Local Stability Analysis of an SVIR Epidemic Model


CAUCHY –Jurnal Matematika Murni dan Aplikasi 
Volume 5(1)(2017), Pages 20-28 
p-ISSN: 2086-0382; e-ISSN: 2477-3344 
 

Submitted: 26 August 2017 Reviewed: 10 October 2017 Accepted: 2 November 2017 
DOI: http://dx.doi.org/10.18860/ca.v5i1.4388 

Local Stability Analysis of an SVIR Epidemic Model 

Joko Harianto1, Titik Suparwati2 

1, 2 Mathematics Department, Cenderawasih University 

 
Email: joharijpr88@gmail.com 

ABSTRACT 

In this paper, we present an SVIR epidemic model with deadly deseases. Initially the basic 
formulation of the model is presented. Two equilibrium point exists for the system; disease free 
and endemic equilibrium point. The local stability of the disease free and endemic equilibrium 
exists when the basic reproduction number less or greater than unity, respectively. If the value of 
𝑅0 less than or equal one then the desease free equilibrium point is locally asymptotically stable, 
and if its exceeds, the endemic equilibrium point is locally asymptotically stable. The numerical 
results are presented for illustration.  

Keywords: SVIR Model, Local Stability, Basic Reproduction Number 

INTRODUCTION 

Health problems are closely associated with a disease. It is motivating humans to study health 
as well as illness. One of the goals of a nation to achieve prosperity in the future is the creation of 
a healthy living environment. One of the indicators of a healthy environment involves a high level 
of quality health services. Quality health care issues need to be addressed systematically, 
effectively and efficiently. Analysis of the spread of a disease is one step to address the problem of 
quality health services. 

The spread of disease due to virus or bacteria that enter into the body will lead to human 
health disorders and will affect the socio-economic development of the community. Efforts to 
prevent a spreading disease can work optimally if several stages such as research, development 
of various diagnostic tools, drugs, and new vaccines, has achieved. 

Mathematical models are one of a useful tool for reviewing or analyse the patterns of a disease 
spread. According to [2] and [4] the spread of infectious diseases can be described mathematically 
through a model, such as SIR and SIRS. The most basic procedure in modelling the spread of 
disease is by using the compartment model. In this case, the population is divided into three 
different classes namely, Susceptible, Infected, and Recovered which then shortened to SIR. 
Vaccination can be considered as the addition of a class naturally into the epidemic base model 
for some types of diseases. 

Along with the development of knowledge in the field of health, control of an epidemic disease 
can be done by vaccination action. Then the SIR model develops with the addition of a vaccinated 
population class.  

Some researchers have proposed models of the dynamics of vaccination. Among others is SVI 
type, which has been developed by [5], [6] and [7]. Several studies have stated that the threshold 

mailto:joharijpr88@gmail.com


Local Stability Analysis of an SVIR Epidemic Model 

Joko Hariantoianto 21 

to determine the occurrence of endemic disease seen from the value of Basic Reproduction 
Number (𝑅0). The disease distribution model will be locally stable if 𝑅0 ≤ 1 and the disease will 
be present if  𝑅0 > 1. Paper [8] discusses the SVIR epidemic model for non-lethal diseases that 
does not involve the individual's mortality rate due to the disease. From these studies, we can 
build some further assumptions related to the SVIR epidemic model that developed in public life. 
In this paper, we analyse the SVIR epidemic model for a deadly disease. We hope the model can 
be useful to analyse the disease outbreaks so that we can take optimal precautions. 

 

METHODS 

This research used literature study with literature sources from some reputable journal. The 
simulation of model SVIR used mathematical software, i.e. Maple 2016. 

 

RESULTS AND DISCUSSION  

2.1. Formulation Model 

The following are assumptions used in modelling: 
1. Closed population (no migration) 
2. Births occur in every sub populations and enter in vulnerable subpopulations. 
3. Natural death occurs on each sub populations at the same rate. 
4. Illness can cause death. 
5. Individuals who have cured cannot return to susceptive class (permanent cure). 
6. Short incubation period. 
7. Vaccination has given to susceptible subpopulations. 
8. We do not distinguish the rate of recovery for a child has been infected and for the adult. 
9. Vaccination will reach the level of immunity over time and finally enters the sub populations 

that heals. 
10. Individuals who have vaccinated become infected if they lose immunity. 
11. Contact between infected and susceptive individuals can lead to disease transmission, and so 

are contact between infected and those have vaccinated. 

