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Chapter 5
A MATHEMATICAL MODEL OF CANCER GROWTH WITH THE
EFFECT OF DELAY IN CELLULAR INTERACTION
INTRODUCTION
In this chapter, we propose and analyze a nonlinear mathematical model to study
growth of cancer and the interaction between cancer and immune cells in the body. The
model is extended by the inclusion of an intracellular delay effect in interaction. We offer
modifications to the model proposed by Banerjee and Sarkar (2008). They studied delayinduced model for cancer-immune interaction and control of malignant cancer growth.
We analyze their model with developed concepts like the Michaelis-Menton interactions
to model biological realities of the immune system. We also assume that the resting cells
or T-helper cells grow not only logistically but also at a rate directly proportional to the
size of the cancer cells as explained earlier in the fourth chapter. Stability of the steady
states of both model systems without and with delay is determined. Conditions for Hopf
bifurcation of the system are determined. Moreover, critical value of delay is also found
which acts as a bifurcation parameter. Numerical investigations are conducted to confirm
the results.
5.1 MATHEMATICAL MODEL
Our model is like a prey–predator system as we discussed in chapter four with
Michaelis-Menton interactions between hunting and resting cells. Following set of
nonlinear ordinary differential equations describe our dynamical system:
96

dT
T 
  1TH ,
 rT 1 
dt
 K1 
dH HR

 d1 H   2TH ,
dt
bR
(5.1.1)

dR
R  HR
 
 cT  sR1 
,
dt
 K2  b  R
with the initial conditions T (0)  0, H (0)  0 and R(0)  0.
Here, T, H and R are the number of cancer, hunting (T-lymphocytes) and resting cells
(T- helper cells), respectively. r, s are their respective intrinsic growth rates of cancer
cells and resting cells. K 1 is the carrying capacity for cancer cell and K 2 is the carrying
capacity of Resting cells.  1 is the rate of loss of cancer cells due to encounter with the
hunting cells and  2 is the rate of loss of hunting cells due to encounter with the cancer
cells. c represents the antigenicity of cancer.  represents the rate of proliferation of
hunting cells due to release of series of stimulating agents from hunting cell and b is a
positive constant. d1 is the natural death rate of hunting cells.
5.2 BOUNDEDNESS
In analogy to the population dynamics, it is very important to observe the consequences
that restrict the growth of the population. In this sense, study of boundedness of the
solution of system around different steady states is very much needed. For this, we find
boundedness of the system in the following lemma:
Lemma 5.2.1: All the solutions of (5.1.1) starting in the positive orthant R3 either
approaches, enter or remain in the subset of R3 defined by
97
K
L


  (T , H , R)  R3 : 0  T  K1 ,0  H  R  where L  cK1  2 ( s   ) 2 . R3

4s


denote the non-negative cone of R 3 including its lower dimensional faces.
Proof: From first equation of the system (5.1.1) we get

dT
T 
.
 rT 1 
dt
 K1 
(5.2.1)
Using standard comparison principle in (5.2.1), we obtain
lim sup T (t )  K1 .
(5.2.2)
t 
Further, adding second and third equations of the system (5.1.1) we have

dH dR
R 
  d1H   2TH .

 cT  sR1 
dt
dt
 K2 
Now using (5.2.2), the following inequality holds for each  ( 0)

d ( H  R)
sR 2 
  ( H  R)  cK1   sR  R 
 (  d1 ) H ,


dt
K
2


 cK1 
where
K2
( s   ) 2  (  d1 ) H ,
4s
K2
(s   ) 2  0 is the maximum value of the function
4s
(5.2.3)
2

 sR  R  sR .

