subpopulation day
-1 

10. 3b  Arrival rates from other countries recovery subpopulation day
-1 

11. c  Death rates due to disease caused by COVID-19. day-1 

12. 1d  Natural death rates are not due to COVID-19 susceptible subpopulation. day
-1 

13. 2d  Natural death rates are not due to COVID-19 infected subpopulation. 
day-1 

14. 3d  Natural death rates are not due to COVID-19 recovery subpopulation. day
-1 

15. 4d  The COVID-19 mortality rates in environment. 
day-1 

16. 1e  Rates of traveling to other countries susceptible subpopulation day
-1 

17. 2e  Rates of traveling to other countries infected subpopulation day
-1 

18. 3e  Rates of traveling to other countries recovery subpopulation day
-1 

19. f  The degree of COVID-19 emergence in environment virus person-1 

20. 1k  Carrying capacity of the susceptible birth subpopulation person
2

 
21. 2k  Carrying capacity of the infected birth subpopulation person

2 

22. 3k  Carrying capacity of the recovery birth subpopulation person
2 

23. 1l  Carrying capacity of the susceptible natural death subpopulation person
2 

24. 2l  Carrying capacity of the infected natural death subpopulation person
2 

25 3l  Carrying capacity of the recovery natural death subpopulation person
2 

26.   The degree of interaction between susceptible subpopulation become infected subpopulation km2.day-1.virus-1 

27.   The degree of interaction betweem infected subpopulation become recovery subpopulation day
-1 

28.   The degree of interaction between recovery subpopulation become susceptible subpopulation. day-1 
 

 

 

In this model it is assumed that: (1) open population, 

means people are free to enter and exit a country/region; 

(2) newborn individuals are included in the susceptible 

subpopulation; individuals who recover from covid are 

not immune; (3) every individual who infected by 

COVID-19 will become infected; (4) After the 

incubation period of two weeks are already considered 

an infected individual; (5) a vaccine for COVID-19 has 

not been found yet, it means that healing is due to good 

individual immunity, not because of vaccine; (6) people 

who recovery from COVID-19 can be a susceptible 

subpopulation again; (7) birth rate and natural mortality 

rate of the infected subpopulation and recovery does not 

exist. 

In this model not analyzed local stability of the 

equilibrium point, and it become our next job. Figure 2 



 

 

 

 Sugiyanto & Abrori – A Mathematical Model of the Covid-19 Cases in … 17 
 

 

is COVID-19 deployment transfer diagram. Based on 

the figure 2, Transfer Diagram, formed the COVID-19 

spread mathematics model Equation (1a) – (1d). 

 

 
 
Figure 1. COVID-19 Distribution between individuals. 

 

 

1 1 1 1
1 1

1 1
   

          
   

dS S S
a S b S R SV e S d S

dt k l
   (1a) 

2 2 2 3
2 2

1 1
   

           
   

dI I I
a I b I SV I cI e I d I

dt k l
   (1b) 

3 3 3 3
3 3

1 1
   

          
   

dR R R
a R b R I R e R d R

dt k l
   (1c) 

2 4
2

1
 

   
 

dV I
fa I d V

dt k
 (1d) 

 

 
 

Figure 2. Transfer Diagram of COVID-19 Deployment Model. 
 

 

 

SIMULATION 

 
In this section, we will discuss about numerical 

simulation and its intepretation by COVID-19 

deployment model. The initial value starts from the 

spread of COVID-19 in Indonesia on March, 12–23, 

2020 that was announced by the Indonesian goverment. 

The initial value of the susceptible is taken from the 

total population of Indonesian divided by the population 

of Indonesian, 𝑆0 =
260,000,000

260,000,000
= 1. The infected value 

was taken from the number of people infected on March 

23, 2020 divided by the population of Indonesian, 𝐼0 =
579

260,000,000
= 0.0000022269. The recovery values is 

taken from the number of people recovered on March 

23, 2020 divided by the total population of Indonesia, 

𝑅0 =
30

260,000,000
= 0.0000001. COVID-19 value is 

taken by the estimation, 𝑉0 =
1,000,000

260,000,000
=

0.0038461538. 
The parameter values are as follows. Indonesia area 

1.905 x 109 km2. In 2019, the birth rate is = 4.4 x 106 

people. In 2019, the mortality rate is = 1.6 x 106 people. 

