CAUCHY –Jurnal Matematika Murni dan Aplikasi 
Volume 5(3) (2018), Pages 102-111 
p-ISSN: 2086-0382; e-ISSN: 2477-3344 

Submitted: 28 August 2018 Reviewed: 08 October 2018 Accepted: 30 November 2018 
DOI: http://dx.doi.org/10.18860/ca.v5i3.5511 

SVIR Epidemic Model with Non Constant Population 

Joko Harianto1, Titik Suparwati2 

1,2Mathematics Department, Cenderawasih University 
Email: joharijpr88@gmail.com 

ABSTRACT 

In this article, we present an SVIR epidemic model with deadly deseases and non constant 
population. We only discuss the local stability analysis of the model. 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 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, Basic Reproduction Number, Numerical results 

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 [8]. Several studies have 
stated that the threshold 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 [9] discusses the SVIR 
epidemic model for non-lethal diseases that does not involve the individual's mortality 

mailto:joharijpr88@gmail.com


SVIR Epidemic Model with Non Constant Population 

Joko Harianto  103 

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 article we consider 
local stability analysis of an SVIR epidemic model with deadly disease and non constant 
population based on SVIR epidemic model developed in [8]. 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  

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 birth occurs on each sub populations at the same rate 
4. Natural death occurs on each sub populations at the different rate. 
5. Illness can cause death. 
6. Individuals who have cured cannot return to susceptive class (permanent cure). 
7. Short incubation period. 
8. Vaccination has given to susceptible subpopulations. 
9. We do not distinguish the rate of recovery for a child has been Infected and for the 

adult. 
10. Vaccination will reach the level of immunity over time and finally enters the sub 

populations that heals. 
11. Individuals who have vaccinated become Infected if they lose immunity. 
12. 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. Let 𝜇1 be the natural death rate for susceptible population, 𝜇2 
be the natural death rate for vaccination population, 𝜇3 be the natural death rate for 
Infected population, 𝜇4 be the natural death rate for recovery population. We assume that 
all newborns enter the susceptible population 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 epidemic model. 
 



SVIR Epidemic Model with Non Constant Population 

Joko Harianto  104 

 

 
                                
 

 
 
 
 
 
 
 
Figure 1. Transfer diagram of the SVIR epidemic model with deadly deseases and non 

constant population 
 
 

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

𝑑𝑡
= 𝜇 −𝜇1𝑆 −𝛽𝑆𝐼 −𝛼𝑆 

𝑑𝑉

𝑑𝑡
= 𝛼𝑆 −𝛽1𝑉𝐼 −𝛾1𝑉 −𝜇2𝑉                                                                                 (1) 

𝑑𝐼

𝑑𝑡
= 𝛽𝑆𝐼 +𝛽1𝑉𝐼 −𝛾𝐼 −𝜇3𝐼 −𝜔𝐼 

𝑑𝑅

𝑑𝑡
= 𝛾1𝑉 +𝛾𝐼 −𝜇4𝑅 

where 𝑆(0) > 0,𝑉(0) > 0,𝐼(0) > 0,𝑅(0) = 0, with 𝑆(0)+𝑉(0)+𝐼(0)+𝑅(0) = 1, and 
all parameters are positive. The parameters with their description are presented in the 
following table. 
 

Table 1. Parameter Description 
Parameter Description 

𝜇 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 
𝜇1 Natural death rate of the susceptible population 
𝜇2 Natural death rate of the vaccination population 
𝜇3 Natural death rate of the Infected population 
𝜇4 Natural death rate of the recovery population 

 
  

𝜇 𝛾𝐼 
S 

𝜇1𝑆 𝜇3𝐼 𝜇4𝑅 𝜔𝐼 

𝛽𝑆𝐼 

I R 

𝛽1𝑉𝐼 

𝛼𝑆 

V 

𝜇2𝑉 

𝛾1𝑉 



SVIR Epidemic Model with Non Constant Population 

Joko Harianto  105 

Equilibrium Point 
As a first step, we define equilibrium points of the system (disease free equilibrium 

point and endemic equilibrium point). 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. 
The equilibrium point of system (1) exists when 

𝑑𝑆

𝑑𝑡
= 0      ,

𝑑𝑉

𝑑𝑡
= 0       ,

𝑑𝐼

𝑑𝑡
= 0       ,       

𝑑𝑅

𝑑𝑡
= 0 

so that: 
               𝜇 −𝜇1𝑆 −𝛽𝑆𝐼 −𝛼𝑆 = 0                                                       (2) 
        𝛼𝑆 −𝛽1𝑉𝐼 −𝛾1𝑉 −𝜇2𝑉 = 0                                                       (3)    
𝛽𝑆𝐼 +𝛽1𝑉𝐼 −𝛾𝐼 −𝜇3𝐼 −𝜔𝐼 = 0                                                       (4) 
                        𝛾1𝑉 +𝛾𝐼 −𝜇4𝑅 = 0                                                       (5) 