K 2 

Note that the right hand side of (5.2.3) is bounded for   d1 . Then we can find a
constant L say, such that
d ( H  R)
  ( H  R)  L.
dt
(5.2.4)
Now using standard comparison principle in inequality (5.2.4), we get
98
lim sup H (t )  R(t )  
t 
L

.
Thus, it suffices to consider solutions in the region  . Solutions of the initial value
problem starting in  and defined by (5.1.1) exist and are unique on a maximal interval
(Hale, 1980). Since solutions remain bounded in the positively invariant region  , the
maximal interval is well posed both mathematically and epidemiologically.
5.3 EQUILIBRIUM ANALYSIS
System (5.1.1) has six equilibrium points given as,
~ ~
E 0 (0,0,0), E1 (0,0, K 2 ), E 2 (Tˆ ,0, Rˆ ), E3 (0, H , R ) and E 4 (T  , H  , R  ).
Existence of equilibria E 0 and E1 are obvious.
Existence of E2 (Tˆ ,0, Rˆ ) :
Equilibrium point E 2 (Tˆ ,0, Rˆ ) can be determined from the system of equations

Tˆ 
  0,
r 1 

K
1

(5.3.1)

Rˆ 
  0.
cTˆ  sRˆ 1 

 K2 
(5.3.2)
Solving (5.3.1) and (5.3.2) we get
K
4cK1 K 2
1
Tˆ  K1 and Rˆ  2 
K 22 
.
2 2
s
~ ~
Existence of E3 (0, H , R ) :
~ ~
Equilibrium point E3 (0, H , R ) is obtained from
99
~
~  d1  0,
bR
R
(5.3.3)
~
~

R  H

s1 
~.

 K2  b  R
(5.3.4)
From (5.3.3) and (5.3.4) we have
~
H
~
R
sb
(   d1 ) 2 K 2
(  d1 ) K 2  d1b,
d1b
.
(   d1 )

b 
.
Where   d1 1 
K
2 

(5.3.5)
Existence of E  (T  , H  , R  ) :
The non–trivial uniform equilibrium point E  (T  , H  , R  ) is solution of
equations

T 
  1TH  0,
rT 1 
 K1 
HR
bR
(5.3.6)
 d1 H   2TH  0,
(5.3.7)

R  HR
 
cT  sR1 
 0.
 K2  b  R
(5.3.8)
From (5.3.6)
H
r 
T 
1 
  f1 (T ),
1  K1 
(5.3.9)
100
(5.3.7) gives R 
(d1   2T )b
 f 2 (T ).
(   d1   2T )
(5.3.10)
Now substituting the value of H and R in (5.3.8) we get,

f (T )  f1 (T ) f 2 (T )

cT  sf 2 (T )1  2
 0.
K 2 
b  f 2 (T )

(5.3.11)
To show existence of E  (T  , H  , R  ) it suffices to show that equation (5.3.11) has a
unique positive solution.
Taking

f (T )  f1 (T ) f 2 (T )
 
G(T )  cT  sf 2 (T )1  2
,
K
b  f 2 (T )
2 

(5.3.12)
we note that

f (0)  f (0) f 2 (0)
G(0)  sf 2 (0)1  2   1
,
K
b

f
(
0
)
2 
2

 r

d1b
sb 
1 
 d1  0,
  

(


d
)
(


d
)
K
1 
1 2 
 1
provided
r
1


d1b
sb 
1 
,
(   d1 )  (   d1 ) K 2 
(5.3.13)
and

f ( K )  f ( K ) f ( K )
G( K1 )  cK1  sf 2 ( K1 )1  2 1   1 1 2 1 ,
K2 
b  f 2 ( K1 )

 cK1 
s(d1   2 K1 )b 
(d1   2 K1 )b 
1 
  0,
(   d1   2 K1 )  (   d1   2 K1 ) K 2 
101

b 
,
if condition   (d1   2 K1 )1 
 K2 
(5.3.14)
is satisfied.
Thus, there exists a T  in the interval 0  T   K1 such that G (T  )  0.
For T  to be unique, we must have



2sf 2' (T ) f 2 (T )
dG

 c  sf 2' (T ) 

f 2' (T )  (b  f 2 (T )) f1' (t ) f 2 (t )  f1 (t ) f 2' (t )  0
2
dx
K2
(b  f 2 (T ))
at T  T  .
(5.3.15)
Thus T  given by G (T  )  0 is unique. After knowing the value of T  , values of H 
and R  can be found from (5.3.9) and (5.3.10), respectively.
5.4 LOCAL STABILITY ANALYSIS
To discuss the local stability of system (5.1.1), we compute the variational matrix.
The signs of the real parts of the eigenvalues of the variational matrix evaluated at a
given equilibrium determine its stability. The entries of general variational matrix are
given by differentiating the right hand side of system (5.1.1) with respect to T , H and R .
The general variational matrix of the system is obtained as:
 
r 1  T   1H  rT
  K1 
K1

V (T , H , R)  
2H



c

 1T
R
bR
 d1   2T

R
bR
102




bH
.