We investigate the basic model formulation by dividing the total population into four 
compartments, such as S(t) for susceptible, V(t) for vaccinated, I(t) for infected and R(t) for 
recovered individuals. Let 𝛽 is the rate of contact that is sufficient to transmit the disease. We also 
assume a constant recovery rate 𝛾. The rate at which the susceptible population is vaccinated is 
𝛼. We assume that there can be disease related death and define 𝜔 to be the rate of disease related 
death, while 𝜇 is the rate of natural death that is not related to the disease. We assume that all 
newborns enter the susceptible class at the constant rate of 𝜇. Let 𝛾1 be the average rate for 
susceptible individuals to obtain immunity and moved into recovered population. We do not 
distinguish the natural immunity and vaccine-induced immunity here because vaccine-induced 
immunity can also last for a long term. We assume that before obtaining immunity the vaccinees 
still have possibility of infection with a disease transmission rate 𝛽1 while contacting with infected 
individuals. 𝛽1 may be assume to be less than 𝛽 because the vaccinating individuals may have 
some partial immunity during the process or they may recognize the transmission characters of 
the disease and hence decrease the effective contacts with infected individuals.  Figure 1 shows 
the details of the population transfer diagram of the SVIR model. 
  



Local Stability Analysis of an SVIR Epidemic Model 

Joko Hariantoianto 22 

𝜔𝑅 

 
 
   
 
 
 
 
 
 
 
 
 

 
Figure.1 Transfer diagram of the SVIR model 
 

The following form presents the system model of differential equations. 
𝑑𝑆

𝑑𝑡
= 𝜇 −𝜇𝑆 −𝛽𝑆𝐼 −𝛼𝑆 

𝑑𝑉

𝑑𝑡
= 𝛼𝑆 −𝛽1𝑉𝐼 −𝛾1𝑉 −𝜇𝑉                                                                                 (1) 

𝑑𝐼

𝑑𝑡
= 𝛽𝑆𝐼 +𝛽1𝑉𝐼 −𝛾𝐼 −𝜇𝐼 −𝜔𝐼 

𝑑𝑅

𝑑𝑡
= 𝛾1𝑉 +𝛾𝐼 −𝜇𝑅 

where 𝑆(0) > 0,𝑉(0) > 0,𝐼(0) > 0,𝑅(0) = 0, 𝑆 +𝑉 +𝐼 +𝑅 = 1,∀ 𝑡 ≥ 0 and all parameters are 
positive. The parameters with their description are presented in Table 1. 
 

Table 1. Parameter Description 
 

Parameter Description 
𝜇 Natural death rate (and equally, the birth rate) 
𝛽 The transmission rate of disease when susceptible individuals 

contact with infected individuals 
𝛼 The rate of migration of susceptible individuals to the vaccination 

process 
𝛽1 The transmission rate for vaccination individuals to be infected 

before gaining immunity 
𝛾1 The average rate for vaccination individuals to obtain immunity and 

move into recovered population 
𝜔 The death rate due to disease 
𝛾 Recovery rate 

 
2.2 . Equilibrium Point 

Next, we will study system dynamics in equation (1). Since the last equation does not 
depend on other equations, we simply study the following system. 

𝑑𝑆

𝑑𝑡
= 𝜇 −𝜇𝑆 −𝛽𝑆𝐼 −𝛼𝑆 

                             
𝑑𝑉

𝑑𝑡
= 𝛼𝑆 −𝛽1𝑉𝐼 −𝛾1𝑉 −𝜇𝑉                                                          (2) 

𝑑𝐼

𝑑𝑡
= 𝛽𝑆𝐼 +𝛽1𝑉𝐼 −𝛾𝐼 −𝜇𝐼 −𝜔𝐼 



Local Stability Analysis of an SVIR Epidemic Model 

Joko Hariantoianto 23 

Furthermore, we performed a SVIR model stability analysis around the equilibrium point 
of the system (2). As a first step, we define system equilibrium points (disease free points and 
endemic points). 