From equation (4), we have  𝐼(𝛽𝑆 +𝛽1𝑉 −𝛾 −𝜇3 −𝜔) = 0, and the solutions are 𝐼 = 0 or 
 𝛽𝑆 +𝛽1𝑉 −𝛾 −𝜇3 −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 (2), we have  𝜇 −𝜇1𝑆 −𝛼𝑆 = 0 ⟺ 𝑆 = 
𝜇

𝛼+𝜇1
 

 From equation (3), we have 𝛼𝑆 −𝛾1𝑉 −𝜇2𝑉 = 0 ⟺ 𝑉 =
𝛼𝑺

(𝛾1+𝜇2)
=

𝛼𝝁

(𝜶+𝝁𝟏)(𝜸𝟏+𝝁𝟐)
   

 From equation (5), we have 𝛾1𝑉 +𝛾𝐼 −𝜇4𝑅 = 0  ⟺ 𝑅 =
𝛾1𝑽

𝜇4
= 

𝛾1𝛼𝜇

𝜇4(𝛼+𝜇1)(𝛾1+𝜇2)
 

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

𝐸0 = (𝑆0,𝑉0, 𝐼0,𝑅0) = (
𝜇

𝛼 +𝜇1
 ,

𝛼𝜇

(𝛼 +𝜇1)(𝛾1 +𝜇2)
  ,0  ,

𝛾1𝛼𝜇

𝜇4(𝛼 +𝜇1)(𝛾1 +𝜇2)
) 

 
b) Situation at  𝑰 > 0,  

This situation is a necessary condition for an endemic equilibrium point. Note that: 

 From equation (2), we have  𝜇 −𝜇1𝑆 −𝛽𝑆𝐼 −𝛼𝑆 = 0 ⟺ 𝑆
∗ = 

𝝁

(𝜶+𝝁𝟏+𝜷𝐼
∗)

 

 From equation (3), we have 

𝛼𝑆 −𝛽1𝑉𝐼 −𝛾1𝑉 −𝜇2𝑉 = 0    ⟺  𝑉
∗ = 

𝛼𝝁

(𝜶+𝝁𝟏+𝜷𝐼
∗)(𝜸𝟏+𝝁𝟐+𝜷𝟏𝐼

∗)
 

From equation (4), if 𝐼 ≠ 0, then  𝛽𝑆 +𝛽1𝑉 −𝛾 −𝜇3 −𝜔 = 0 , consequently, with 
substitute 𝑆∗ and 𝑉∗ we have: 

         
𝛽𝜇

(𝛼 +𝜇1 +𝛽𝐼
∗)
+

𝛽1𝛼𝜇

(𝛼 +𝜇1 +𝛽𝐼
∗)(𝛾1 +𝜇2 +𝛽1𝐼

∗)
−(𝜇3 +𝛾 +𝜔) = 0   

⟺ (𝜇3 +𝛾 +𝜔)−
𝛽𝜇

(𝛼 +𝜇1 +𝛽𝐼
∗)
−

𝛼𝜇𝛽1
(𝛼 +𝜇1 +𝛽𝐼

∗)(𝛾1 +𝜇2 +𝛽1𝐼
∗)
= 0 

⟺                                                  (𝜇3 +𝛾 +𝜔)−
𝛽𝜇

(𝛼 +𝜇1 +𝛽𝐼
∗)
−
𝛽1𝛼𝜇

𝑘
= 0 

⟺                                       (𝜇3 +𝛾 +𝜔)𝑘 −𝛽𝜇(𝛾1 +𝜇2 +𝛽1𝐼
∗)−𝛼𝜇𝛽1 = 0 

with 𝑘 = (𝛼 +𝜇1 +𝛽𝐼
∗)(𝛾1 +𝜇2 +𝛽1𝐼

∗) = 𝛽𝛽1𝐼
∗ +[(𝛼 +𝜇1)𝛽1 +(𝛾1 +𝜇2)𝛽]𝐼

∗ −𝛽1𝛼𝜇 
and then we have 



SVIR Epidemic Model with Non Constant Population 

Joko Harianto  106 

𝐴1 𝐼
∗2 +𝐴2𝐼

∗ +𝐴3 (1−(
𝛽𝜇

(𝛼 + 𝜇
1
)(𝜇

3
+ 𝛾 + 𝜔)