(b  R) 2

R  sR
bH 

 
s1 

 K 2  K 2 (b  R) 2 
0
The variational matrix V ( E0 ) at equilibrium point E0 is given by
0
0
r

V ( E0 )  0  d1 0.
c
0
s 
From V ( E0 ), we note that two characteristic roots namely, r and s are positive implying
that E0 is unstable.
The variational matrix V ( E1 ) at equilibrium point E1 is given by


r
K2

V ( E1 (0,0, K 2 ))  0
b  K2

c




0
0


b 

 0 .
  d1 1 
 K 2 


K 2

 s 
b  K2

From (5.3.5) we note that here also, two characteristic root namely, r
K2
b  K2
and


b 
 are positive implying that E1 is unstable.
  d1 1 
K
2



The variational matrix V ( E 2 (Tˆ ,0, Rˆ )) at equilibrium point E2 is given by

 r

ˆ
ˆ
V ( E 2 (T ,0, R))   0


c

 1K1
ˆ
R
 d1   2 Kˆ 1
b  Rˆ
Rˆ

b  Rˆ


0

.
0


cK1 sRˆ 


K 2 
Rˆ
Here we see that two eigenvalues of the variational matrix, namely  r and 
are negative. Hence stability or unstability of the equilibrium point
103
cK1 sRˆ

K2
Rˆ
E 2 depends on
Rˆ
(d1   2 K1 )b
and
 d1   2 Kˆ 1 . It can be easily observed that E 2 is stable if Rˆ 
(  d1   2 K1 )
b  Rˆ
(d1   2 K1 )b
it is unstable if Rˆ 
.
(   d1   2 K 1 )
~ ~
~ ~
The variational matrix V ( E3 (0, H , R )) at equilibrium point E3 (0, H , R ) is given
by


~
 r  1 H

~ ~
~
V ( E3 (0, H , R ))     2 H


 c

0
0
~

~
bR
R


0

~

bH
.
~
(b  R ) 2

~
~~ 
sR
HR


~ 2
K 2 (b  R
) 
~
~ ~
From V ( E3 (0, H , R )), we note that one characteristic root (r   1 H ) is positive
since
r

d1b
sb 
~
~ ~
1 
  0 , from (5.3.13). Thus, E3 (0, H , R ) is
r  1 H  1  
  1 (   d1 )  (   d1 ) K 2  
an unstable equilibrium point.
The variational matrix V ( E  (T  , H  , R  )) of the system is obtained as:


  rT

K1

V ( E  (T  , H  , R  ))    2 H 


 c

 1T 
0

R
b  R




bH 
.


2
(b  R )





cT
sR
H R 






2
K
2
R
(b  R ) 
0
The characteristic equation for the variational matrix V ( E  (T  , H  , R )) is given by
104
3  M12  M 2  M 3  0,
(5.4.1)
where
M1 
cT  sR 
H  R rT 



,


2
K
K
2
1
R
(b  R )
M2 
b 2 H  R  rT   cT  sR 
H  R 



K1  R 
K 2 (b  R  ) 2
(b  R  ) 3


    T H ,
1 2


 cT  sR 
rT b 2 H  R   1T cbH 
H  R 
M3 

  1 2T  H  


 
K 2 (b  R  ) 2
K1 (b  R  ) 3
(b  R  ) 2
 R

.


