International Journal of Modern Nonlinear Theory and Application, 2012, 1, 47-50
http://dx.doi.org/10.4236/ijmnta.2012.12006 Published Online June 2012 (http://www.SciRP.org/journal/ijmnta) 47
Fluctuating Role of Param eters in the Analysis of the
Continues and Discrete Version of a
Susceptible-Incubated-Infected Model
Prasenjit Das1, Debasis Mukherjee2, Kalyan Das3*, A. Sabarmathi4
1Baikunthapur High School, Maipith, India
2Department of Mathematics, Vivekananda College, Kolkata, India
3Department of Mathematics, National Institute of Food Technology Entrepreneurship and Management,
HSIIDC Industrial Estate, Kundli, India
4Department of Mathematics, School of Advanced Sciences, VIT University, Vellore, India
Received March 16, 2012; revised April 15, 2012; accepted April 30, 2012
ABSTRACT
The article concentrates on the role of fluctuating parameters for removable population from the incubated class in a
susceptible-incubated-infected model. The discrete analogous of this model is also investigated. Conditions for local
asymptotic stability are derived for both the disease free and endemic cases. Numerical simulations are performed to
validate the theoretical results.
Keywords: Incubated Class; Stability; Schur-Cohn Criterion
1. Introduction
A rigorous study of mathematical models on biology
shows that discrete-time models described by difference
equations are more justified than the continuous-time
models when the size of the popu lation is rarely small or
populations have non-overlapping generations. Further,
epidemiological data for infectious diseases is collected
in discrete from. Difference equation models give richer
dynamics than continuous ones. Many authors [1-9] stu-
died and analyzed the global stability, dynamics and
chaotic behavior of various disease models. In the year
2009, G. Q. Sun [10] analyzed the Predator cannibalism
can give rise to regular spatial pattern in a predator–prey
system. In a recent works on disease model Dhar and
Sharma [11] investigated the role of incubation period
and showed that Hopf-bifurcation has occurred for cer-
tain threshold value of disease contact rate. In the study,
they have considered logistic growth rate of susceptible
populations. But in the present article we have analyzed
the same model by considering constant recruitment rate
of susceptible population along with its discrete version.
The main thrust of our paper is to highlight the role of
removable population from the incubated class. The pa-
per is organized in the following manner. In Section 2 we
present our model. The Section 3 deals with local behav-
ior of the continuous-time model. Asymptotic behavior
of discrete version is carried out in Section 4. In Section
5 numerical examples and simulations are given and we
round up the paper in Section 6.
2. Model Equation
Keeping in view that incubation period is a time period
from exposure to onset of disease; we have considered
that susceptible population instead of joining infected
class directly will go through an intermediate class termed
as incubated class. Let us consider total population is
divided into three classes namely the susceptible class,
incubated class and infected class. The continuous model
system is described by the following equations:
1
dS
b
dS SD D
dt
dI SD I
dt
dD ID
dt



 


(1)
Here


St,


It and are the number of sus-
ceptible population, incubated population and infected
population respectively at time . The parameter

Dt
t
b
is
the recruitment rate of susceptible, and is its natural
death rate. The parameter d

represents the disease con-
tact rate.

is the fraction of diseased population re-
*Corresponding a uthor.
C
opyright © 2012 SciRes. IJMNTA

P. DAS ET AL.
48
covery from disease while

and

respectively rep-
resent the total removable population from incubated
class and from diseased class, which include death due to
natural death and disease induced death. Further 1

is
the fraction of incubated class population that will go to
the diseased class.
3. Analysis of the Continuous Model
3.1. Equilibria
The model system (1) has the following two equilibria: 1 )
Disease free equilibrium point

S,0

1
E,0 where
b
Sd  and, 2) Endemic Equilibrium point
where

** *
2
ES,I,D

*
1
S




,


1 *
11
b
d
I

 




 

and

*1
1
b
d
D









. Further is feasible if
2
E
11
b
d





 or 11
b
d





 .
3.2. Local Stability
We now state the stability behavior of the model system
(1) at the endemic equilibrium point .
2
Theorem1. The endemic equilibrium point is sta-
E
2
E
ble if 1




 and is unstable if 1





ad
.
Proof. The characteristic equation of the model system
(1) at the point is given by:
where
2
E
 
32
123
aaa   0*
1D

 

**
D
 

aS
,


*
ad  
2D1
  and 3




1
E
.
The result follows by the Routh-Hurwich criteria. Fur-
ther it is important to note that when is stable for
1
b
d




 then is unstable. On the contrary while
is unstable for
2
E
1
E1
d
b




 then is stable.
2
E
n
SD
4. Analysis of Discrete Model
In this section we investigate the dynamics of the discrete
analogues of the continuous model (1). The model is de-
scribed as follows:
n1 nnn
n1 nnn
n1n1n
SSbD
IIS I
DD D
n
n
n
dS
D
I





