Chapter - 4
A GENERALISED PREY-PREDATOR MODEL OF CANCER
GROWTH WITH THE EFFECT OF IMMUNOTHERAPY
INTRODUCTION
Cancer is one of the greatest killers in many countries and the control of
cancer growth requires special attention. In the present chapter, we study a
generalized mathematical model on cancer growth and its treatment by
immunotherapy as a deterministic prey-predator like model. We assume that the prey
is cancer cells and predators are immune cells. Two types of Immune cells are
considered namely resting cells or T-helper cells and hunting cells or cytotoxic Tlymphocytes. When T-helper cells find malignant cancer cells, they release a series of
stimulating agents (cytokines, IL-2, interferon gamma etc.) that activate the hunting
cells to kill cancer cells. The process of natural immune attack against cancer cells is
not always sustainable and therefore several techniques and methodologies have been
developed to enhance natural immune response against cancer. This method of
treatment of cancer is called immunotherapy. In this therapy, blood is drawn from
cancer patients and then immune cells are expanded in number artificially which are
again put back into the bloodstream.
In this chapter, we modify the model given by Nani and Freedman (2000) by
considering induction of primary immune response against cancer in resting cells or
T- helper cells as a function of cancer cells in the body. The model is analyzed for
stability of equilibria using stability theory of differential equations. Moreover, the
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numerical simulation of the proposed model is also performed by using fourth order
Runge - Kutta method.
4.1 MATHEMATICAL MODEL
Mathematical modeling of the actual phenomenon in the cancer immune cell
interactions is very difficult. We present here a very basic and general mathematical
model to discuss the interaction among cancer and immune cells of the body. We
study a three dimensional model with cancer cells x(t ) , hunting cells y (t ) and resting
cells z (t ) using a system of nonlinear ordinary differential equations. Each equation
of the system represents the rate of change of a variable with respect to time. Thus,
the final form of mathematical model is,
dx
 B( x)  D( x)  h( x, y ),
dt
dy
 Q1  f ( y, z )  1d1 ( y )  h( x, y ),
dt
(4.1.1)
dz
 Q2   ( x)  f ( y, z )   2 d 2 ( z ),
dt
with,
x(t 0 )  x0  0, y(t0 )  y0  0, z (t0 )  z0  0.
We assume that the cancer cells are proliferating at the rate B (x ) defined as
the birth rate and dying at the rate D(x) . Q1 and Q2 are the proliferation rate of
hunting and resting cells respectively due to external infusion of immune cells during
immunotherapy. 1 ,  2 ,  and  are the positive constants. In addition, our model
is based on following hypothesis given below:
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H1: There do not exist negative solutions x(t ) , y (t ) and z (t ) for non-small t, since
they are physically unacceptable, so that
x(t )  0 y (t )  0 and z (t )  0 t  0.
H2:The term f ( y , z ) represents the rate of proliferation of hunting cells due to
release of series of stimulating agents from by resting cells. It is characterized by
f y ( y, z )  0, y  0, z  0,
f z ( y, z)  0, y  0, z  0,
f (0, z )  0, f ( y,0)  0
H3: h( x, y ) represents the cancer cell destruction by hunting cells due to stimulation
by resting cells. It may be assumed that
hx ( x, y)  0, x  0, y  0,
h y ( x, y )  0, x  0, y  0,
h y (0, y)  0, y  0
hx (0, y)  0, y  0, h(0, y )  0, h( x,0)  0 .
H4: d1 ( y) and d 2 ( z ) represent the elimination of hunting and resting cells
respectively. It satisfies following conditions:
d1, 2 (0)  0,

d1 (0)  0, y  0,
d 2 (0)  0, z  0
H5:  (x) is the induction of primary immune response against cancer in resting cells
that is assumed to be a function of cancer cells population in the body. It satisfies
following conditions:
 (0)  0,
  ( x)  0, x  0,
  ( x)  0, x  0.
H6: The birth and death rates of cancer cells are based on following assumptions:
B(0)  D(0), B( x)  0, D( x)  0, B (0)  0, D(0)  0, and there exist a value
K  0 such that B( K )  D( K ) and B ( K )  D( K ).
Note: Here f y , f z , hx , hy represent partial derivatives of functions f ( y, z ) and h( x, y )
with respect to the variables y , z , x respectively. B( x), D( x) and   (x ) are the total
derivatives of B( x), D( x) and  (x) with respect to x respectively.
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4.2 BOUNDEDNESS
Here we show that system (4.1.1) is bounded.
Theorem 4.1: All solutions of system (4.1.1) with initial values in R3 are bounded in
Q1