By Routh – Hurwitz criteria, if
M 1 , M 3  0 and M 1 M 2  M 3 ,
(5.4.2)
then all roots of equation (5.4.1) have negative real parts and E  is a locally
asymptotically stable equilibrium.
Remark 1: Condition for local stability of equilibrium point E2 implies that population
of cancer cells rise to its maximum value K1 if density of resting cells is less than a fixed
value given by
(d1   2 K1 )b
.
(   d1   2 K 1 )
5.5 MATHEMATICAL MODEL WITH TIME DELAY
We now focus our attention to the effect of time delay in release of series of
stimulating agents from resting T-cells and activation or proliferation of hunting cells.
We consider discrete time delay in and denote it by  . Then model (5.1.1) transforms to
the following delay model:
105

dT
T 
  1TH ,
 rT 1 
dt
 K1 
dH HR (t   )

 d1H   2TH ,
dt b  R(t   )
(5.5.1)

dR
R  HR (t   )
 
 cT  sR1 
,
dt
 K 2  b  R(t   )
with initial conditions   (1, 1, 2 ) defined in the space,


C    C ([  ,0], R03 ) : 1 ( )  M ( ), 1 ( )  N ( ), 2 ( )  Z ( ) ,
3
where, M ( )  0, N ( )  0, Z ( )  0,   C([ ,0]; C ([ ,0], R0
) is the space of vector
valued
continuous
functions
and
is
a
mapping
from
[  ,0]
to
3
,
R0
R03  (M , N , Z )  R3 : M , N , Z  0. C is positively invariant and in this region, the
usual existence, uniqueness and continuation results hold for system (5.5.1).
5.6 STABILITY ANALYSIS WITH TIME DELAY
The determination of stability in case of delay differential equations is analogous
to the ordinary differential equations. We perturb the system around equilibrium
point E * (T  , H  , R  ) to get the following linearized system of differential equations:
dT1
T
 rT1
 1T  H1 ,
dt
K1
dH1 H  R1 (t   ) H  R  R1 (t   )


  2T1H  ,


2
dt
bR
(b  R )
106
(5.6.1)
 2 R   H R  H  R (t   ) H  R  R (t   )
dR1
1
1
1

 cT1  sR1 1 


.


)2
dt
K 2  b  R
b

R
(
b

R


Where T1 (t )  T (t )  T  , H 1 (t )  H (t )  H  and R1 (t )  R (t )  R  .
To explore local stability of mathematical model with delay we determine
variational matrix of the system (5.6.1) at E  (T  , H  , R  ),


 rT
  K
1


V ( E  )    2 H 


 c


 1T 
0

R 
b  R


0



bH e  
.
(b  R  ) 2

 2 R   bH e   


s 1 

K 2  (b  R  ) 2 



The characteristic equation for the variational matrix E  (T  , H  , R  ) is given by
3  A12  A2   A3  ( B12  B2   B3 )e   0,
where
A1 
 2R 
rT 
,
 s 1 

K1
K2 


A2  
srT   2 R  
1
  1 2T  H  ,


K1
K2


 2R 
,
A3  s 1 2T  H  1 

K2 


B1 
bH 
,
(b  R ) 2
107
(5.6.2)
B2 
rT bH 
b 2 H  R

,
K1 (b  R ) 2 (b  R ) 3
B3 
rT b 2 H  R 1T cbH 
bH 

 1 2T  H 
.

3

2

2
K1 (b  R )
(b  R )
(b  R )
It is well known that signs of the real parts of the solutions of (5.6.2) characterize the
stability behavior of system (5.6.1) and hence of (5.6.1). Therefore, substituting
    i in (5.6.2) we obtain real and imaginary parts, respectively, as
 3  3 2  (  2   2 )( A1  B1e   cos  )  2B1e   sin 
  ( A2  B2e
 
