Biology, Medicine, & Natural Product Chemistry  ISSN 2089-6514 (paper) 
Volume 10, Number 2, October 2021 | Pages: 123-127 | DOI: 10.14421/biomedich.2021.102.123-127 ISSN 2540-9328 (online) 
 
 

 

 

Stability Analysis of Mathematical Modeling of Interaction between 

Target Cells and COVID-19 Infected Cells 

 
Sugiyanto1*, Mansoor Abdul Hamid2, Alya Adianta1, Hanny Puspha Jayanti1, Muhammad Ja'far Luthfi

3
 

1Mathematics Department, Faculty of Science and Technology, Universitas Islam Negeri Sunan Kalijaga, Indonesia. 
2School of Food Science & Nutrition, Universiti Malaysia Sabah, Malaysia. 

3Department of Biological Education, Faculty of Tarbiyah and Education, Universitas Islam Negeri Sunan Kalijaga, Indonesia. 

 

Corresponding author* 
sugiyanto@uin-suka.ac.id 

 

 
Manuscript received: 19 October 2021. Revision accepted: 30 October, 2021. Published: 02 November, 2021. 

 

 

Abstract 

 

The stability analysis in this mathematical model was related to the infection of the Coronavirus Disease 2019 (Covid-19). In this 

mathematical model there were two balance points, namely the point of balance free from Covid-19 and the one infected with Covid-19. 

The stability of the equilibrium point was influenced by all parameters, i.e. target cells die during each cycle, number of t arget cells at 𝑡′ = 
0, target cells infected during each cycle based on virion unit density, effective surface area of the network, the ratio of the number of 

virus particles to the number of virions, infected cells die during each cycle, the number of virus particles produced by each infected cell 

during each cycle, and virus particles die during each cycle. In the simulation model, immunity is divided into high, medium and low 

immunity. For high, moderate and low immunity, respectively, the highest number of target cells is in high, medium and low immunity, 

whereas for the number of infected cells and the number of Covid-19, it is in the opposite sequence of the number of target cells. 

 

Keywords: Coronavirus Disease 2019; Equilibrium point stability; target cells and infected cells. 

 

 

INTRODUCTION 
 

Coronavirus Disease 2019 (Covid-19) was first known 

to infect residents in Wuhan City, China, and was 

notified by the Chinese Government to WHO in 

December 2019 (Sugiyanto & Abrori, 2020). Covid-19 

belongs to subfamily Orthocoronavirinae, family 

Coronaviridae, and order Nidovirales (Tan et. al., 2020). 

About 80% of Covid-19 illness show mild symptoms 

and 20% have severe symptoms. Some of the 20% 

patients who contract Covid-19 develop severe 

pneumonia, sometimes with acute respiratory distress, 

which can lead to organ failure and death. 

The stability analysis of mathematical modeling is 

used to determine the recovery period of Covid-19 

patients. There are many factors that determine a person 

would get into mild, severe or severe symptoms. We can 

classify these symptoms into three things depend on the 

immunity of the Covid-19 patient. In this modeling, 

categorization were done using the T – I – V model. The 

target cell subpopulation (T) is cells in several organs, 

such as the lungs, heart, arteries, intestines and kidneys. 

The Infected cell subpopulation (I) is a cell that is 

infected through a receptor on the surface called 

Angiotensin Converting Enzyme 2 (ACE2) (Diaz, 

2020). Target cells were epithelial cells in all of these 

organs. This target cell was ACE2. The conversion of 

angiotensin II (vasoconstruction peptide) to angiotensin 

1-7 (vasodilator) was catalyzed by ACE2 (Zhang et. al., 

2020). 83% of normal lung cells express ACE2, namely 

type II alveolar epithelial cells (AECII), which make 

these cells viral reservoirs. The spike protein (shaped 

like a nail) stuck to the surface of the SARS-CoV virus 

(Zoufaly et. al., 2020). The ACE2 enzyme attaches to 

the cell membranes of several organs (Bourgonje et. al, 

2020).  