3
( x, y, z )  R ,0  x(t )  K ,0  y (t )    ,1  0,


1
the region  defined by   

0  z (t )  Q2   ( K ) ,  0

2


2
.
Where,

~
~

1  max  max f ( y, z )  1 min d1 ( y) ,
y
y, z y, z


~
2   2 min d 2 ( z)  0 .
z
Proof: Let x0  0 , considering first equation of model (4.1.1) we have
dx
 B( x)  D( x)  h( x, y ),
dt
dx
 B( x)  D( x).
dt
But by hypothesis there exist a value K  0 such that B( K )  D( K ) . Thus,
x(t )  max( K , x0 )
We note that
dx
 0 for x  K and hence,
dt
lim sup x(t )  K .
t 
Let us now consider second equation of system (4.1.1),
dy
 Q1  f ( y, z )  1d1 ( y )  h( x, y ),
dt
71
dy
 Q1  f ( y, z )  1d1 ( y ),
dt
~
~
 Q1  y max f ( y, z )  y1 min d1 ( y),
y
y, z
~
~
where f ( y, z )  yf ( y, z ), d1 ( y )  yd1 ( y ).
Now,


~
~
dy
 Q1  y max  max f ( y, z )  1 min d1 ( y) .
dt
y
y, z y, z


~

~
Let 1  max  max f ( y, z )  1 min d1 ( y)  ,
y
y, z y, z

~
~
We assume that max f ( y, z)  1 min d1( y) or 1  0.
y
y, z
Q
Thus, we have y (t )   1  y0 e1t , y0  0.
1
 Q

thus, y (t )  max   1 , y0 ,
 1

Q
or lim sup y(t )   1 , 1  0, y0  0.
1
t 
Similarly, if z 0  0 , third equation of the model gives,
dz
 Q2   ( x)  f ( y, z )   2 d 2 ( z ),
dt
~
dz
 Q2   ( K )   2 z min d 2 ( z ),
dt
~
where, d 2 ( z )  zd 2 ( z ) .
~
Now, if  2   2 min d 2 ( z )  0,
we have,
z (t ) 
Q2   ( K )
2
 z0 e  2 t ,
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 Q   (K )

, z0 ,
this implies that z (t )  max  2
2


and hence,
lim sup z (t ) 
t 
Q2   ( K )
2
.
This proves the boundedness of the system.
4.3 EQUILIBRIUM ANALYSIS
Equilibrium points of the system are obtained by solving right hand side of
equations given in (4.1.1). There are two possible equilibrium points of the system
(4.1.1):
Cancer
free
equilibrium
point
E (0, y, z )
and
Interior
Equilibrium
point E  ( x  , y  , z  ) .
Existence of E (0, y, z ) :

In this case, the system is restricted to R yz
. The equilibrium point E (0, y, z ) is
obtained by solving following system of differential equations:
dy
 Q1  f ( y, z )  1d1 ( y ),
dt
(4.3.1)
dz
 Q2  f ( y, z )   2 d 2 ( z ),
dt
with y (t 0 )  y0  0, z (t0 )  z0  0 .
Theorem 4.3.1: Let
P1  max f ( y, z )  0,
(4.3.2)
y, z
73


P2  min 1 min d1 ( y), 2 min d 2 ( z )   0.
y, z
y
z

(4.3.3)
Then,
lim sup  y (t )  z (t )  
t 
Q1  Q2  (1   ) P1
.
P2
Proof: Let us choose a function
M (t )  y (t )  z (t ).
(4.3.4)
Differentiating (4.3.4) with respect to t, we have
dM d ( y  z )

 Q1  Q2  f ( y, z )  f ( y, z )  1d1 ( y )   2 d 2 ( z )
dt
dt
~
~
 Q1  Q2  (1  ) max f ( y, z )  1 y min d1 ( y)   2 z min d 2 ( z )
y, z
y
z


~
~
 Q1  Q2  (1  ) P1  ( y  z )1 min d1 ( y),  2 min d 2 ( z ) 
z
 y

 Q1  Q2  (1   ) P1  MP2
This implies that ,
Q  Q2  (1   ) P1
lim sup M (t )  1
,
P2
t 
or
Q  Q2  (1   ) P1
lim sup  y(t )  z (t )   1
P2
t 
.
Thus, we have shown that system (4.3.1) is dissipative under conditions (4.3.2) and
(4.3.3).

