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Chapter 6
MATHEMATICAL MODELING AND ANALYSIS OF CANCER
VIROTHERAPY
INTRODUCTION
In this chapter, we study a nonlinear mathematical model for the treatment of
cancer by using oncolytic viruses. The interaction between the growing cancer and the
replicating oncolytic virus is highly complex and nonlinear. Thus, to precisely define
the conditions that are required for successful therapy, mathematical models are
needed. Our model builds upon the model of Novozhilov (2006) with a modified
functional response. Novozhilov presented a mathematical model that describes the
interaction between cancer cells (uninfected cancer cells and infected cancer cells)
with ratio dependent functional response between them. We consider a more realistic
type of functional response involving measure of immune response to suppress
interaction among oncolytic virus and cancer cells. Further, we extend the model to
determine the effect of virus specific immune response to cancer cells by the inclusion
of a separate variable for immune response.
6.1 MATHEMATICAL MODEL
Our model consist of two types of cancer cells x and y growing in logistic
fashion. x is the size of uninfected cancer cell population and y is the size of cancer
cell population infected with oncolytic virus. Separate equation for viruses is not
considered. Cancer cells, infected with oncolytic viruses, further cause infection in
other cancer cells through bursting of cancer cells and release of multiple oncolytic
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virus that are ready to infect other cancer cells susceptible to infection. Based on
these assumptions model takes the following form:
dx
bxy
 x y
 r1 x1 
,

dt
K  x ya

dy
bxy
 x y
 r2 y1 
 y ,

dt
K  x ya

(6.1.1)
With initial conditions: x(0)  x0  0 and y(0)  y0  0.
Here r1 and r2 are the maximum per capita growth rates of uninfected and infected
cancer cells correspondingly; We assume that r1  r2 , that is cells infected with
oncolytic virus grow slowly as compared to uninfected cancer cells that are
susceptible to infection, K is the carrying capacity, b is the transmission rate of
oncolytic virus infection (this parameter also includes the replication rate of the
viruses) ; The expression
by
, displays saturation effect. Parameter a measures
x ya
the extent to which immune response provides protection to cancer cells from
oncolytic virus.  is the rate of infected cell killing by the viruses (cytotoxicity). All
the parameters of the model are supposed to be nonnegative.
6.2 BOUNDEDNESS
We find the region of attraction in the following lemma.
Lemma 6.2: All the solutions of (6.1.1) starting in the positive orthant R 2 either
approaches, enter or remain in the subset of R 2 defined by


  ( x, y)  ( R2 : 0  x  y  K ,
where R 2
denote the non-negative cone of R 2 including its lower dimensional
faces.
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Proof: From system (6.1.1) we get:
dx dy
 x y

 (r1 x  r2 y )1 
  y ,
dt dt
K 

dx dy
 x y

  ( x  y )1 
,
dt dt
K 

where   max( r1, r2 ).
Then by usual comparison theorem, we get the following expression as t  ,
lim sup x(t )  y (t )  K .
t 
Thus, it suffices to consider solutions in the region  . Solutions of the initial value
problem starting in  and defined by (6.1.1) exist and are unique on a maximal
interval (Hale, 1980). Since solutions remain bounded in the positively invariant
region  we infer that the maximal interval is well posed both mathematically and
epidemiologically.
6.3 EQUILIBRIUM ANALYSIS
System
E2 (0,
(6.1.1) possess
the following equilibria
E 0 (0,0),
E1 ( K ,0),
K
(r2   )) and interior equilibrium point E3 ( x  , y  ) .
r2
We will discuss existence of each equilibrium point separately as given
below:
Existence of E 0 (0,0) : The existence of trivial equilibrium point E 0 (0,0) is obvious.
This equilibrium point implies the complete elimination of cancer. Biologically, it
means that both infected and uninfected cancer cells can be eliminated with time, and
complete recovery is possible because of the virus therapy.
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Existence of E1 ( K ,0) : The existence of equilibrium point E1 ( K ,0) is obvious. This
equilibrium implies the failure of virus therapy. Biologically, it means that both
infected and uninfected cancer cells tend to the same state, as they would have been
reached without virus administration.
Existence of E2 (0,
E2 (0,
K
(r2   )) : It can be checked out that the equilibrium point
r2
K
(r2   )) exists if r2   . This equilibrium implies complete infection of
r2
cancer cells and stabilization of cancer load to a finite minimal size
K
(r2   ). Biologically, it gives a situation in which cancer load can be reduced to
r2
lower size if cancer is detected at an initial stage.
Existence of E3 ( x  , y  ) . To see the existence of interior equilibrium point
E3 ( x  , y  ) we note that x  , y  are positive solutions of the system of algebraic
equations given below:
by
 x y
r1 1 
 0,

K  x ya

(6.3.1)
bx
 x y
r2 1 
   0.