In general, according to [1] and [3] there exist two equilibrium points of epidemic model, 
i.e. the disease free equilibrium point and the endemic equilibrium point. The two equilibrium 
points are reviewed based on the existence of the disease in a population with a continuous time. 
The disease-free equilibrium point is the point at which the disease is unlikely to spread in an area 
because the infected population is equal to zero (I=0) for 𝑡 → ∞. While the endemic equilibrium 
point is the point at which the disease must spread (I>0) for 𝑡 → ∞, in a defined closed area. 
Theorem 1 
There are two equilibrium points in the system (2), namely: 

a) The disease-free equilibrium point, that is, 

𝐸0 = (𝑆0,𝑉0, 𝐼0) = (
𝜇

𝛼 +𝜇
 ,

𝛼𝜇

(𝛼 +𝜇)(𝛾1 +𝜇)
  ,0) 

b) The endemic equilibrium point, that is, 

𝐸∗ = (𝑆∗,𝑉∗, 𝐼∗) = (
𝜇

(𝛼 +𝜇 +𝛽𝐼∗)
 ,

𝛼𝜇

(𝛼 +𝜇 +𝛽𝐼∗)(𝜇 +𝛾1 +𝛽1𝐼
∗)
  , 𝐼∗), 

where  𝐼∗ is the positive root of 𝐴1𝐼
2 +𝐴2𝐼 +𝐴3(1−𝐶) = 0, for C > 1, and 

𝐴1 = (𝛾 +𝜇 +𝜔)𝛽𝛽1 > 0 

𝐴2 = ( 𝛾 +𝜇 +𝜔)(( 𝛼 +𝜇)𝛽1  + (𝛾1 + 𝜇)𝛽))−𝛽1𝛽𝜇   

𝐴3=(𝛾 +𝜇 +𝜔)( 𝛼 +𝜇)( 𝛾1 + 𝜇) > 0 

Proof: 

The equilibrium point of system (2) exists when 
𝑑𝑆

𝑑𝑡
= 0, 

𝑑𝑉

𝑑𝑡
= 0, 

𝑑𝐼

𝑑𝑡
= 0,  so that: 

𝜇 −𝜇𝑆 −𝛽𝑆𝐼 −𝛼𝑆 = 0                                                      (3) 
𝛼𝑆 −𝛽1𝑉𝐼 −𝛾1𝑉 −𝜇𝑉 = 0                                               (4)    
𝛽𝑆𝐼 +𝛽1𝑉𝐼 −𝛾𝐼 −𝜇𝐼 −𝜔𝐼 = 0                                       (5) 

From equation (5), we have  𝐼(𝛽𝑆 +𝛽1𝑉 −𝛾 −𝜇 −𝜔) = 0, and the solutions are 𝐼 = 0 or  𝛽𝑆 +
𝛽1𝑉 −𝛾 −𝜇 −𝜔 = 0, consequently, we have to situation, namely: 
a) Situation at 𝑰 = 𝟎 

It is the necessary condition for the disease-free equilibrium point. Note that, 

 From equation (3),  𝜇 −𝜇𝑆 −𝛼𝑆 = 0 ⟺ 𝑺 = 
𝝁

𝜶+𝝁
 

 From (4), 𝛼𝑆 −𝛾1𝑉 −𝜇𝑉 = 0 ⟺ 𝑽 = 
𝛼𝝁

(𝜶+𝝁)(𝜸𝟏+𝝁)
   

So we get the equilibrium point of disease-free, that is,  

𝐸0 = (𝑆0,𝑉0, 𝐼0) = (
𝜇

𝛼 +𝜇
 ,

𝛼𝜇

(𝛼 +𝜇)(𝛾1 +𝜇)
  ,0) 

b) Situation at 𝑰 ≠ 𝟎, or 𝑰 > 0,  
This situation is a necessary condition for an endemic equilibrium point. Note that: 

 From equation (3),  𝜇 −𝜇𝑆 −𝛽𝑆𝐼 −𝛼𝑆 = 0 ⟺ 𝑺∗ = 
𝝁

(𝜶+𝝁+𝜷𝐼∗)
 

 From (4): 

𝛼𝑆 −𝛽1𝑉𝐼 −𝛾1𝑉 −𝜇𝑉 = 0    ⟺  𝑽
∗ = 

𝛼𝝁

(𝜶+𝝁+𝜷𝐼∗)(𝜸𝟏+𝝁+𝜷𝟏𝐼
∗)