Lemma 4.3.1: Let us assume that ( y , z )  R yz
such that
Q1  1d1 ( y )  1 Q2   2 d 2 ( z )  0

as t   .
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Then the Equilibrium point E (0, y, z ) exists.
Proof: Equating the right hand side of system (4.3.1) to zero we have,
Q1  f ( y, z)  1d1( y)  0,
(4.3.5)
Q2 f ( y, z)   2d 2 ( z)  0.
(4.3.6)
We have shown that system (4.3.1) is dissipative under conditions (4.3.2) and (4.3.3)
in theorem (4.3.1).
Now from (4.3.5) and (4.3.6) we have,
f ( y, z )  Q1  1d1 ( y ) 
iff 1d1 ( y )  Q1 
1

iff Q1  1d1 ( y )  
1

Q2   2 d 2 ( z ),
Q2   2 d 2 ( z )  0,
1

(4.3.7)
Q2   2 d 2 ( z )  0
This proves the lemma.
Existence of Interior Equilibrium point E  ( x  , y  , z  ) :
E  ( x  , y  , z  ) is the equilibrium point of the system (4.1.1) if it satisfies its right
hand side, that is
B( x  )  D( x  )  h( x  , y  )  0
(4.3.8)
Q1  f ( y  , z  )  1d1 ( y  )  h( x  , y  )  0
Q2   ( x  )  f ( y  , z  )   2 d 2 ( z  )
(4.3.9)
(4.3.10)
within the region  .
We will prove the existence of E  ( x , y , z  ) by persistence analysis in section 4.4.
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4.4 LOCAL STABILITY ANALYSIS
We now discuss local stability of the system (4.1.1) about its equilibrium
points. To do so we explore variational matrix due to linearization of (4.1.1) about
equilibrium points and compute its eigenvalues. Negative eigenvalues of the
variational matrix about an equilibrium point implies local asymptotic stability of that
equilibrium point. General variational matrix of the system about an arbitrary
equilibrium point is given by,
 B( x)  D( x)  hx ( x, y )

V (E)  
  h x ( x, y )

 ( x)



f z ( y, z )
.
 f z ( y, z )   2 d 2 ( z )
 h y ( x, y )
f y ( y, z )  1d1 ( y )  h y ( x, y )
0
 f y ( y, z )
4.4.1 Local Stability of Cancer Free Equilibrium Point E (0, y, z ) :
Using hypothesis H1-H6, the variational matrix of the system due to
linearization of the system (4.4.1) about E (0, y, z ) is expressed as,
 B(0)  D(0)  hx (0, y )

V (E )  
 hx (0, y )

 (0)



f y ( y , z )  1d1 ( y )
f z ( y, z )
. (4.4.1)
 f y ( y , z )
 f z ( y , z )   2 d 2 ( z )
0
0
The eigenvalues of the variational matrix V (E ) are given by
1  B(0)  D(0)  hx (0, y ),
(4.4.2)
and the quadratic equation
2   f z ( y, z )  1d1 ( y )   2 d 2 ( z )  f y ( y, z )