 

 

 

STABILITY ANALYSIS 
 

The Mathematical Model obtained in System (1) refers 

to Du and Yuan's (2020) paper. 

 
𝑑𝑇

𝑑𝑡′
= (𝑑𝜏)𝑇0 − (𝑑𝜏)𝑇 −

(𝑘𝜏)

𝐴𝛼
𝑉𝑇 (1a) 

 
𝑑𝐼

𝑑𝑡′
=

(𝑘𝜏)

𝐴𝛼
𝑉𝑇 − (𝛿𝜏)𝐼 (1b) 

 
𝑑𝑉

𝑑𝑡′
= (𝑝𝜏)𝐼 − (𝑐𝜏)𝑉 (1c) 

 

 

Description of the target cell subpopulation, Covid-

19 infected cells, virus population and parameters are 

shown in Table 1. 

https://doi.org/10.14421/biomedich.2021.102.123-127


 

 

 

124 Biology, Medicine, & Natural Product Chemistry 10 (2), 2021: 123-127 
 

 

Table 1. Target cell subpopulation, Covid-19 infected cells, virus 
population and parameters. 

 

No. Symbol Explanation  Unit 

1 𝜏 Average cycle time for viral 
replication 

𝑑𝑎𝑦 

2 𝑡′ = 𝑡/𝜏 Number of virus replication 
cycles   

- 

3 𝑇 Number of target cells at 𝑡′ 𝑐𝑒𝑙𝑙 
4 𝐼 Number of infected cells at 𝑡′ 𝑐𝑒𝑙𝑙 
5 𝑉 Number of virus particles at 𝑡′ 𝑣𝑖𝑟𝑢𝑠 
6 (𝑑𝜏) Target cells die during each 

cycle 

- 

7 𝑇0 Number of target cells at 𝑡
′ = 0 𝑐𝑒𝑙𝑙 

8 (𝑘𝜏) Target cells infected during 
each cycle based on virion unit 

density 

- 

9 𝐴 Effective surface area of the 
network 

𝑚𝑚2 

10 𝛼 The ratio of the number of 
virus particles to the number of 

virions 

𝑣𝑖𝑟𝑢𝑠
/𝑚𝑚2 

11 (𝛿𝜏) Infected cells die during each 
cycle 

- 

12 (𝑝𝜏) The number of virus particles 
produced by each infected cell 

during each cycle 

- 

13 (𝑐𝜏) Virus particles die during each 
cycle 

- 

 
Theorem 1. Equilibrium Point 

There are two equilibrium points of System (1), namely: 

free from the Covid-19 virus and infected with the 

Covid-19 virus. The Covid-19 virus-free equilibrium 

point is  

𝐸𝑃0 = (𝑇, 𝐼, 𝑉) = (𝑇0, 0,0). 
 

The equilibrium point for contracting the Covid-19 virus 

is  

𝐸𝑃1 = (𝑇, 𝐼, 𝑉) = (𝑎1, 𝑎2, 𝑎3), 
 

where 

𝑎1 =
𝐴𝛼(𝛿𝜏)(𝑐𝜏)

(𝑘𝜏)(𝑝𝜏)
,      

𝑎2 =
(𝑘𝜏)(𝑑𝜏)𝑇0(𝑝𝜏)− (𝛿𝜏)(𝑐𝜏)(𝑑𝜏)𝐴𝛼

(𝑝𝜏)(𝛿𝜏)(𝑘𝜏)
, 

𝑎3 =
(𝑘𝜏)(𝑑𝜏)𝑇0(𝑝𝜏)− (𝛿𝜏)(𝑐𝜏)(𝑑𝜏)𝐴𝛼

(𝛿𝜏)(𝑐𝜏)(𝑘𝜏)
. 

 

Proof. 