 

From (5), if 𝐼 ≠ 0, then  𝛽𝑆 +𝛽1𝑉 −𝛾 −𝜇 −𝜔 = 0 , consequently, with substitute 𝑺
∗ and 𝑽∗ 

we have: 
𝛼𝜇𝛽1

(𝛼 +𝜇 +𝛽𝐼∗)(𝛾1 +𝜇 +𝛽1𝐼
∗)
 = 𝜇 +𝛾 +𝜔 −

𝛽𝜇

(𝛼 +𝜇 +𝛽𝐼∗)
   

⟺ 𝛽1𝛽(𝜇 +𝛾 +𝜔) 𝐼
∗2 +((𝜇 +𝛾 +𝜔)(( 𝛼 +𝜇)𝛽1 +(𝜇 +𝛾1)𝛽)−𝛽1𝛽𝜇) 𝐼

∗ + 



Local Stability Analysis of an SVIR Epidemic Model 

Joko Hariantoianto 24 

(𝜇 + 𝛾 + 𝜔)(𝜇 + 𝛾1)(𝛼 + 𝜇)(1− (
𝛽𝜇

(𝛼 +𝜇)(𝜇 + 𝛾 + 𝜔)
+

𝛼𝛽1𝜇

(𝛼 + 𝜇)(𝜇 + 𝛾1)(𝜇 + 𝛾 + 𝜔)
)) = 0 

By applying:  
 𝐴1 = (𝛾 +𝜇 +𝜔)𝛽1𝛽 > 0 
𝐴2 =( 𝛾 +𝜇 +𝜔)(( 𝛼 +𝜇) 𝛽1  + (𝛾1 + 𝜇)𝛽)− 𝛽1𝛽𝜇 
𝐴3=(𝛾 +𝜇 +𝜔)( 𝛼 +𝜇)( 𝛾1 + 𝜇) > 0 

𝐶 =
𝛽𝜇

(𝛼 +𝜇)(𝜇 +𝛾 +𝜔)
+

𝛼𝛽1𝜇

(𝛼 +𝜇)(𝜇 +𝛾1)(𝜇 +𝛾 +𝜔)
 , 

the above equation becomes:  
𝐴1𝐼

∗2 +𝐴2𝐼
∗ +𝐴3(1−𝐶) = 0                                                                                (6) 

with the roots of equation (6),  

𝐼1,2
∗ =

−𝐴2 ±√𝐴2
2 −4𝐴1𝐴3(1−𝐶)

2𝐴1
 

where  𝐼∗ > 0 and 𝐶 > 1. 
Thus, we obtain an endemic equilibrium point, that is,  

𝐸∗ = (𝑆∗,𝑉∗, 𝐼∗) = (
𝜇

(𝛼 +𝜇 +𝛽𝐼∗)
 ,

𝛼𝜇

(𝛼 +𝜇 +𝛽𝐼)(𝜇 +𝛾1 +𝛽1𝐼
∗)
  , 𝐼∗). 

2.3. Basic Reproduction Number (𝑹𝟎)  
The equilibrium point stability analysis on the SVIR model depends on the value of basic 

reproduction number 𝑅0 (the number of susceptible individuals who are then infected if they 
interact with patients in an entirely vulnerable population). A selection of this number is by 
observing the state of the endemic equilibrium point. 

Consider Eq. (6), an endemic equilibrium point only applicable to positive roots (𝐼∗ > 0) 
when C > 1. In conclusion, the value of a positive endemic equilibrium point lies in the value of C, 
so that it can be defined the value basic reproduction number, that is, 

𝑅0 =
𝛽𝜇

(𝛼 +𝜇)(𝜇 +𝛾 +𝜔)
+

𝛼𝛽1𝜇

(𝛼 +𝜇)(𝜇 +𝛾1)(𝜇 +𝛾 +𝜔)
 

2.4. LOCAL STABILITY OF EQUILIBRIUM POINT 

 We will determine the stability of the equilibrium point of the SVIR model around the 
disease-free equilibrium and the endemic equilibrium point. Equations (3), (4), and (5) are 
nonlinear equations, so by using Taylor series at each equilibrium point, there will be a system of 
linear differential equations. 
Theorem 2 
Defined:  