K  x ya

(6.3.2)
Now from (6.3.1), we get
y
 2r1 x  r1K  ar1  Kb 
2r1 x  r1K  ar1  Kb2  4r12 ( K  x)( x  a)
2r1
 f ( x)  0
Putting value of y in (6.3.2) we have
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(6.3.3)
bx
 x  f ( x) 
r2 1 
   0.

K

 ( x  f ( x)  a )
(6.3.4)
To show existence of x  it suffices to show that equation (6.3.4) has a unique positive
solution.
Taking,
bx
 x  f (x 
G ( x)  r2 1 
,

K  ( x  f ( x)  a )

(6.3.5)
We note that
G (0)  r2    r2
f (0)
 0,
K
provided
f (0) 

r2 1 
 
K 

(6.3.6)
and
G( K ) 
bK
   0,
K a
provided

bK
.
K a
(6.3.7)
Thus, there exists a x  in the interval 0  x  K such that
G( x  )  0.
(6.3.8)
For x  to be unique, we must have
dG
 0 for x  x̂.
dx
(6.3.9)
It is easy to find that (6.3.9) holds if
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b
bx(1  f ( x))
 1  f ( x) 
 r2 
at x  x  .

2
x  f ( x)  a
K

 ( x  f ( x)  a)
Corresponding value of y  is given by y   f ( x ).
6.4 LOCAL STABILITY ANALYSIS
To discuss the local stability of equilibrium points we compute the variational
matrix of system (6.1.1). The signs of the real parts of the eigenvalues of the
variational matrix evaluated at a given equilibria determine its stability. The entries of
general variational matrix are given by differentiating the right hand side of system
(6.1.1) with respect to x, y . The matrix is given by
V (E ) 
 
x  y  r1 x
by
bxy
r1 1  K   K  ( x  y  a) 

( x  y  a) 2
 

r y
by
bxy
 2 


K
( x  y  a) ( x  y  a) 2

rx

bxy
bx
 1 


K ( x  y  a) ( x  y  a) 2


x  y  r2 y
bxy
bx

r2 1 




K  K
( x  y  a) ( x  y  a) 2


We denote the variational matrix corresponding to E i by V ( Ei ), i  0,1,2,3
To explore local stability of trivial equilibrium point, we compute variational
matrix of E 0 . The variational matrix of equilibrium point E 0 is given by
0 
r
V ( E0 )   1
.
 0 r2   
(6.4.1)
Eigenvalues of V ( E 0 ) are given by   r1 ,   r2   . E 0 is an unstable equilibrium
point since both the eigenvalues of the matrix are positive.
Now, to study the stability behavior of E1 , we compute the variational matrix
V ( E1 ) corresponding to E1 as follows:
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bK 

 r1  r1  K  a 
V ( E1 )  
.
bK
 0
 
K a


(6.4.2)
From (6.4.2) we observe that eigenvalues of the matrix V ( E1 ) are given by    r1
and  
bK
bK
  . Existence of interior equilibrium point infers that  
. Thus,
K a
K a
V ( E1 ) has a positive eigenvalue, consequently E1 is a saddle point in y-direction.
Remark: As measure of immune response of the individual to the virus is very less as
compared

to
carrying
capacity
of
tumor
cells
i.e.
a  K ,
inequality
bK
   b. It implies that death rate of infected cells due to the virus attack is
K a
less than the rate of transmission of infection from virus to the uninfected cells.
However, if   b then all the infected cells would die without having time to infect
other cells and cancer cells grow unaffectedly.
 K

The variational matrix of equilibrium point E2  0, (r2   )  is given by
 r2

bK (r2   )
 r1
 r  K (r   )  ar
2
2
M2   2
 r    bK (r2   )
 2
K (r2   )  ar2


.
 (r2   )

0
(6.4.3)
From (6.4.3) we observe that eigenvalues of the matrix V ( E2 ) are given by

r1
bK (r2   )
  (r2   ). Thus V ( E 2 ) has negative eigenvalues

r2 K (r2   )  ar2 and
and E 2 is stable equilibrium point if
K (r2   )
r1
bK (r2   )
or a 
(br2  r1 )  a  .