From Equation (1a) and 
𝑑𝑇

𝑑𝑡′
= 0, we get 

 

𝑇 =
(𝑑𝜏)𝑇0𝐴𝛼

(𝑑𝜏)𝐴𝛼+(𝑘𝜏)𝑉
 (2) 

 

From Equation (1c) and 
𝑑𝑉

𝑑𝑡′
= 0 obtained 

 

𝐼 =
(𝑐𝜏)

(𝑝𝜏)
𝑉 (3) 

 

From 
𝑑𝐼

𝑑𝑡′
= 0 and substituting equations (2) and (3) into 

equation (1), we get 

 

𝑉 = 0 (4) 

or 𝑉 =
(𝑑𝜏)[(𝑘𝜏)𝑇0(𝑝𝜏)− (𝛿𝜏)(𝑐𝜏)𝐴𝛼]

(𝛿𝜏)(𝑐𝜏)(𝑘𝜏)
= 𝑎3 (5) 

From Equation (2) and Equation (4), we get  

𝑇 = 𝑇0.  (6) 

From Equation (3) and Equation (4), we get  

𝐼 = 0.  (7) 

From Equations (6), (7) and (4) it is proven that the 

Covid-19 virus-free equilibrium point is 𝐸𝑃0.  
If Equation (5) is substituted into Equation (2), then 

we get  
 

𝑇 =
𝐴𝛼(𝛿𝜏)(𝑐𝜏)

(𝑘𝜏)(𝑝𝜏)
= 𝑎1 (8) 

If Equation (8) is substituted into Equation (3), then we 

get 
 

𝐼 =
(𝑘𝜏)(𝑑𝜏)𝑇0(𝑝𝜏)− (𝛿𝜏)(𝑐𝜏)(𝑑𝜏)𝐴𝛼

(𝑝𝜏)(𝛿𝜏)(𝑘𝜏)
= 𝑎2 (9) 

From Equations (8), (9) and (5) it is proven that the 

equilibrium point for contracting the Covid-19 virus is 

𝐸𝑃1. ■ 
From Theorem 1 it can be conveyed, if there is no 

Covid-19 virus then someone will be safe or someone is 

virus free, and if there is a virus then a person's healing 

point is influenced by all parameters. Virus-free can be 

achieved if there is no person carrying the virus or 

complying with health procedures such as wearing a 

mask, keeping a distance and washing hands as often as 

possible. When a person gets a virus, only the immune 

(target cells) can fight the infected cells.  

 

Theorem 2. Existence of the Equilibrium Point 

Existence 𝐸𝑃0 fulfilled in any non-negative number 
parameter and existence 𝐸𝑃1 fulfilled if  
 

(𝑘𝜏)𝑇0(𝑝𝜏) −  (𝛿𝜏)(𝑐𝜏)𝐴𝛼 > 0. 
 

Proof. 

From Theorem 1, that existence 𝐸𝑃0 and 𝐸𝑃1 proven. ■ 
From Theorem 2 it can be seen that all parameters do 

not affect the existence of the equilibrium point 𝐸𝑃0. All 
parameters are target cells die during each cycle, number 

of target cells at 𝑡′ = 0, target cells infected during each 
cycle based on virion unit density, effective surface area 

of the network, the ratio of the number of virus particles 

to the number of virions, infected cells die during each 

cycle, the number of virus particles produced by each 

infected cell during each cycle, and virus particles die 

during each cycle. This means that if a person is not 

exposed to the Covid-19 virus, the target cells would not 

affected or the condition of a person is healthy without 
the virus. For someone who is infected with the virus, all 

parameters affect the existence of the equilibrium point 

𝐸𝑃0. This means that a person's condition will remain 



 

 

 

 Sugiyanto et al. – Stability Analysis of Mathematical Modeling of Interaction …  125 
 

 

healthy or even die depending on the target cells 

working well or not.  

 

Theorem 3. Stability of the Equilibrium Point 

(1) If √((𝛿𝜏) − (𝑐𝜏))
2
+ 4

(𝑘𝜏)(𝑝𝜏)𝑇0

𝐴𝛼
− ((𝛿𝜏) + (𝑐𝜏)), 

then the equilibrium point 𝐸𝑃0 is locally 
asymptotically stable. 