There are 365 days in one year.  
6

3 1
1 9

4.4 10
6.3 10

1.905 10 365

 
 

x
a x day

x x
 

6
1 3 1

1 9

1.6 10
2.3 10

1.905 10 365

  
 

x
d day x day

x x
 

From the assumption of natural birth and death that 

the infected subpopulation does not exist, then the 

parameter 𝑎2 = 0 and 𝑑2 = 0. From the assumption of 
natural birth and death recovery subpopulation does not 

exist, then the parameter 𝑎3 = 0 dan 𝑑3 = 0. Because 
the Indonesia country has suspended its arrival visa and 

the other countries have also applied for a visa 

suspension, then the parameter 𝑏1 = 𝑏2 = 𝑏3 = 𝑒1 =
𝑒2 = 𝑒3 = 0. 

From the Table 2, we take the highest gradient that is 

from March 17–18, 2020 are 14 souls per day 

(Zonautara, 2020). It means, the death rates due to the 

diseases caused by COVID-19 is the highest mortality 

divided by Indonesian population or 𝑐 =
14

260,000,000
=

0.0000000538 𝑝𝑒𝑟𝑠𝑜𝑛−2𝑑𝑎𝑦−1. Value 260,000,000 
are taken from the Indonesian population. 

The value of COVID-19 mortality rates in the 

environment ( d ) be accepted 𝑑4 =
2

260,000,000
=

7.7𝑥10−9 𝑑𝑎𝑦−1 (BNPB, 2020). 
The estimated value of COVID-19 emergence rate in 

the environment (𝑓) be accepted 𝑓 =
260,000

260,000,000
=

1𝑥10−3 𝑣𝑖𝑟𝑢𝑠 𝑝𝑒𝑟𝑠𝑜𝑛−1𝑑𝑎𝑦−1. 
The value of the level of interaction between 

susceptible subpopulation become infected 

subpopulation (𝛼) taken from the slope of the highest 
infected divided by Indonesian population. The highest 

slope is on March, 18–19, 2020 or March, 20–23, 2020 

there were 81 people died in one day. So, the value of 𝛼 
is 81 divided ten times of 579 people were infected (ten 

fold was infected but its unknown, because it’s not 

COVID-19 tested),  

𝛼 =
81

579𝑥10
= 3.6𝑥10−3 𝑑𝑎𝑦−1𝑣𝑖𝑟𝑢𝑠−1. 

The value of the level of interaction between 

recovery subpopulation become susceptible 



 

 

 

18 Biology, Medicine, & Natural Product Chemistry 9 (1), 2020: 15-19 
 

 

subpopulation (𝛽) taken from the highest slope recovery 
divided by the population of Indonesian. The highest 

slope was on March 21–22, 2020 as many as 9 people 

died in one day. So, the 𝛽
 

value is 9 divided by a 

hundred multiple of 30 people who recover (one 

hundred multiples are recovery but not known, because 

it does not test by COVID-19), 𝛽 =
9

30𝑥100
=

3𝑥10−3𝑑𝑎𝑦−1. The value of the level of interaction 
between recovery subpopulation become susceptible 

subpopulation (𝛾) taken the same as 𝛽. The value of 
carrying capacity of natural birth and death in the 

susceptible sub population taken with an estimated 

number of deaths of 500,000 people divided Indonesian 

population, that is  

3
1 2 3 1 2 3

500, 000
1.923 10 .

260, 000, 000


      k k k l l l x  

To summarize the parameter values, we shown in Table 

3. 

 
 

Table 2. The data that died in Indonesia because of COVID-19 
(Zonautara, 2020). 

 

No. 
By the 

date 

The 

number of 

people died 

The 

number of 

people 

recovered 

The 

number of 

people 

infected 

1. 12 March 

2020 

1 4 34 

2. 13 March 

2020 

4 4 69 

3. 14 March 

2020 

5 5 96 

4. 15 March 

2020 

5 5 117 

5. 16 March 

2020 

5 8 134 

6. 17 March 

2020 

5 9 172 

7. 18 March 

2020 

19 11 227 

8. 19 March 

2020 

25 15 308 

9. 20 March 

2020 

32 17 369 

10. 21 March 

2020 

38 20 450 

11 22 March 

2020 

48 29 514 

12 23 March 

2020 

49 30 579 

 

 
 

Table 3 is multiplied by 260,000,000 people; this 

number is taken from the Indonesian population. 