r2
K (r2   )  ar2
r1r2
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consequently, E 2 is a saddle point if a 
K (r2   )
(br2  r1 )  a  .
r1r2
Where, br2  r1 because r2  r1 (by assumption) and b   (existence of E  . )
Remark: Stability condition a  a  of equilibrium E 2 implies that all the cancer cells
may become infected if measure of immune response to cancer cells infected with
oncolytic virus is less than a particular value a  .
Variational matrix of E3 ( x  , y  ) is given by

P

V ( E3 )  
by 
Q  

( x  y   a)
where, P   r1
P 
bx


( x  y   a) 



Q

(6.4.4)
x
bx y 
y
bx  y 

, Q   r2

.
K ( x  y   a) 2
K ( x  y   a) 2
From variational matrix V ( E 3 ), we find that eigenvalues are   where
2
 bx y  (r  r )
r x   r2 y  1  r1 x   r2 y  
ab 2 x  y  
1
2 
   1

 4

 ( x   y   a) K ( x   y   a) 3 
2K
2 
K




The signs of the real parts of   and   are negative. This implies that E3 is always
locally asymptotically stable if it exists.
6.5 GLOBAL STABILITY ANALYSIS
We prove global stability of (6.1.1) using Bendixon-Dulac criteria.
If we choose D( x, y)  1 xy as Dulac function and denoting
bxy
 x y
r1 x1 
 P ( x, y )

K  x ya

121
bxy
 x y
r2 y1 
 y  Q ( x, y )

K  x ya

then
( PD) ( PD)
1
r1x  r2 y   0,


x
x
Kxy
(x,y) Ω .
This implies that interior equilibrium point E3 ( x  , y  ) is globally asymptotically
stable.
6.6 MODELING THE EFFECT OF ONCOLYTIC VIRUS ON CANCER
CELLS IN THE PRESENCE OF VIRUS-SPECIFIC IMMUNE CELLS
In this section, we consider the role of virus specific immune response in the
cancer cells and oncolytic virus interaction. The model has three variables: two
variables x and y for cancer cells uninfected and infected with oncolytic virus
respectively and the third variable is for virus specific immune cells z . The Cytotoxic
T Lymphocyte
response (CTL) or immune response is modeled according to
predator-prey dynamics. In addition to the assumptions made in modeling the system
(6.1.1), we make the following assumptions:

Upon exposure to virus as a foreign particle in the body, the CTL proliferates and
kill oncolytic viruses replicating inside the cancer cells.

There are many different functional responses that can be used to model immune
response dynamics; however, the exact functional response that would be
applicable is currently not known. So, we use here a relatively simple bilinear
functional response. It is directly proportional to the population of oncolytic virus
inside the body or the density of cancer cells infected with oncolytic virus, y , and
the population of CTL, z .
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
We analyse a density dependent growth of immune response that declines with an
increase in the ratio of CTL to virus load and stops if that crosses a threshold. In
biological terms, such dynamics may be due to dilution of cytokines or spatial
interference if the CTL population becomes much larger than the number of

infected cells. It is expressed as cz1 

z 
, a similar growth rate has been taken
y  w 
in (Hassel, 1981), for describing predator interference in ecology. In the absence
of infection, the CTL may persist at background level, corresponding to the
population of recirculating CTLs. The parameter  determines the level of CTLs
in the absence of infection and may be connected to immunological memory.
Based on these assumption model takes the following form:
dx
x y
bxy
 r1 x(1 
)
,
dt
K
x ya
dy
x y
bxy
 r2 y (1 
)
 y  yz,
dt
K
x ya

dz
z 
  dz.
 cz1 
dt
 y  w
(6.6.1)
With initial conditions: x(0)  x0  0, y(0)  y 0  0 and z (0)  z 0  0.
 is the cytotoxicity rate of CTLs on interaction with cancer cells containing virus.
c is the intrinsic growth rate,  is the intraspecific competition rate and d is the cell
lysis rate of immune cells or CTLs. All the parameters of the model are supposed to
be nonnegative.
6.7 BOUNDEDNESS OF SOLUTIONS
Lemma 6.7.1.: All the solutions of the system (6.6.1) starting in the positive orthant
R3 either approaches, enter or remain in the subset of R3 defined by
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(c  d )( K  w)


1  ( x, y, z )  R3 : 0  x  y  K ,0  z 
,c  d 
c


where R3 denote the non-negative cone of R 3 including its lower dimensional faces.
Proof: From system (6.6.1) we get:
dx(t ) dy (t )
 x y