+
𝛽1𝛼𝜇

(𝛼 + 𝜇
1
)(𝛾

1
+ 𝜇

2
)(𝜇

3
+ 𝛾 + 𝜔)

)) = 0 

where:  
 𝐴1 = (𝜇3 + 𝛾 + 𝜔)𝛽1𝛽 > 0 

𝐴2 = (𝜇3 + 𝛾 + 𝜔)[( 𝛼 +𝜇1) 𝛽1  + (𝛾1 + 𝜇2)𝛽]− 𝛽1𝛽𝜇 

𝐴3 = (𝜇3 + 𝛾 + 𝜔) ( 𝛼 +𝜇)( 𝛾1 + 𝜇) > 0 

Let 

𝐶 =
𝛽𝜇

(𝛼 + 𝜇
1
)(𝜇

3
+ 𝛾 + 𝜔)

+
𝛽1𝛼𝜇

(𝛼 + 𝜇
1
)(𝛾

1
+ 𝜇

2
)(𝜇

3
+ 𝛾 + 𝜔)

 , 

So, the above equation becomes:  
𝐴1𝐼

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

with the roots of equation (6), i.e. 

𝐼1,2
∗
=
−𝐴2 ±√𝐴2

2
−4𝐴1𝐴3(1−𝐶)

2𝐴1
 

In this case, the proportion of the Infected population must be positive. So, the 
existence of the endemic equilibrium point of Infected population depends on the value 
of C. From equation (6) we can have positive root (𝐼∗ > 0) if 𝐶 > 1. 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 +𝜇1)(𝜇3 +𝛾 +𝜔)
+

𝛽1𝛼𝜇

(𝛼 +𝜇1)(𝛾1 +𝜇2)(𝜇3 +𝛾 +𝜔)
 

 
In epidemiology, the basic reproduction number shows about the disease spread and 
control. If 𝑅0 < 1, then the disease free equilibrium is stable, the disease dies out from 
population. If 𝑅0 > 1 the endemic equilibrium exists, and disease permanently exist in the 
population. 
It is observed that 𝐼∗ exist if 𝑅0 > 1. 
Thus, we obtain an endemic equilibrium point if 𝑅0 > 1 , that is,  

𝐸∗ = (𝑆∗,𝑉∗, 𝐼∗,𝑅∗) = (
𝜇

(𝛼 +𝜇1 +𝛽𝐼
∗)
 ,

𝛼𝜇

(𝛼 +𝜇1 +𝛽𝐼)(𝛾1 +𝜇2 +𝛽1𝐼
∗)
  , 𝐼∗,

𝛾1𝑉
∗ +𝛾𝐼∗

𝜇4
). 

The existence of disease free equilibrium point is not depends on 𝑅0 condition. Thus  we 
conclude that, 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 point 𝐸0 and the 
endemic equilibrium point 𝐸∗. 
 
Local Stability Analysis 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. 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 −𝜇1𝑆 −𝛽𝑆𝐼 −𝛼𝑆 

                             
𝑑𝑉

𝑑𝑡
= 𝛼𝑆 −𝛽1𝑉𝐼 −𝛾1𝑉 −𝜇2𝑉                                                          (7) 



SVIR Epidemic Model with Non Constant Population 

Joko Harianto  107 

𝑑𝐼

𝑑𝑡
= 𝛽𝑆𝐼 +𝛽1𝑉𝐼 −𝛾𝐼 −𝜇3𝐼 −𝜔𝐼 

Furthermore, we performed a SVIR model stability analysis around the equilibrium point 
of the system (7). To discuss the local stability of system (7), we compute the Jacobian 
matrix of system (7). The signs of the real parts of the eigenvalues of the Jacobian Matrix 
evaluated at a given equilibrium determine its stability. The entries of general Jacobian 
matrix are given by differentiating the right hand side of system (7) with respect to 𝑆,𝑉,𝐼. 
The disease free stability and the endemic stability are presented in the following results. 
 
Theorem 1 
Defined:  

𝑅0 =
𝛽𝜇

(𝛼 + 𝜇
1
)(𝜇

3
+ 𝛾 + 𝜔)

+
𝛼𝛽1𝜇

(𝛼 +𝜇1)(𝛾1 +𝜇2)(𝜇3 +𝛾 +𝜔)
 

i. If 𝑅0 < 1, then the equilibrium point 𝐸0 is locally asymptotically stable and if     
𝑅0 > 1, then the equilibrium point 𝐸0 is unstable. 

ii. If 𝑅0 > 1, then the endemic equilibrium 𝐸
∗ is locally asymptotically stable. 