 f z ( y , z )   2 d 2 ( z )  1d1 ( y )  f y ( y , z )  f y ( y, z ) f z ( y , z )  0
By the Routh Hurwitz criteria, the eigenvalues of variational matrix V (E ) have
negative real parts if conditions,
76
f z ( y, z )  1d1 ( y)   2d 2 ( z )  f y ( y, z )  0,
(4.4.3)
f z ( y, z )   2d2 ( z )1d1 ( y)  f y ( y, z )f y ( y, z ) f z ( y, z )  0 hold.
(4.4.4)
Thus, if conditions (4.4.2), (4.4.3) and (4.4.4) are satisfied then the equilibrium point
E (0, y, z ) is locally asymptotically stable equilibrium point.
Remark 1: We note that the equilibrium point E (0, y , z ) is a hyperbolic saddle point
if 1  B(0)  D(0)  hx (0, y)  0 and 2 , 3  0 . In other words, we can say that
E (0, y , z ) is repelling in x  direction in this case. And E (0, y , z ) is hyperbolic source
if 1  B(0)  D(0)  hx (0, y)  0 and 2 , 3  0 .
Remark 2: Equilibrium point E (0, y , z ) demonstrate the scenario in which all the
cancer cells are killed. In this case, immune system expel the cancer cells thoroughly
out of the body.
Let us now determine the existence of interior equilibrium point,
Suppose equilibrium point E (0, y , z ) exists and is unique hyperbolic point repelling
x  direction. Further assume that neither periodic nor homoclinic orbits exist in the
planes of R3 that is,
T
 B(0)  D(0)  hx (0, y)dt  0
0
and system (4.1.1) is bounded then by the definition of uniform persistence given by
Butler et al. (1986), Freedman and Rai (1995,1987), Nani and Freedman (2000),
lim inf x(t )  0,
t 
lim inf y (t )  0,
t 
77
lim inf z (t )  0.
t 
In particular, the system (4.1.1) exhibit uniform persistence and a positive interior
equilibrium of the form E  ( x , y , z  ) exists.
We now study the linearized stability of this equilibrium point.
4.4.2 Local Stability of Interior Equilibrium point E  ( x  , y  , z  ) :
Using hypothesis H1-H6, the variational matrix of the system due to
linearization of the system (4.1.1) about E  ( x  , y  , z  ) is expressed as
 B( x  )  D( x  )  hx ( x  , y  )

 hy ( x , y  )
0


V (E  )  
  hx ( x  , y  )
f y ( y  , z  )  1d1 ( y  )
f z ( y , z )
 (4.4.5)

 
 
 

 ( x )
 f y ( y , z )
 f z ( y , z )   2 d 2 ( z )

The eigenvalues of the variational matrix V ( E  ) are given by the cubic equation
3  A12  A2   A3  0
(4.4.6)
Where,
A1   B( x  )  D( x  )  hx ( x  , y  )  f z ( y  , z  )  f y ( y  , z  )  hy ( x  , y  )  1d1 ( y  )   2 d 2 ( z  )



A2  f z ( y  , z  ) 1d1 ( y  )  hy ( x  , y  )   2 d 2 ( z  )  f y ( y  , z  )  1d1 ( y  )  hy ( x  , y  )


  B( x  )  D( x  ) f z ( y  , z  )  f y ( y  , z  )  1d1 ( y  )   2 d 2 ( z  )  hy ( x  , y  )

 hx ( x  , y  ) f z ( y  , z  )  f y ( y  , z  )  1d1 ( y  )   2 d 2 ( z  )



 h y ( x  , y  )  f y ( y  , z  ) 




 f ( y , z )   d  ( z ) 
z
2 2



   d ( y )
A3   B( x  )  D( x  )  hx ( x  , y  ) 
 1 1



 f ( y  , z  ) f ( y  , z  )

z
y






78



 hx ( x  , y  )hy ( x  , y  ) f z ( y  , z  )   2 d 2 ( z  )   ( x  ) f z ( y  , z  )hy ( x  , y  )
By the Routh Hurwitz criteria, the eigenvalues of variational matrix V ( E  ) have
negative real parts if
A1  0, A3  0
and A1 A2  A3  0.
(4.4.7)
Thus, if conditions given in (4.4.7) are satisfied then the interior equilibrium point
E  ( x  , y  , z  ) is locally asymptotically stable equilibrium point.
4.5 GLOBAL STABILITY ANALYSIS
In this section, we derive global stability of the equilibrium points by choosing
the Lyapunov function and finding conditions for its derivative with respect to time to
be negative definite. We use following two lemmas to prove global stability of the
system used in by (Nani and Freedman, 2000).
Lemma 4.5.1: Liapunov function V expressed as V  X T AX where,
 x1 
x 
 2
X   .  , X T  x1
 
.
 x n 
x2 . . xn   R n ,
and A be a symmetric n  n matrix over R is negative definite if ,
1. X T AX is negative definite,
2. X T AX is negative if A is negative definite,
3. A is negative definite if the eigenvalues of polynomial g (, A)  A  I n  0
has negative real parts.
Frobenius in 1876 gave an alternative method to prove Lyapunov function V to be
negative definite in the following lemma (Nani and Freedman, 2000):
79
Lemma 4.5.2: (Frobenius 1876) Let
 x1 
x 
 2
X   .  , X T  x1
 