cos  )B2 e
 
sin   A3  B3e
 
cos   0,
3 2   3  2 ( A1  B1e   cos  )  (  2   2 ) B1e   sin 
  ( A2  B2 e
 
cos  )  B2 e
 
sin   B3e
 
(5.6.3)
(5.6.4)
sin   0.
A necessary condition for a stability change of E  (T  , H  , R  ) is that the characteristic
equation (5.6.2) has purely imaginary solutions. Hence, to obtain the stability criterion,
we set   0 in (5.6.3) and (5.6.4) to obtain
  2 ( A1  B1 cos  )  B2 sin   A3  B3 cos   0,
(5.6.5)
  3   2 B1 sin    ( A2  B2 cos  )  B3 sin   0.
(5.6.6)
Eliminating  between (5.6.5) and (5.6.6) we get
~
~
~
 6  M 1 4  M 2 4  M 3  0,
(5.6.7)
where.
~
M 1  A12  2 A2  B12 ,
~
M 2  A22  2 A1 A3  2B1 B3  B22 ,
108
~
M 3  A32  B32 .
Substituting  2  P in (5.6.7), we get a cubic equation given by
~
~
~
 ( P)  P 3  M 1P 2  M 2 P  M 3  0.
(5.6.8)
Now (5.6.8) will have a positive root if
~
~
M 1  0 and M 3  0.
(5.6.9)
Since, the existence condition for interior equilibrium point E  (T  , H  , R  ) holds true,
~
~
we have the condition for M 1 to be positive and M 3 to be negative. Thus, we can say that
there is a unique positive  0 satisfying (5.6.8), that is, the characteristic equation (5.6.2)
has a pair of purely imaginary roots of the form  i 0 . From (5.6.5) and (5.6.6), we
have,
 k 
  2 B ( 2  A2 )  ( 02 B1  B3 )( 02 A1  A3 )  2k
arccos  0 2 0
.

0
( 02 B1  B3 ) 2   02 B22

  0
1
(5.6.10)
For   0, E  (T  , H  , R  ) is stable if (5.5.2) holds. Hence by Butler’s lemma
(Freedman and Rao, 1983), E  (T  , H  , R  ) remains stable for    0 where  0   0 at
k  0.
We also observe that the conditions for Hopf-bifurcation (Hale and Lunel,1993)
are satisfied if condition (5.6.9) holds, that is,
 d (Re  ) 
 0.
 d 

  
0
This signifies that there exists at least one eigenvalue with positive real part for    0 .
109
5.7 HOPF BIFURCATION ANALYSIS
Let   i0 be the purely imaginary root of equation (5.6.2). We wish to
determine the direction of motion of  as  is varied.
  d  1 
 d (Re  ) 
So, we determine sign 
.
 sign Re   
 d   i0
  d    i
0
Now, differentiating (5.6.2) with respect to  , we get
(3
2
 2 A1  A2 )  e   (2 B1  B2 )  e   ( B12  B2   B3 )
dd
 ( B12  B2   B3 )e    ,
this implies that
 d 
 
 d 


1

(32  2 A1  A2 )
( B12  B2   B3 )e   
(32  2 A1  A2 )
 (3  A12  A2   A3 )
(23  A12  A3 )
 (  A1  A2   A3 )
3
2
2



2 B1  B2

,
 ( B12  B2  B3 ) 
2 B1  B2

B12  B3



,
 ( B12  B2   B3 ) 

.
 ( B1  B2   B3 ) 
2
2
Thus,
  d  1 
sign Re   
  d  
  i 0
 
(23  A12  A3 )
B12  B3
 
 sign Re 

 
,
   (3  A12  A2   A3 )2 2 ( B12  B2   B3 )  
  i 0
 

( A3  A102 )  i 203
B102  B3

,

sign Re

2
2
3
2



0
  ( A10  A3 )  i (0  A20 ) ( B3  B10 )  iB20 
1
110
 ( A3  A102 )( A102  A3 )  2 3 ( 3  A20 ) ( B  2  B )( B  B  2 ) 
0 0
3
3
1 0 ,

sign 
 1 0
2
2
2
3
2
2
2

0
( A10  A3 )  (0  A20 )
( B3  B10 )  ( B20 ) 2 
1

 ( A  A102 )( A102  A3 )  203 (03  A20 )  ( B102  B3 )( B3  B102 ) 
sign  3
,
02
( B3  B102 ) 2  ( B20 ) 2