 
Proof: 
(i). The Jacobian Matrix of system (2) is 

𝐽(𝑆,𝑉,𝐼) = (

−(𝛼 +𝜇1 +𝛽1𝐼) 0 −𝛽𝑆

𝛼 −(𝛾1 +𝜇2 +𝛽1𝐼) −𝛽1𝑉

𝛽𝐼 𝛽1𝐼 𝛽𝑆 +𝛽1𝑉 −(𝜇3 +𝛾 +𝜔)
) 

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

𝛼+𝜇1
 ,

𝛼𝜇

(𝛼+𝜇1)(𝛾1+𝜇2)
  ,0) is 

𝐽(𝐸0) = (

−(𝛼 +𝜇1) 0 −𝛽𝑆0
𝛼 −(𝛾1 +𝜇2) −𝛽1𝑉0
0 0 𝛽𝑆0 +𝛽1𝑉0 −(𝜇3 +𝛾 +𝜔)

) 

The characteristic equation is 
⟺ [𝜆 +(𝛼 +𝜇1)][𝜆 +(𝛾1 +𝜇2)][ 𝛽𝑆0 +𝛽1𝑉0 −(𝜇3 +𝛾 +𝜔)] = 0 
where 
𝜆1 = −(𝛼 +𝜇1) < 0 
𝜆2 = −(𝛾1 +𝜇2)< 0 
𝜆3 =  𝛽𝑆0 +𝛽1𝑉0 −(𝜇3 +𝛾 +𝜔) = (𝜇3 +𝛾 +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.  
(ii). 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∗) = (

−(𝛼 +𝜇1 +𝛽𝐼
∗) 0 −𝛽𝑆

𝛼 −(𝛾1 +𝜇2 +𝛽1𝐼
∗) −𝛽1𝑉

𝛽𝐼∗ 𝛽1𝐼
∗ 𝛽𝑆∗ +𝛽1𝑉

∗ −(𝜇3 +𝛾 +𝜔)
) 

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

𝛼 +𝜇1 +𝛽𝐼
∗ = 

𝜇

𝑆∗
 

𝛾1 +𝜇2 +𝛽1𝐼
∗ = 

𝛼𝑆∗

𝑉∗
 

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

∗ −𝜇3 −𝛾 −𝜔 = 0,  
and the matrix become  



SVIR Epidemic Model with Non Constant Population 

Joko Harianto  108 

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        (8) 
where 

𝑎1 =
𝜇

𝑆∗
+
𝛼𝑆∗

𝑉∗
> 0 

𝑎2 =
𝛼𝜇

𝑉∗
+𝛽1

2
𝑉∗𝐼∗ + 𝛽2𝑆∗𝐼∗ > 0 

𝑎3 = 𝛼𝛽1𝛽𝑆
∗𝐼∗ +

𝛼𝛽2𝑆∗
2
𝐼∗

𝑉∗
+
𝜇𝛽1

2
𝑉∗𝐼∗

𝑆∗
> 0 

hence, 

𝑎1𝑎2 −𝑎3  = (
𝜇

𝑆∗
+
𝛼𝑆∗

𝑉∗
)(
𝛼𝜇

𝑉∗
+𝛽1

2
𝑉∗𝐼∗ + 𝛽2𝑆∗𝐼∗)−(𝛼𝛽1𝛽𝑆

∗𝐼∗ +
𝛼𝛽2𝑆∗

2
𝐼∗

𝑉∗
+
𝜇𝛽1

2
𝑉∗𝐼∗

𝑆∗
) 

=
𝛼𝜇2

𝑆∗𝑉∗
+𝜇𝛽2𝐼∗ +

𝛼2𝜇𝑆∗

𝑉∗
2
+𝛼𝛽1

2𝑆∗𝐼∗ −𝛼𝛽1𝛽𝑆
∗𝐼∗ 

=
𝛼𝜇2

𝑆∗𝑉∗
+(𝛼 +𝜇1 +𝛽𝐼

∗)𝛽2𝑆∗𝐼∗ +
𝛼2𝜇𝑆∗

𝑉∗
2
+𝛼𝑆∗𝐼∗(𝛽 −𝛽1)

2 +𝛼𝛽1𝛽𝑆
∗𝐼∗ > 0. 