(2) If √((𝛿𝜏) − (𝑐𝜏))
2
+ 4

(𝑘𝜏)(𝑝𝜏)(𝑎1)

𝐴𝛼
− ((𝛿𝜏) +

(𝑐𝜏)) < 0, then the equilibrium point 𝐸𝑃1 is locally 
asymptotically stable. 

 

Proof. 
For example, in System (1) it is written 
 

𝑓1 =
𝑑𝑇

𝑑𝑡′
= (𝑑𝜏)𝑇0 − (𝑑𝜏)𝑇 −

(𝑘𝜏)

𝐴𝛼
𝑉𝑇 (10a) 

𝑓2 =
𝑑𝐼

𝑑𝑡′
=

(𝑘𝜏)

𝐴𝛼
𝑉𝑇 − (𝛿𝜏)𝐼 (10b) 

𝑓3 =
𝑑𝑉

𝑑𝑡′
= (𝑝𝜏)𝐼 − (𝑐𝜏)𝑉  (10c) 

 

Jacobian matrix function 𝑓 from System (10) written can 
be obtained by first performing the partial derivation of 

the functions 
 

𝑓1 = (𝑇, 𝐼, 𝑉 )  (11a) 

𝑓2 = (𝑇, 𝐼, 𝑉  ) (11b) 

𝑓3 = (𝑇, 𝐼, 𝑉  ) (11c) 
 

as follows. 

(i). Partial derivative 𝑓1 with respect to 𝑇, 𝐼, 𝑉 namely: 
𝜕𝑓1

𝜕𝑇
= −(𝑑𝜏) −

(𝑘𝜏)

𝐴𝛼
𝑉;     

𝜕𝑓1

𝜕𝐼
= 0;     

𝜕𝑓1

𝜕𝑉
= 0; 

(ii). Partial derivative 𝑓2 with respect to 𝑇, 𝐼, 𝑉 namely: 
𝜕𝑓2

𝜕𝑇
=

(𝑘𝜏)

𝐴𝛼
𝑉;     

𝜕𝑓2

𝜕𝐼
= −(𝛿𝜏);     

𝜕𝑓2

𝜕𝑉
=

(𝑘𝜏)

𝐴𝛼
𝑇; 

(iii). Partial derivative 𝑓3 with respect to 𝑇, 𝐼, 𝑉 namely: 
𝜕𝑓3

𝜕𝑇
= 0;     

𝜕𝑓3

𝜕𝐼
= (𝑝𝜏);     

𝜕𝑓3

𝜕𝑉
= −(𝑐𝜏); 

 

The Jacobian matrix is 

𝐽(𝑇, 𝐼, 𝑉) =

[
 
 
 
 −(𝑑𝜏) −

(𝑘𝜏)

𝐴𝛼
𝑉 0  0

(𝑘𝜏)

𝐴𝛼
𝑉 −(𝛿𝜏)

(𝑘𝜏)

𝐴𝛼
𝑇

0 (𝑝𝜏) −(𝑐𝜏)]
 
 
 
 

 

(1) For 𝑬𝑷𝟎, we get 

𝐽(𝑇0, 0,0) =

[
 
 
 
−(𝑑𝜏) 0  0

0 −(𝛿𝜏)
(𝑘𝜏)

𝐴𝛼
(𝑇0)

0 (𝑝𝜏) −(𝑐𝜏) ]
 
 
 
 

We find the eigenvalues of 𝐽(𝑇0, 0,0) that is 𝜆𝑖, for 𝑖 =
1,2,3, where 

 0 , 0, 0 0.J T I   

We get the eigenvalues of the Jacobian Matrix which is 

represented by  

 

         
  

         
  

1

2

2

2

3

0

0

1
,

,

4

4

2

1
.