 206  ( A12  2 A2  B12 )04  ( B32  A32 ) 
sign

.
02
( B3  B102 ) 2  ( B20 ) 2


1
1
As A12  2 A2  B12 and B32  A32 are positive by virtue of condition (5.6.9), thus we have
 d (Re  ) 
 0.
 d 
 0
Therefore, we conclude that the transversality condition holds and hence Hopf bifurcation
occurs at    0 .
5.8 NUMERICAL SIMULATION
In this section, we present numerical simulation of the system using MATLAB
software. The numerical solutions of the ordinary differential equation are obtained by
using fourth order Runge-Kutta method under the following set of parameters [Banerjee
and Sarkar (2008), Usman, Jackson and Cunningham (2005)]:
r  0.18 , K1  5 X 10 6 , K 2  1X 10 7 , 1  1.101 10 7 , d1  0.0412 ,  2  3.422  10 10 ,
  6.2 10 9 , s  0.0245 , c  5 X 10 5 , b  1000
(5.8.1)
The endemic equilibrium values for this set of parameters are:
T   4.9783 X 10 6 , H   7.08445 X 103 , R  2.24668 X 103 .
Eigenvalues corresponding to endemic equilibria E  (T  , H  , R  ) are obtained as:
111
 0.00867  0.04148i and  0.17906. Since all the eigenvalues corresponding to
E  (T  , H  , R  ) are negative, therefore E  (T  , H  , R  ) is locally asymptotically
stable equilibrium point.
Keeping other parameters fixed we changed the values of parameters  and c,
and found some interesting observations for the system. On varying  (rate of
proliferation of hunting cells on being activated by resting cells) and keeping other
parameters fixed it was found in Fig. 1 that number of cancer cells increase for low value
of  . Low value of  corresponds to slow rate of proliferation of hunting cells. In this
case, number of hunting cells produced will be lesser than the population of hunting cells
required to suppress cancer growth in the body. Hence, increase in cancer cells in the
body occurs.
In Fig. 2, variation of cancer population with time for different values of
c (antigenicity of cancer cells), is displayed. It is found from the figure that as
antigenicity of cancer increases cancer cell population decreases. Further, we have
observed that on an increase in c cancer cell population first decrease abruptly and then
tend to a smaller equilibrium value. Moreover, we observe from the figure that higher is
the value of c more abrupt decrease in cancer cell population takes place. Thus, for
higher antigenicity of cancer cells, numbers of cancer cells reduce for a while but after
some time their number again starts rising and reach an equilibrium level that is lower
than that for lower antigenicity of cancer cells. Further, we have observed in Fig.3 that
for higher value of cancer antigenicity, c  25 X 105 onwards, cancer cell population
show periodic oscillations. It first decreases to zero, remains at zero level for some time
112
and then further rises abruptly to the endemic equilibrium level. It corresponds to the
recurrence of cancer even for high antigenicity of cancer.
x 10
5.2
6
5
4.8
T(t)
4.6

= 1.2x10

= 6.2x10
-2
-2
4.4
4.2
4
0
200
400
600
800
1000
Time(t)
Fig. 1, Variation of T(t) with time for different