𝑅0 =
𝛽𝜇

(𝛼 +𝜇)(𝜇 +𝛾 +𝜔)
+

𝛼𝛽1𝜇

(𝛼 +𝜇)(𝜇 +𝛾1)(𝜇 +𝛾 +𝜔)
 

i. If 𝑅0 ≤ 1, then there is exist one (unique) equilibrium of system (1), that is, the disease-
free equilibrium point 𝐸0 . Furthermore, if  𝑅0 > 1, then there are two equilibrium points 
of system (1), that is, the disease-free equilibrium points 𝐸0  dan 𝐸

∗. 
ii. If 𝑅0 ≤ 1, then the equilibrium point 𝐸0 is locally asymptotically stable and if     𝑅0 > 1, 

then the equilibrium point 𝐸0 is unstable. 
iii. If 𝑅0 > 1, then the endemic equilibrium 𝐸

∗ is locally asymptotically stable. 
 
Proof: 
(i) We have discussed the existence of the equilibrium point of disease-free in Theorem 1. In the 
discussion, we obtain a point of equilibrium when I = 0 and is singular if 𝑅0 ≤ 1. The equilibrium 
point is the disease-free equilibrium point. 
Next, let's look at the equation 𝐴1𝐼

2 +𝐴2𝐼 +𝐴3(1−𝑅0) = 0, if 𝑅0 > 1, then                    𝐴3(1−𝑅0) <
0  and 𝐴1 > 0, consequently, the equation has two real roots (positive and negative). Therefore, a 
necessary condition that ensures the existence and unique equilibrium 𝐸∗ has been fulfilled. 
Furthermore, when 𝑅0 ≤ 1, we will show that the above quadratic equation has no positive roots. 
Note that, 



Local Stability Analysis of an SVIR Epidemic Model 

Joko Hariantoianto 25 

𝑅0 ≤ 1 ⟹      𝛽𝜇 ≤ (𝛼 +𝜇)(𝛾 +𝜇 +𝜔)    ⟹ 𝐴2 ≥ ( 𝛾 +𝜇 +𝜔)(( 𝛼 +𝜇) 𝛽1  + (𝛾1 + 𝜇)𝛽)−
 𝛽1(𝛼 +𝜇)(𝛾 +𝜇 +𝜔) > ( 𝛾 +𝜇 +𝜔)(𝛾1 + 𝜇)𝛽 > 0 
So if 𝐴1 > 0, 𝐴3(1−𝑅0) > 0 then the function of 𝐴1𝐼

2 +𝐴2𝐼 +𝐴3(1−𝑅0) will rise and 𝐴1𝐼
2 +

𝐴2𝐼 +𝐴3(1−𝑅0) > 𝐴3(1−𝑅0) ≥ 0, for 𝐼 > 0, which means the quadratic function has no 
positive roots.  
(ii) The Jacobian Matrix of system (2) is 

𝐽(𝑆,𝑉,𝐼) = (

−𝜇 −𝛼 −𝛽𝐼 0 −𝛽𝑆
𝛼 −𝜇 −𝛾1 −𝛽1𝐼 −𝛽1𝑉
𝛽𝐼 𝛽1𝐼 𝛽𝑆 +𝛽1𝑉 −𝜇 −𝛾 −𝜔

) 

So the Jacobian matrix around the point 𝐸0 = (𝑆0,𝑉0, 𝐼0) = ( 
𝜇

𝛼+𝜇
 ,

𝛼𝜇

(𝛼+𝜇)(𝛾1+𝜇)
  ,0) is 

𝐽(𝐸0) = (

−𝜇 −𝛼 0 −𝛽𝑆0
𝛼 −𝜇 −𝛾1 −𝛽1𝑉0
0 0 𝛽𝑆0 +𝛽1𝑉0 −𝜇 −𝛾 −𝜔

) 