2

d

k p T
c c

A

k p T
c c

A



 
   



 
   









 

 
   
  

 
   
 



 



 



 

We know that ((𝛿𝜏) − (𝑐𝜏))
2

≥ 0 and 
(𝑘𝜏)(𝑝𝜏)𝑇0

𝐴𝛼
> 0, so 

that ((𝛿𝜏) − (𝑐𝜏))
2
+ 4

(𝑘𝜏)(𝑝𝜏)𝑇0

𝐴𝛼
> 0.  

Since the parameters are greater than zero, we get

  1 20, 0,    

and because √((𝛿𝜏) − (𝑐𝜏))
2
+ 4

(𝑘𝜏)(𝑝𝜏)𝑇0

𝐴𝛼
−

((𝛿𝜏) + (𝑐𝜏)) < 0, then we get 𝜆3 =
1

2
[−((𝛿𝜏) −

(𝑐𝜏)) ±

√((𝛿𝜏) − (𝑐𝜏))
2
− 4 ((𝛿𝜏)(𝑐𝜏) −

(𝑘𝜏)(𝑝𝜏)𝑇0

𝐴𝛼
)] < 0. 

We get all negative eigenvalues, so that 𝐸𝑃0 is locally 
asymptotically stable. 

 

(2) For 𝑬𝑷𝟏, we get 

𝐽(𝑎1, 𝑎2, 𝑎3) =

[
 
 
 
 −(𝑑𝜏) −

(𝑘𝜏)

𝐴𝛼
(𝑎3) 0  0

(𝑘𝜏)

𝐴𝛼
(𝑎3) −(𝛿𝜏)

(𝑘𝜏)

𝐴𝛼
(𝑎1)

0 (𝑝𝜏) −(𝑐𝜏) ]
 
 
 
 

 

We find the eigenvalues of 𝐽(𝑎1, 𝑎2, 𝑎3) that is 𝜆𝑖, for 
𝑖 = 1,2,3, where 

 1 2 3, 0.,J a a a I   

We get the eigenvalues of the Jacobian Matrix which is 

represented by  

 
 

 

         
   

         
   

3

1

1

1

2

2

2

3

,

1
4 ,

2

1
4

2

k
d a

A

k p
c c

A

k p

a

a
c c

A






 
   



 
   








 
  

 

 
   
  

 
   
 



 

 







 

We know that ((𝛿𝜏) − (𝑐𝜏))
2

≥ 0 and 
(𝑘𝜏)(𝑝𝜏)𝑇0

𝐴𝛼
> 0, so 

((𝛿𝜏) − (𝑐𝜏))
2
+ 4

(𝑘𝜏)(𝑝𝜏)𝑇0

𝐴𝛼
> 0.  

Since the parameters are greater than zero, we get  

𝜆1 < 0, 𝜆2 < 0, 



 

 

 

126 Biology, Medicine, & Natural Product Chemistry 10 (2), 2021: 123-127 
 

 

and because √((𝛿𝜏) − (𝑐𝜏))
2
+ 4

(𝑘𝜏)(𝑝𝜏)(𝑎1)

𝐴𝛼
−

((𝛿𝜏) + (𝑐𝜏)) < 0, then we get  

         
   2

3

11
4 0

2
c

A

ak p
c

 
   




 
   






 



. 

We get all negative eigenvalues, so that 𝐸𝑃1 is locally 
asymptotically stable.■ 

From Theorem 3 the stability point is affected by all 

parameters. This means that a person will recover 

depending on the target cells that work. The better the 

target cells work, the healthier the person would be and 

those who have been infected with Covid-19 will 

recover. 

 
 

SIMULATION 
 

The parameters in this simulation are taken from Du and 

Yuan's (2020) paper. Table 2 shows the parameter 

values. In this simulation, we replace the symbol (𝑝𝜏) 
with 𝑏. This is because in Matlab there is no Insert 
Legend that can be written (𝑝𝜏). 
 
Table 2. Parameter values for simulation. 