 






(2)
The fixed points of system (2) are as the follows: 1)
1b
E,0,0
d






and 2) : where

**
S ,I*
2
E ,D*
1
S




,



1 *
11
b
d
I

 

 

 and

*1
1
b
d
D



 

 provided
11
b
d





 or 11
b
d





 .
We study asymptotic stability of the system of differ-
ence Equation (2) with the help of Schur-Cohn criterion
[2]. That is, asymptotic stability of the system of differ-
ence equations can be found with the eigenvalues of Ja-
cobian matrix of system (2).
4.1. Stability Analysis of Fixed Point
The Jacobian matrix of system (2) at the state variable is
given by



1
dD0S
J S,I,DDS
0




  






We now state the local behavior of boundary fixed
points.
Theorem 2. The fixed point is locally stable if
1
E
1
b
d




 and is unstable if 1
b
d




.
Proof: The characteristic equation of the Jacobian ma-
trix at is
1
E

21b
d0
d







and the proof is obvious. Before stating the Theorem 3
we use the following theorem:
Theorem: Schur-Cohn criterion [2]: The zeros of the
characteristic polynomial

kk1
1k
p
pp  

 
where the pi’s and only if the following results hold: are
real numbers lie inside the unit disk if
1)


p
10, 2)

k
1p1 0

 and 3) the




k1k1

 matrices
a)
k
1k
k1
k2 k3kk12
10 000p
p1000 p
pp 1
1
p
pp

 

 
 
 
 

 
 
 


 

b) a real positive innerwise.
Theorem 3. The fixed point is asymptotically sta-
ble if the condition
2
E2
213

3
p
pp p1

 is satisfied.
Proof: The characteristic equation of the above Jaco-
bian matrix at is
2
E32
123
p
pp  0 where,
11
p
a3

, 212
p
32a a

 and 33
p
a. The result is
followed by the application of Schur-Cohn criterion.
5. Numerical Examples and Simulations
We now provide some examples to discuss the dynamics
of the system with some parameter values. Choosing b =

P. DAS ET AL.
49
0.4, ,
d0.050.02

, 0.01

, 0.02

, β1 = 0. 01 ,
and 0.02

 we observe that the system (1) has two
equilibrium points namely the disease free equilibrium
point (8, 0, 0) and the endemic equilibrium point (2, 20,
10). From numerical simulations we observe that the
endemic equilibrium is stable for both the continuous
case (Figure 1(a)) and in discrete case (Figure 1(b)).
Considering the parameter values
b
0.25, d1

, α
= 0.5, 0.5


, 0.1


, 10.3

 and 1


0.25
, we ob-
serve that the disease free fixed point

is
stable for both the continuous time model (Figure 2(a))
and discrete time model (Figure 2(b)).
,0,0
Considering the parameter values
b
0.25, d1

,α
= 1, , , and
11

10.5


0.3



. We get the
endemic fixed point


0.3,0.25,0.25 . From numerical
simulations (see Figures 3(a) and (b)) it is observed that
(a) (b)
Figure 1. (a) Stable Endemic equilibrium for continuouscase; (b) Stable Endemic equilibrium for discrete case.
(a) (b)
Figure 1. (a) Disease free stable fixed point for continuous model; (b) Disease free stable fixed point for discrete model.
(a) (b)
Figure 2. (a) Unstable endemic equilibrium point for continuous time model; (b) Unstable endemic equilibrium point for dis-
rete time model. c

P. DAS ET AL.
50
the endemic equilibrium point is unstable for both the
continuous time model and discrete time model.
6. Discussion
The analytical results and numerical findings of the paper
suggest the removable rate of population from incubated
class (β) which plays an important role on the dynamics
of the system. The disease free equilibrium approaches to
the endemic equilibrium when β is above a certain thre-
shold value and on the contrary the endemic equilibrium
approaches to the disease free equilibrium below this
thresh- old of β. So the disease outbreak can be under
control with the parameter β. On the other hand in paper
[11] authors have shown that the disease transfer rate
from susceptible to incubated population as a bifurcation
parameter. In brief, we can conclude that though the dis-
ease is endemic in nature initially, still in the long run, it
would be possible to control the disease and even if it
may also be eradicated from the society based on the
number of rem ova ble po p ulation from incubated class.
7. Acknowledgements
Authors are thankful to the reviewers for their valuable
comments and suggestions to improve this paper.
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