 (r1 x  r2 y )1 
  y ,
dt
dt
K 

dx(t ) dy (t )
 x y

  ( x  y )1 
,
dt
dt
K 

where   max( r1 , r2 ) , then by usual comparison theorem, we get the following
expression as t  , lim sup x(t )  y (t )  K .
t 
Similarly, from third equation of system (6.6.1) we have,
lim sup z (t ) 
t 
(c  d )( K  w)
, c  d.
c
Thus, it suffices to consider solutions in the region 1
6.8 EQUILIBRIUM ANALYSIS
System (6.6.1) possess the following equilibria:
 K

(c  d ) w 
(c  d ) w 


E0 (0,0,0), E1  0,0,
, E2 ( K ,0,0), E3  K ,0,
, E4  0, (r2   ),0 ,
c 
c 


 r2

E5 0, y , z , E6 ( xˆ, yˆ ,0) and E7 ( x , y , z  ).
(c  d ) 

 r2   



c
 , z  (c  d ) K (r2   )  r2 w 
Where y  K 


 (c  d ) K 

 r2 c   (c  d ) K 
 r2 

c


and
xˆ, yˆ
are
positive solutions of the first two equations of system of algebraic equations (6.6.1)
124
and E7 ( x , y , z  ) is the interior equilibrium. Existence conditions of these
equilibria are discussed as under:
Existence of E0 (0,0,0) : Existence of trivial equilibrium point E 0 (0,0,0) is obvious.
This equilibrium point implies the complete elimination of cancer and absence of
active immune cells in the case of no virus specific antigenicity. Biologically, this
equilibrium point means that both infected and uninfected cancer cells can be
eliminated with time, and complete recovery is possible because of the virus therapy.
(c  d ) w 
(c  d ) w 


Existence of E1  0,0,
 : Existence of equilibrium point E1  0,0,
 is
c 
c 


also obvious. It corresponds to the elimination of cancer cells due to active
proliferation of CTLs under the condition c  d . Thus, existence of this equilibrium
point implies that active proliferation of immune cells can be helpful in eliminating
cancer.
Existence of E2 ( K ,0,0) : This equilibrium point implies failure of virus therapy. It
means that that uninfected and infected cancer cells tend to the same state, as they
would have been reached without virus administration and virus specific antigenicity.
(c  d ) w 

Existence of E3  K ,0,
 : It corresponds to failure of virus therapy and
c 

immune response in killing cancer cells. It may be due to excessive killing of virus by
immune cells resulting in the death of infective cells without having time to infect
other cells.
125
 K

Existence of E4  0, (r2   ),0  : It can be checked out that the equilibrium point
 r2

 K

E4  0, (r2   ),0  exists if r2   . This equilibrium implies complete infection of
 r2

cancer cells and stabilization of cancer load to a finite minimal size
K
(r2   ) in the
r2
absence of virus specific antigenicity. Here cancer load, x  y is given by
K
(r2   )
r2
from which we observe that as cytotoxicity rate of virus increases, cancer load
decreases but it should not exceed growth rate of cancer cells infected with oncolytic
virus.
Existence of E5 0, y , z  ,
w(c  d ) 

 r2   

c
 and z  (c  d ) K (r2   )  r2 w  : It corresponds to virus
y  K
 r c   (c  d ) K 
 (c  d ) K 

 2

 r2 

c


replication in the presence of virus specific immune response. In this case cancer load
w(c  d ) 

 r2   

c


. From the expression of cancer load we
is given by x  y  y  K
 (c  d ) K 

 r2 

c


infer that cancer load decreases with the increase in  . Biologically, it can be
interpreted that cancer load decrease with the increase in the rate at which immune
cells destroy cancer cells infected with oncolytic virus.
Existence of E6  xˆ , yˆ ,0  : Proof for the existence of interior equilibrium point
E6  xˆ , yˆ ,0  is equivalent to the proof for existence of E3 ( x  , y  ) in section 6.3.
126
Existence of E7 ( x , y , z  ) : Similarly, to see the existence of interior equilibrium
point E7 ( x , y , z  ) we note that x  , y  and z  are positive solutions of the system
of algebraic equations given below:
by
 x y
r1 1 
0

K  x ya

(6.8.1)
bx
 x y
r2 1 
   z  0

K  x ya

(6.8.2)

z 
d 0
c1 
y  w 

(6.8.3)
Now from (6.8.1), we get
y
 2r1 x  r1K  ar1  Kb 
2r1x  r1K  ar1  Kb2  4r12 ( K  x)( x  a)
2r1
 f ( x)  wc  d  .
Equation (6.8.3) gives z 
c
 f ( x)  0.
Putting value of y and z in (6.8.2) we have
bx

 x  f (x 
r2 1 
    f ( x)  wc  d   0.