.
 x n 
x2 . . xn   R n ,
and let A be a symmetric n  n matrix over R . Then the real quadratic form X T AX
is negative definite if A is negative definite. In particular, a necessary and sufficient
condition for the real symmetric matrix A to negative definite is that the principal
minors of A starting with that of the first order be alternately negative and positive.
4.5.1 Global Stability of Cancer Free Equilibrium Point:
Let us choose the Liapunov function
1
1
V  x  k1 ( y  y ) 2  k 2 ( z  z ) 2
2
2
(4.5.1)
Derivative of V with respect to time t is given by
dV dx
dy
dz

 k1 ( y  y )  k 2 ( z  z ) .
dt dt
dt
dt
(4.5.2)
Using B( x)  D( x)  xg( x) ,
h( x, y)  xh1 ( x, y),
h1 ( x, y)  yh2 ( x, y)
Using system (4.1.1) in equation (4.5.2), we have
dV
 xg ( x)  yh2 ( x, y )   k1 ( y  y )Q1  f ( y, z )  1d1 ( y )  h( x, y ) 
dt
 k2 ( z  z )Q2   ( x) f ( y, z)   2d 2 ( z),
 xg ( x)  x yh2 ( x, y)  yh2 ( x, y)  xyh2 ( x, y)  k1 ( y  y) f ( y, z)  f ( y, z )
 1k1( y  y)d1( y)  d1( y)  xk1( y  y) yh2 ( x, y)  yh2 ( x, y)
80
 xk1 ( y  y) yh2 ( x, y)   2 k 2 ( z  z )d 2 ( z)  d 2 ( z )  k 2 ( z  z ) f ( y, z)  f ( y, z )
 k2 ( z  z ) ( x),
writing (4.5.3) as
(4.5.3)
dV
 X T AX ,
dt
(4.5.4)
 x 


where X   y  y  , A is a real symmetric matrix defined as A  aij
,
1 i, j  3
 z  z 
 
with

 a11
1
A   a12
2
1 a
 2 13
1
a12
2
a22
1
a23
2
1

a13 
2

1
a23 ,
2

a33 

and V  a11x 2  a12 x( y  y)  a13 ( x  x )( z  z )  a22 ( y  y) 2
 a23 ( y  y )( z  z )  a33 ( z  z ) 2 ,
where,
a11 
g ( x)  yh2 ( x, y )
,
x
 yh ( x, y)  yh2 ( x, y ) 
  k1 yh2 ( x, y),
a12   2
yy


k  ( x)
a13  2
,
x
 k d ( y )  d1 ( y ) 
a22  1 1 1
,
y y
k  f ( y, z )  f ( y , z )  k 2  f ( y, z )  f ( y , z ) 
a23  1

,
zz
yy
81
 k d ( z)  d 2 ( z )
a33   2 2 2
zz
.
Thus, by Frobenius theorem and hermiticity of matrix A , the matrix A and hence the
quadratic form (4.5.3) is negative definite if the following criteria hold,
A1  a11  0, A2 
a11
1
a12
2
1
a12
2
a11
A3  det A 
1
a12
2
1
a13
2
a 22
1
a 23
2
1
a12
2
 0 and
a 22
1
a13
2
1
a 23  0.
2
(4.5.5)
a33
Thus, we have the following theorem for the global stability of cancer free
equilibrium point:
Theorem 4.5.1: The cancer free equilibrium point
E (0, y, z ) is globally
asymptotically stable if conditions (4.5.5) are satisfied.
Remark: Global asymptotic stability of cancer free equilibrium E (0, y, z ) gives the
criteria for total success of therapy in eliminating cancer cells from human body. In
such cases immune system fights well with the cancer cells such that they are not able
to proliferate and spread in the human body.
4.5.2 Global stability of Interior equilibrium point
Let us consider a Lyapunov function
V1  x  x  x ln
x
1
1
 l1 ( y  y  ) 2  l2 ( z  z  ) 2 .
2
x 2
Derivative of V1 with respect to time t is given by
82
(4.5.6)
dV1
1 dx
dy
dz
 ( x  x )
 l1 ( y  y  )  l2 ( z  z  ) ,
dt
x dt
dt
dt
(4.5.7)
Again using B( x)  D( x)  xg( x) ,
h( x, y)  xh1 ( x, y),
h1 ( x, y)  yh2 ( x, y)
Using system (4.1.1) in equation (4.5.7), we have
dV1 ( x  x )
xg( x)  xh1( x, y)  l1( y  y )Q1  f ( y, z)  1d1( y)  h( x, y)