It can be seen that all of the coefficients 𝑎𝑖 > 0,𝑖 = 1,2,3 are positive. The Routh-Hurwitz 
criteria that are necessary and sufficient for the local asymptotic stability of the endemic 
equilibrium are that the coefficient are positive and the Hurwitz determinants 𝐻𝑖 are all 
positive [7]. For a third degree polynomial, the Hurwitz determinants criteria are 𝐻1 =
𝑎1 > 0, 𝐻2 = 𝑎3 > 0 and 𝐻3 = 𝑎1𝑎2 −𝑎3 > 0. By Routh-Hurwitz criteria, roots of 
equation (7) have negative real parts if 𝑎1𝑎2 −𝑎3 > 0. Note that all term in 𝑎1𝑎2 −𝑎3 is 
always positive. Thus, the Routh-Hurwitz criteria satisfied for all parameter values. 
Therefore, by Routh-Hurwitz criteria the endemic 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 reach the value 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 reach the value 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. 

 

NUMERICAL RESULTS 
To substantiate the above analitycal findings, the model is studied numerically. We 

will discuss the results of the model by taking numerical example. We consider the 
parameters 𝜇 = 0,03,   𝜇1 = 0,04 , 𝜇2 = 0,035 ,  𝜇3 = 0,06, 𝜇4 = 0,045 , 𝛽 = 0,5,  𝛽1 =
0,05, 𝛾 = 0,1 ,𝛾1 = 0,1 , 𝛼 = 0,2 , 𝜔 = 0,1 and the basic reproduction number is         



SVIR Epidemic Model with Non Constant Population 

Joko Harianto  109 

𝑅0 = 0,28 < 1. The diseases free equilibrium point for this set of parameters are 
(0.125,0.185,0 ,0.41). The numerical results are shown in Figure 1 and 2. Figure 1, 
illustrate the fact, when the basic reproduction number less than unity, the infected 
proportion will reach to zero so that the desease free equilibrium point is locally 
asymptotically stable.  

 
 

 
 
 
 
 
 
 
 
 
 

Figure 2.  Basic reproduction number 𝑅0 versus infected proportion 𝐼(𝑡) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

 
 
 
Figure 3. The dynamical behavior of system (1), when 𝑅0 < 1 and the initial conditions 

are 𝑆(0) = 0.5 ,𝑉(0) = 0.3 , 𝐼(0) = 0.2 ,𝑅(0) = 0.   
We consider the parameters 𝜇 = 0,03,   𝜇1 = 0,04 , 𝜇2 = 0,035 ,  𝜇3 = 0,06, 𝜇4 =

0,045 , 𝛽 = 0,5,  𝛽1 = 0,05, 𝛾 = 0,1 ,𝛾1 = 0,1 , 𝛼 = 0,02 , 𝜔 = 0,1 and the basic 
reproduction number is  𝑅0 = 1,17 > 1. The endemic equilibrium point for this set of 
parameters are (0.315,0.046,0.07 ,0.26). The numerical results are shown in Figure 3.  
 



SVIR Epidemic Model with Non Constant Population 

Joko Harianto  110 

 
Figure 4. The dynamical behavior of system (1), when 𝑅0 > 1 and the initial conditions 

are 𝑆(0) = 0.5 ,𝑉(0) = 0.3 , 𝐼(0) = 0.2 ,𝑅(0) = 0. 
 

CONCLUSIONS 

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

𝑅0 =
𝛽𝜇

(𝛼 + 𝜇
1
)(𝜇

3
+ 𝛾 + 𝜔)

+
𝛼𝛽1𝜇

(𝛼 +𝜇1)(𝛾1 +𝜇2)(𝜇3 +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,𝑅0) = (
𝜇

𝛼+𝜇1
 ,

𝛼𝜇

(𝛼+𝜇1)(𝛾1+𝜇2)
  ,0  ,

𝛾1𝛼𝜇

𝜇4(𝛼+𝜇1)(𝛾1+𝜇2)
). 

Conversely, if  𝑅0 > 1, then there are two equilibrium points, that is, 𝐸0 and an endemic 

equilibrium point 𝐸∗ = (𝑆∗,𝑉∗, 𝐼∗,𝑅∗) = (
𝜇

(𝛼+𝜇1+𝛽𝐼
∗)
 ,

𝛼𝜇

(𝛼+𝜇1+𝛽𝐼)(𝛾1+𝜇2+𝛽1𝐼
∗)
  , 𝐼∗,

𝛾1𝑉
∗+𝛾𝐼∗

𝜇4
) 

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∗, 𝐼∗and 𝑅∗. 

  



SVIR Epidemic Model with Non Constant Population 

Joko Harianto  111 

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