The characteristic equation is 
⟺ [−(𝜇 +𝛼)−𝜆][−(𝜇 +𝛾1)−𝜆][ 𝛽𝑆0 +𝛽1𝑉0 −𝜇 −𝛾 −𝜔 −𝜆] = 0 
where 
𝜆1 = −(𝜇 +𝛼) < 0 
𝜆2 = −(𝜇 +𝛾1)< 0 
𝜆3 =  𝛽𝑆0 +𝛽1𝑉0 −𝜇 −𝛾 −𝜔 = (𝛾 +𝜇 +𝜔)(𝑅0 −1) 
It is clear if 𝑅0 < 1 then all the eigenvalues of 𝐽(𝐸0) are negative, hence the point 𝐸0 is locally 
asymptotically stable. Meanwhile, if 𝑅0 > 1, then there is a positive eigenvalue, hence the point 𝐸0 
is unstable.  
(iii) It is clear that the endemic point 𝐸∗ exist when  𝑅0 > 1,  so the Jacobian matrix around the 
point 𝐸∗ = (𝑆∗,𝑉∗, 𝐼∗) is 

𝐽(𝐸∗) = (

−𝜇 −𝛼 −𝛽𝐼∗ 0 −𝛽𝑆
𝛼 −𝜇 −𝛾1 −𝛽1𝐼

∗ −𝛽1𝑉
𝛽𝐼∗ 𝛽1𝐼

∗ 𝛽𝑆∗ +𝛽1𝑉
∗ −𝜇 −𝛾 −𝜔

) 

We can modify the element 𝑎11,𝑎22, 𝑎33, from matrix above into 

−𝜇 −𝛼 −𝛽𝐼∗ = − 
𝜇

𝑆∗
 

−𝜇 −𝛾1 −𝛽1𝐼
∗ = − 

𝛼𝑆∗

𝑉∗
 

Based on the evaluation results around the endemic equilibrium point we are getting 
 𝛽𝑆∗ +𝛽1𝑉

∗ −𝜇 −𝛾 −𝜔 = 0,  
and the matrix become  

𝐽(𝐸∗) =

(

 
 
− 
𝜇

𝑆∗
0 −𝛽𝑆

𝛼 − 
𝛼𝑆∗

𝑉∗
−𝛽1𝑉

𝛽𝐼∗ 𝛽1𝐼
∗ 0 )

 
 

 

The characteristic equation of matrix 𝐽(𝐸∗) is 
𝜆3 +𝑎1𝜆

2 +𝑎2𝜆 +𝑎3 = 0 
where 

𝑎1 =
𝜇

𝑆∗
+
𝛼𝑆∗

𝑉∗
> 0 

𝑎2 =
𝛼𝑆∗

𝑉∗
+𝛽1

2
𝑉∗𝐼∗ + 𝛽2𝑆∗𝐼∗ > 0 

𝑎3 = 𝛼𝛽1𝛽𝑆
∗𝐼∗ +

𝛼𝛽2𝑆∗
2
𝐼∗

𝑉∗
+
𝜇𝛽1

2
𝑉∗𝐼∗

𝑆∗
> 0 

hence, 

𝑎1𝑎2 −𝑎3  =
𝛼𝜇2

𝑆∗𝑉∗
+(𝜇 +𝛽𝐼∗)𝛽2𝑆∗𝐼∗ +

𝛼2𝜇𝑆∗

𝑉∗2
+𝛼𝐼∗𝑆∗(𝛽 −𝛽1)

2 +𝛼𝛽1𝛽𝑆
∗𝐼∗   >    0 



Local Stability Analysis of an SVIR Epidemic Model 

Joko Hariantoianto 26 

has been fulfilled. According to the Routh-Hurwitz criterion, all the eigenvalues of the above 
characteristic equations have a negative real part. So, the equilibrium point  𝐸∗ = (𝑆∗,𝑉∗, 𝐼∗) is 
locally asymptotically stable.  

Based on the results, we will interpret the results biologically as follows, 
 If 𝑅0 ≤ 1, then for 𝑡 → ∞ and (𝑆,𝑉,𝐼) that close enough to 𝐸0 = (𝑆0,𝑉0, 𝐼0), the solution of 

system (1) will move to 𝐸0 = (𝑆0,𝑉0, 𝐼0). It means that if 𝑅0 ≤ 1,  then for the number of 
susceptible, vaccinated and infected individuals close to                            𝐸0 = (𝑆0,𝑉0, 𝐼0), the 
disease will not plague and tends to disappear indefinitely time. This condition is then called 
asymptotically stable around the equilibrium point    𝐸0 = (𝑆0,𝑉0, 𝐼0), and we interpret as a 
disease-free equilibrium point. 