dt
x
 l2 ( z  z  )Q2   ( x) f ( y, z )   2 d 2 ( z ) ,


 ( x  x  )g ( x)  ( x  x  ) yh2 ( x, y)  y h2 ( x, y  )  ( x  x  ) y h2 ( x, y  )




 l1 ( y  y ) f ( y, z )  f ( y  , z  ) l2 ( z  z  ) ( x)   ( x )  2l2 ( z  z  )d 2 ( z )  d 2 ( z  ) 
(4.5.8)
 l2 ( z  z  ) f ( y, z )  f ( y , z  ),
 1l1 ( y  y  ) d1 ( y)  d1 ( y  )  l1 ( y  y  ) h( x, y)  h( x , y  )
writing (4.5.8) as
dV1
 X T BX ,
dt
(4.5.9)
 x  x 


where X   y  y  , B is a real symmetric matrix defined as B  bij
,
1 i, j  3

z  z 


 
with

 b11
1
B   b12
2
1 b
 2 13
and
1
b12
2
b22
1
b23
2
1

b13 
2

1
b23 ,
2

b33 

dV1
 b11 ( x  x ) 2  b12 ( x  x )( y  y  )  b13 ( x  x )( z  z  )  b22 ( y  y  ) 2
dt
83
 b23 ( y  y  )( z  z  )  b33 ( z  z  ) 2 .
Here,
b11 
g ( x)  y h2 ( x, y  )
x  x
,
 yh2 ( x, y )  y h2 ( x, y  ) 
 h ( x, y )  h ( x  , y  ) 