 

No. Parameter Value 

1 𝜏 7 

2 𝑑𝜏 2 × 10−4 
3 𝑇0 10

8 

4 
(𝑘𝜏)

𝐴𝛼
𝑇0 0.075 

5 𝛿𝜏 0.4 

6 𝑐𝜏 0.4 

7 𝐼0 10 

8 𝑉0 100 
 
 

 
Figure 1. Changes in the number of target cells against the presence of the 
Covid-19 virus. 

 
Target cells reflect the number of cells in people with 

three conditions, namely: low, moderate and high 

immunity conditions. Figure 1, Figure 2 and Figure 3 

represent person with high immunity ((𝑝𝜏) = 𝑏 = 50), 
moderate immunity ((𝑝𝜏) = 𝑏 = 100), and low 
immunity ((𝑝𝜏) = 𝑏 = 150). Person with good 
immunity shows the target cell from 10,000,000 cells in 

13.09 days to 101,800 cells. Person with moderate 

immunity shows the target cell from 10,000,000 cells in 

8,514 days to 104,300 cells. Person with low immunity 

shows the target cell from 10,000,000 cells in 2,398 

days to 105,000 cells. The order of decline in target cells 

from the longest to the fastest is good, medium and low 

immunity. Table 3 describes the descending order of the 

target cells. 

 
Table 3. Target cell decrease. 

 

No. Immunity  

Initial 

amount 

(cell) 

Total Ten 

Thousand 

(cell) 

Time 

(day) 

1 High 10,000,000 101,800 13.09 

2 Medium 10,000,000 104,300 8.514 

3 Low 10,000,000 105,000 2.398 

 

 

 
 

Figure 2. Changes in the number of infected cells against the presence of 

the Covid-19 virus. 

 

Figure 2 shows the peak number of infected cells 

differed between individuals with high, moderate and 

low immunity. A person with low immunity on day 

6,155 the number of infected is 7.146 × 107 cell. A 
person with moderate immunity on day 7,685 the 

number of infected is 6.65 × 107 cell. A person with 
high immunity on day 11.46 the number of infected is 

5.655 × 107 cell. Briefly, this explanation is in Table 4. 
 

Table 4. Increase in the number of infected cells. 
 

No. Immunity  
Highest number of cells 

(cell) 

Time 

(day) 

1 High 5.655 × 107 11.46 

2 Medium 6.65 × 107 7.685 

3 Low 7.146 × 107 6.155 
 
 

 
 

Figure 3. Changes in the number of virus particles. 



 

 

 

 Sugiyanto et al. – Stability Analysis of Mathematical Modeling of Interaction …  127 
 

 

Figure 3 shows the number of viruses with high, 

medium and low immunity conditions. For someone 

with high immunity the maximum virus count on day 

13.19 is 4.31 × 108 virus. For someone with moderate 
immunity the maximum virus count on 9,492 days is 

8.946 × 109 virus. For a person with low immunity the 
maximum viral load on day 8,068 is 1.356 × 1010 
virus. Table 5 describes the amount of virus in the 

condition of a person with high, medium and low 

immunity. 

 
Table 5. Increase in the number of virus particles. 

 

No. Immunity 
Highest number of 

viruses (virus) 
Time (day) 

1 High 4.31 × 108 13.19 

2 Medium 8.946 × 109 9.492 

3 Low 1.356 × 1010 8.068 

 

 

CONCLUSION 
 

The stability of being free of the Covid-19 virus and 

infected with the virus is influenced by all parameters. 

The number of target cells, virus-infected cells and virus 

particles is affected by a person's immunity. If a person 

has high immunity, the number of target cells would 

decrease slowly. Vice versa, if a person has low 

immunity, then the number of target cells will drop 

rapidly. In a person having low immunity, the infected 

cells and viruses will quickly increase in number 

compared to the one with high immunity. 

 

Conflicts of Interest: MJL is on the editorial board of 

the Biology, Medicine, & Natural Product Chemistry, 

and was recused from this article’s review and decision. 

The authors declare that there are no conflicts of 

interest. 

 

 

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