for the set of parameters same as (5.8.1)
x 10
6
5
4.9
4.8
c = 5x10
T(t) 4.7
-5
c = 10x10
c = 15x10
4.6
c = 20x10
-5
-5
-5
4.5
4.4
4.
0
200
400
600
800
Time(t)
Fig. 2, Variation of T(t) with time for different
for the set of parameters same as (5.8.1)
113
c
1000
5
x 10
6
4
3
T(t)
2
1
0
-1
c  25 X 10 5
0
1000
2000
Time(t)
3000
4000
Fig. 3, Variation of tumor cell population with time for high valve of c
for the set of parameters same as (5.8.1)
It is widely known that the introduction of time delays into the predator prey
system may be a cause for the periodic oscillations of populations and can make the
behavior of the model more complex. In this chapter, we have tried to understand the
effect of time delay in proliferation of hunting cells due to release of series of stimulating
agents from resting cells with Michaelis-Menton interactions to model biological realities
of the immune system. From the section 5.6, we see that under some conditions the
positive equilibrium loses its stability and a periodic solution through Hopf bifurcation
occurs when the delay passes through a critical value. This implies that the time delays
are able to alter the dynamics of system significantly. To verify the results numerically,
~
~
~
~
we calculated the values of M 1 and M 3 and find that M 1  0 and M 3  0 for the same
114
set of parameters given in (5.8.1). Solving (5.6.8), we see that there exists a simple
positive root given by, 0  0.0495. The value of  where stability switch occurs,
which corresponds to the transition from the stable steady state to stable oscillatory state,
is obtained as  0  7.9528. This is the critical value of delay and can be easily obtained
using (5.6.10).  0 is found to be a bifurcation parameter as for time delay below  0 ,
system is stable and for values above  0 , system is unstable.
6
6
x 10
7400
5.5
7200
H(t)
T(t
5
)
7000
4.5
4
0
2000
4000
6000
Time (t)
6800
8000
0
2000
(a)
4000
6000
Time (t)
8000
(b)
2600
2600
2400
2400
R(t)
R(t)
2200
2200
2000
2000
0
1800
2000 4000 6000 8000 10000
Time (t)
7400
(c)
7200
7000
6800
H(t)
5
4.95
6
4.9
T(t)x 10
(d)
Fig.4, (a-c) Time evolution of all the cells for the system around E
*
 0
and (d) phase portrait depicting stable limit cycle for the set of parameters same as (5.8.1)
115
In Fig. 4 (a-d), we have shown that system remains stable for    0 , (  7.3 ) as
each cell population converges to its equilibrium value. Fig. 4(d) shows the stable limit
cycle approaching the equilibrium point and hence display the stable nature of the system
for    0 . Further, we have observed that for    0 (  8), E  (T  , H  , R  ) remains in
the stable oscillatory mode, with higher amplitude of oscillations. It is illustrated
graphically in Fig.5(a-c).
6
4.9795
x 10
7400
4.9785
7200
H (t)
T
7000
(t)
4.9775
6800
3000
4000
5000
Time (t)
6000
3000
4000
Time (t)
(a)
5000
6000
(b)
2600
2800
2400
R (t)
R(t)
2200
2400
2000
1800
3000
2000
7200
4000
5000
Time (t)
6000
5
4.95 x 106
6800
H (t)
(c)
4.9 T (t)
(d)
Fig.5, (a-c) Time evolution of all the cells for the system around
   0 and (d) phase portrait
depicting the unstable limit cycle for the set of parameters same as (5.8.1)
116
Phase portrait of the system for    0 , (  8) is displayed in Fig. 5(d), from
which we see an unstable limit cycle growing out of E  (T  , H  , R  ) .
5.9 CONCLUSION
Here, a mathematical non-linear differential equation model for cancer and
immune response has been considered and the stability and instability of the equilibrium
points of the system are studied using the linear stability approach. To substantiate the
analytical findings, the model is studied numerically and for which the system of
differential equation is integrated using fourth order Runge-Kutta method.
We have found that population of cancer cells rise to its maximum value K1 , that
is the carrying capacity of the cancer cells, if density of resting cells is less than a fixed
value. Hence, attempts should be made to keep density of resting cells always more than
this value to control the cancer. In addition, we have found that the key role is being
played by the parameters  (rate of proliferation of hunting cells) and c (antigenicity of
cancer cells) in our model. From numerical simulation, we observed that for low value of
 and keeping other parameters constant, number of cancer cells increase. Moreover, we
have also shown the variation of cancer population with time for different values of
c (antigenicity of cancer cells). We have observed that on an increase in antigenicity of
cancer cells, cancer cell population first decrease abruptly and then tend to a smaller
equilibrium value. Further, we observed that higher is the value of c more abrupt
decrease in cancer cell population takes place. This implies that cancer cell population is
smaller for high antigenicity of the cancer
117
The model is extended by the inclusion of an intracellular delay effect. We have
analyzed the stability of the steady states of model systems with delay, and conducted the
numerical investigation to confirm the results. We have also derived the conditions for
the existence of a Hopf bifurcation such that the steady state loses its stability at    0 . A
bifurcation parameter  0 is found. Numerically we have shown that the critical value of
delay is  0  7.9528 . We found that system remains stable for    0 , (  7.3) , however,
for    0 i.e. for   8, we find that system stability is lost and cells population
oscillates periodically.
118