,
b12  
 l1






y

y
x

x






l  ( x)   ( x )
b13  2
,
x  x


 l d ( y )  d1 ( y  )
b22   1 1 1
,
y  y




l f ( y , z )  f ( y  , z  ) l 2 f ( y , z )  f ( y  , z  )
b23  1

,
z  z
y  y
b33  

 2l2 d 2 ( z)  d 2 ( z  )
zz


.
Thus, by Frobenius theorem and hermiticity of matrix B , the matrix B and hence the
quadratic form (4.5.9) is negative definite if the following criteria hold,
B1  b11  0, B2 
b11
B3  det B 
1
b12
2
1
b13
2
b11
1
b12
2
1
b12
2
b22
1
b23
2
1
b12
2
 0 and
b22
1
b13
2
1
b23  0.
2
(4.5.10)
b33
Thus, we have the following theorem for the global stability of interior
equilibrium point:
84
Theorem 4.5.2: The interior equilibrium point
E  ( x , y  , z  )
is globally
asymptotically stable if condition (4.5.10) are satisfied.
Remark: Global stability of equilibrium point E  ( x  , y  , z  ) infers that cancer cells
proliferate and attain a particular equilibrium level in the human body. In this case,
although therapy is not able to eliminate cancer cells from the body yet it is effective
in controlling the cancer cells and reducing them to a lowest possible limit. Thus,
immunotherapy would be most effective in the case it is able to reduce the number of
cancer cells in the body to lowest equilibrium value.
4.6 NUMERICAL SIMULATION
Let us consider the following system to justify the analytical findings
dx
 rx (1  x)  a1 xy,
dt
dy
 Q1  b1 yz  1 y  a2 xy,
dt
(4.6.1)
dz
 Q2  x  b2 yz   2 z ,
dt
with,
x(0)  x0  0, y(0)  y0  0, z (0)  z0  0.
Hence,
B( x)  D( x)  rx (1  x),
h( x, y)  a1xy,
f ( y, z)  b1 yz,
d1 ( y)  y, d 2 ( z)  z,  ( x)  x, a1  a2 , b1  b2
(4.6.2)
We note that all the hypothesis H1-H6 hold for (4.6.2). Choosing the
following values of the parameters in system (4.6.1):
r  5, a1  0.5, b1  0.1, a2  0.2, b2  0.2, Q1  1,
85
Q2  2,   0.8, 1  0.2 and  2  0.1
(4.6.3)
We find that all the equilibria of system (4.1.1) exist and are given by,
E (0, 9.7562, 0.9750) and E* (0.0926, 9.0744, 1.0831) .
The characteristic equation corresponding to E * is given by
3  2.48832  1.1484  0.1870  0.
Roots of this equation are -1.9480, -0.2701+0.1517i, -0.2701-0.1517i. This implies
that E * is a locally asymptotically stable equilibrium point owing to negative real
parts of the eigenvalues of characteristic equation.
Further, to show changes occurring in populations with time under different
conditions, figures have been plotted between dependent variables and time for
different parameter values. In Figs 1 and 2, global stability of the system is displayed
by plotting the graphs in x  y plane and x  z plane respectively. It is observed from
the figures that whatever initial value of equilibrium point is taken, trajectory always
moves towards the equilibrium point. Thus, global stability of the system is ensured.
Fig. 1, Graph of
x versus y for different initial starts for the set of parameters same as (4.6.3)
86
Fig. 2, Graph of
x versus z for different initial starts for the set of parameters same as (4.6.3)
In Fig 3, variation of cancer cell population with time for different
proliferation rate of hunting cells due to external infusion of immune cells is given. It
is evident from the figure that cancer cell population decrease as proliferation rate of
hunting cells, denoted by Q1 , increases and for a particular value of Q1 , cancer
population vanish. It may be because lymphocytes are cytotoxic to cancer cells and
increase in their proliferation causes reduction in number of cancer cells as enhanced
population of lymphocytes kill more cancer cells. In this way, cancer cell population
can be reduced largely and hence can be controlled.
In Fig. 4, variation of cancer cell population with time for different
recruitment rate of resting cells due to external infusion and different rate of induction
of immune response in resting cells due to cancer antigens is determined. It is
observed that cancer cell population decrease with increase in recruitment of resting
cells during immunotherapy in the presence of antigenicity of cancer cells. However,
if antigenicity is zero i.e., there is very little or no induction of immune response in
87
resting cells due to presence of cancer cells then cancer cell population rise to a large
value. This implies that immunogenic cancers are easy to control to a lower level than
non-immunogenic cancers. On the other hand, if rate of antigenicity is higher cancer
cell population can be controlled.
Fig. 3, Graph of
Fig. 4, Graph of
x versus t for different Q1 and other values of parameters are same as (4.6.3)
x versus t for different Q2 and  and other values of parameters are same as (4.6.3)
88
4.7 CONCLUSION
This chapter considers a generalized mathematical model to discuss the effect
of immune response to cancer cells using nonlinear differential equations. The model
is analyzed using stability theory of differential equations and numerical simulation. It
is found that model has two equilibrium points. Conditions for local and global
stability of these equilibrium points are determined. Cancer free equilibrium point
E (0, y, z ) gives the criteria for total success of therapy in eliminating cancer cells
from human body. This case implies that immune system fights so well with the
cancer cells that they are not able to proliferate and spread in the human body and
hence cancer can be cured. Interior equilibrium point E  ( x  , y  , z  ) demonstrates
the case of how the cancer cells proliferate and attain a particular equilibrium level in
the human body. In this case, although therapy is not able to eliminate cancer cells
from the body yet it is effective in controlling the cancer cells and reducing them to a
lowest possible limit.
To substantiate the analytical findings, the model is studied numerically for a
particular case using fourth order Runge-Kutta method. Local stability conditions are
verified for a set of hypothetical parameter values. Global stability of the interior
equilibrium point is displayed graphically. Numerically, it is observed that cancer cell
population is very sensitive to proliferation rate of hunting cells due to external
infusion of immune cell during immunotherapy. A little increase in the numerical
value of proliferation rate of lymphocytes produces a considerable decrease in the
equilibrium level of cancer cell population. It is further observed that cancer cell
population decrease with increase in induction of immune response in resting cells
due to cancer antigens. However, if antigenicity is zero i.e., cancer is non89
immunogenic then cancer cell population rise to a large value. This implies that
immunogenic cancers are easy to control to a lower level than non-immunogenic
cancers. On the other hand, it is observed that if rate of antigenicity is higher cancer
cell population can be controlled considerably to a lower level.
90