 If 𝑅0 > 1, then for  𝑡 → ∞ and (𝑆,𝑉,𝐼) that close enough to 𝐸
∗ = (𝑆∗,𝑉∗, 𝐼∗), the solution of 

system (1) will move to 𝐸∗ = (𝑆∗,𝑉∗, 𝐼∗). This means if 𝑅0 > 1, then for the number of 
susceptible, vaccinated and infected individuals close to 𝐸∗ = (𝑆∗,𝑉∗, 𝐼∗), the disease will 
plague but does not reach extinction in an infinite time. This condition is then called 
asymptotically stable around the equilibrium point 𝐸∗ = (𝑆∗,𝑉∗, 𝐼∗), and we interpret as an 
endemic equilibrium point. 

 

2.5. NUMERICAL RESULTS 
In this section, we investigate the numerical solution of the system (1) by using the Runge-

Kutta order four scheme. The state variables are chosen with same initials conditions. The 
numerical results are shown in Figure 1 and 2. Figure 1, illustrate the fact, when the basic 
reproduction number less than unity, when the value of basic reproduction number exceeds than 
unity, Figure 2 illustrate this fact.  

 

 
 

Figure 1. The dynamical behavior of system (1), for same initial conditions and different 
parameters:  when 𝑅0 = 0,22 ≤ 1,the desease free equilibrium point is locally 
asymptotically stable where  𝜇 = 1, 𝛽 = 10,𝛽1 = 2,𝛾 = 4,𝛾1 = 8,𝛼 = 10,𝜔 = 0,01. 

 



Local Stability Analysis of an SVIR Epidemic Model 

Joko Hariantoianto 27 

 
 

Figure 2. The dynamical behavior of system (1), for same initial conditions and different 
parameters:  when 𝑅0 = 3,17 > 1, endemic equilibrium is locally asymptotically stable 
where  𝜇 = 1, 𝛽 = 20,𝛽1 = 15,𝛾 = 1,𝛾1 = 2,𝛼 = 10,𝜔 = 0,01.  
 

CONCLUSION 

From the results of endemic equilibrium point analysis, we define the basic reproduction 
number parameter, that is, 

𝑅0 =
𝛽𝜇

(𝛼 +𝜇)(𝜇 +𝛾 +𝜔)
+

𝛼𝛽1𝜇

(𝛼 +𝜇)(𝜇 +𝛾1)(𝜇 +𝛾 +𝜔)
 

Furthermore, 𝑅0 is a necessary condition for the existence of the two points of equilibrium as well 
as its local stability. 

When 𝑅0 ≤ 1, there is only one (unique) equilibrium point of disease-free, that is, 𝐸0 =

(𝑆0,𝑉0, 𝐼0) = ( 
𝜇

𝛼+𝜇
 ,

𝛼𝜇

(𝛼+𝜇)(𝛾1+𝜇)
  ,0). Conversely, if 𝑅0 > 1, then there are two equilibrium points, 

that is, 𝐸0 and an endemic equilibrium point                                                       𝐸
∗ = (𝑆∗,𝑉∗, 𝐼∗) =

(
𝜇

(𝛼+𝜇+𝛽𝐼∗)
 ,

𝛼𝜇

(𝛼+𝜇+𝛽𝐼)(𝜇+𝛾1+𝛽1𝐼
∗)
  , 𝐼∗) where 𝐼∗ is the positive root of the equation 𝐴1𝐼

2 +𝐴2𝐼 +

𝐴3(1−𝑅0) = 0.  
Results of local stability analysis indicates if 𝑅0 ≤ 1, then the disease free equilibrium point 

𝐸0 is locally asymptotically stable. It means if the terms 𝑅0 ≤ 1 fulfilled, then in a long time, there 
would have no spread of disease in susceptible and vaccinated subpopulations, or in other words, 

the epidemic will stop. Conversely, if 𝑅0 > 1 then the endemic equilibrium point E


 is locally 
asymptotically stable. It means, in a long time, the disease will always exist in the population with 
the proportion of each subpopulation is equal to 𝑆∗,𝑉∗,dan 𝐼∗. 

 

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Local Stability Analysis of an SVIR Epidemic Model 

Joko Hariantoianto 28 

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