K  ( x  f ( x)  a)
c

(6.8.4)
To show existence of x  , it suffices to show that equation (6.8.4) has a unique
positive solution.
Taking
H ( x)  r2 (1 
x  f (x
bx

)
    f ( x)  wc  d ,
K
( x  f ( x)  a )
c
we note that
f (0) 


H (0)  r2 1 
     f (0)  wc  d   0 (Using (6.3.6)),
K 
c

and H ( K ) 
bK
w(c  d )
 
0
K a
c
127
(6.8.5)
provided,
bK
w(c  d )

 .
K a
c
(6.8.6)
Thus, there exists a x  in the interval 0  x   K such that H ( x  )  0.
For x  to be unique, we must have
dH
 0 for x  x .
dx
Corresponding value of y  and z  is given by


f ( x  )  w c  d 
y   f ( x  ) and z  
.
c
6.9 LOCAL STABILITY ANALYSIS
We compute the variational matrix of system (6.6.1) to discuss the local
stability of equilibrium points. The signs of the real parts of the eigenvalues of the
variational matrix evaluated at a given equilibria determine its stability. The entries of
general variational matrix are given by differentiating the right hand side of system
(6.6.1) with respect to x, y and z . We denote the variational matrix corresponding to
E i by V ( Ei ), i  01,2,3,4,5,6,7.
To explore local stability of trivial equilibrium point, we compute variational
matrix of E 0 . The variational matrix of equilibrium point E 0 is given by
0
r1

V ( E0 )   0 r2  
 0
0
0 
0 .
c  d 
(6.9.1)
Here, the three eigenvalues of V ( E0 ) are given by 1  r1 ,  2  r2   and 3  c  d .
It can be easily observed that all the three eigenvalues are positive if equilibrium
points E 4 and E1 (or E3 ) exist respectively. Thus E 0 is an unstable equilibrium
point.
128
Now, to study the stability behavior of E1 we compute the variational matrix
V ( E1 ) corresponding to E1 as follows:
0
r1

V ( E1 )   0 r2  
 0
0
0 
0 .
 (c  d )
(6.9.2)
From (6.9.2) we observe that eigenvalues of the matrix V ( E1 ) are given by
1  r1 ,  2  r2   and 3  (c  d ). Here  3 is negative but  2 is positive by virtue
of existence condition of E 4 . 1 is also a positive eigenvalue. Therefore E1 is a saddle
point if equilibrium point E 4 exists.
The variational matrix of equilibrium point E2 ( K ,0,0) is given by
bK

 r1  r1  K  a

bK
V ( E2 )   0

K a

0
 0


0 

0 

c  d

(6.9.3)
From (6.9.3) we observe that eigenvalues of the matrix V ( E2 ) are given by
1   r1 ,  2 
bK
Ka
  and 3  c  d . We observe that  2 and  3 are positive if
equilibrium points E 6 and E1 (or E3 ) exist respectively. Thus E 2 is a saddle point.
(c  d ) w 

Variational matrix of E3  K ,0,
 is given by
c 


 r1

V ( E3 )   0


 0

bK
K a
bK
w(c  d )
 
K a
c
2
(c  d )
c
 r1 



0 .


 (c  d ) 

0
129
(6.9.4)
From variational matrix
2 
V ( E 3 ),
we find that eigenvalues are
1   r1 ,
bK
w(c  d )
and 3  (c  d ).
 
K a
c
We observe that 2 
bK
w(c  d )
 
 0 if equilibrium point E 7 exists. Thus
K a
c
E3 is a saddle point and it is unstable equilibrium point.
 K

Variational matrix of E4  0, (r2   )  is given by
 r2

bK (r2   )
 r1
 r  K (r   )  ar
2
2
2

bK
(
r


)
2
V ( E 4 )   r2   

K (r2   )  ar2

0



0 

 (r2   )
0 .