Lockdown case is the level of interaction between 

susceptible subpopulation become infected 
subpopulation, the degree of interaction between 

infected subpopulation becomes recovery subpopulation, 

and the degree of the interaction between recovery 

subpopulation being a susceptible subpopulation doesn 

not exist, it means the value of 0     . 

 
Table 3. The parameter value without COVID-19 deployment lockdown 

in Indonesia. 

 

No. Symbol Value No. Symbol Value 

1. 1a  
3

6.3 10


x  13. 2e  0 

2. 2a  0 14. 3e  0 

3. 3a  0 15. f  
3

1 10


x  

4. 1b  0 16. 1k  
3

1.923 10


x  

5. 2b  0 17. 2k  
3

1.923 10


x  

6. 3b  0 18. 3k  
3

1.923 10


x  

7. c  85.38 10


x  19. 1l  
3

1.923 10


x  

8. 1d  
3

2.3 10


x  20. 2l  
3

1.923 10


x  

9. 2d  0 21. 3l  
3

1.923 10


x  

10. 3d  0 22.   
3

3.6 10


x  

11. 4d  
9

7.7 10


x  23.   
3

3 10


x  

12. 1e  0 24.   
3

3 10


x  

 

 
 

 
 

Figure 3. Diagram of Trajectory Subpopulation Infected between Under 

Lockdown and Without Lockdown. 

 
Figure 3 shows the case trajectory diagram under and 

without lockdown in infected subpopulation. In the case 

of not lockdown on day-0 showed 579 people infected. 

On 30th for the case to be 677,600 people infected. On 

60th showed 738,900 people infected. On day-0 of 

lockdown case showed 579 people infected and the 30th 

day showed 579 people infected. Because the incubation 

period of 14 days, then in the case of lockdown on the 

15th day no one has been infected. 
 
 

CONCLUSION 
 
Covid-19 is a vicious virus, because its spread from 

modeling result is very fast. From the Indonesia 

government’s data on March 12–23, 2020 up to 579 



 

 

 

 Sugiyanto & Abrori – A Mathematical Model of the Covid-19 Cases in … 19 
 

 

people have been infected, while from the model when 

not in lockdown of 579 people become 738,900 people 

have been infected. This is 1,276 times in two months. 

Under lockdown, it can minimize casualties because 

COVID-19 does not spread. Lockdown is a solution to 

reduce deaths due to COVID-19. 
 
 

REFERENCES 
 
BNPB, 2020 in https://www.covid19.go.id/ketahui-cara-

mengurangi-risiko/ downloaded on March 20, 2020. 

Chen, H., Guo, J., Wang, C., Luo, F., Yu, X., Zhang, W., ... & 

Liao, J. (2020). Clinical characteristics and intrauterine 

vertical transmission potential of COVID-19 infection in nine 

pregnant women: a retrospective review of medical records. 

The Lancet.   

Ku, C. C., Ng, T. C., & Lin, H. H. (2020). Epidemiological 

benchmarks of the COVID-19 outbreak control in China after 

Wuhan’s lockdown: a modelling study with an empirical 

approach. Available at SSRN 3544127.  

Liu, Y., Gayle, A. A., Wilder-Smith, A., & Rocklöv, J. (2020). 

The reproductive number of COVID-19 is higher compared to 

SARS coronavirus. Journal of travel medicine. 

Pan, F., Ye, T., Sun, P., Gui, S., Liang, B., Li, L., ... & Zheng, C. 

(2020). Time course of lung changes on chest CT during 

recovery from 2019 novel coronavirus (COVID-19) 

pneumonia. Radiology, 200370. 

Zonautara, 2020 in https://zonautara.com/2020/03/22/tabel-

sebaran-virus-corona-per-provinsi-22-maret-2020/ 

downloaded on March 20, 2020. 

 

 



 

THIS PAGE INTENTIONALLY LEFT BLANK