0
c  d


0
(6.9.5)
We see that E 4 is a saddle point if E1 or E3 exists, as its one of the three eigenvalues
  c  d is positive. Thus E 4 is an unstable equilibrium point.
Now, to study the stability behavior of E5 0, y , z  we compute the variational
matrix V ( E5 ) corresponding to E5 as follows:

r1 1  y   by
  K  ya

y
by
V ( E5 )   r2 
K ya


0


0
 r2
y
K
cz 2
( y  w) 2

0 


 y .

cz 


y  w 
(6.9.6)
From (6.9.6) we observe that one of the eigenvalues of the matrix V ( E5 ) given by


1  r1 1 
y
by
is positive as it is the growth rate of x at E5 , implying that

K ya
E5 is unstable.
130
The variational matrix of equilibrium point E6 ( xˆ , yˆ ,0) is given by

Pˆ


byˆ
V ( E6 )  Qˆ 

xˆ  yˆ  a

0


Pˆ 
bxˆ
xˆ  yˆ  a

0 

 yˆ .

c  d


Qˆ
0
(6.9.7)
xˆ
bxˆyˆ
yˆ
bxˆyˆ
Where Pˆ  r1 
and Qˆ  r2 
.
K ( xˆ  yˆ  a) 2
K ( xˆ  yˆ  a) 2
From V ( E6 ) we observe that one of the eigenvalues given by 3  c  d is positive.
Thus E 6 is locally asymptotically unstable equilibrium point.
Variational matrix of E7 ( x , y , z  ) is given by

P



by 
V ( E7 )  Q  

x  y   a


0


P 
bx 
x  y   a
Q

R z 
y  w

0 


 y  .


 
R


(6.9.8)
Where,
x
bx  y 
y
bx  y 
cz 


P  r1

, Q  r2

.
and R   
K ( x   y   a) 2
K ( x   y   a) 2
y w

The characteristic equation for the variational matrix given in (6.9.8) is
3  M12  M 2   M 3  0,
(6.9.9)
Where,
M 1  ( P   Q   R  ),
r x   r2 y 
cz 
 1

 0.
K
y  
131

 


 

bx 
by 
 R y z
 Q 

M 2  P Q   Q  R   R  P    
  P   
,






(
y


)
x

y

a
x

y

a







cz  (r1 x   r2 y  )
( y   )K
2

cz  y 
( y   ) 2

ab 2 x  y 
( x   y   a) 2

(r1  r2 )bx y 
K ( x   y   a)
 0, for r1  r2 .

 

bx 
by 
P  R  y  z 
 Q 
  P Q  R  
M 3  R   P   
,






x

y

a
x

y

a
(
y


)



2


(r1  r2 )bx y 
r1 x y  z 
ab 2 x  y 
bx y  z 

.









3



2 

( y   ) K ( x  y  a) ( x  y  a)
K ( y   ) ( x  y  a) ( y  w) 


cz 
By Routh – Hurwitz criteria, if
M 1 , M 3  0 and M 1M 2  M 3 ,
(6.9.10)
then all roots of equation (6.9.9) have negative real parts and E7 ( x , y  , z  ) is
locally asymptotically stable equilibrium point. However, in our case M 1  0 for all
parameter values thus condition (6.9.10) reduces to the condition,
0  M 3  M 1M 2 .
(6.9.11)
6.10 NUMERICAL SIMULATION
The model is studied numerically to substantiate the above analytical findings.
The system of differential equations is integrated using fourth order Runge-Kutta
method. First we consider system (6.1.1) with the help of following hypothetical
parameters:
r1  5, K  10, b  0.03, a  0.05, r2  0.5,   0.003.
The equilibrium points for this set of parameters are:
E 0 (0,0), E1 (10,0), E2 (0,9.94) and E3 (0.05, 9.93).
132
(6.10.1)
It is found that all the conditions for local and global stability of interior
equilibrium point are satisfied for above parameter values. Figures have been plotted
between dependent variables and time for different parameter values to show changes
occurring in population level of cells with time considering initial values (1,1). Fig.1
shows the dynamics of cancer cells uninfected and infected with oncolytic virus. It is
observed from the figure that infected cancer cell population first rise and then attain a
constant equilibrium value whereas uninfected cancer cell population first rise
abruptly and then decrease due to virus infection and cytotoxicity to attain its
equilibrium value.
Fig.1, Variation of x(t ), y(t ) with time for parameter values given in (6.10.1)
In Fig. 2, variation of cancer cells with time for different replication rate of
oncolytic virus, b , is determined. It is observed that as b increases, infected cancer
cells increase and uninfected cancer cells decrease. This behavior of cancer cells
supports the fact that replication of oncolytic virus within cancer cells and their
transmission to other cancer cells is responsible for the infection among more cancer
133
cells and hence increase in the number of infected cancer cells. In this process number
of uninfected cancer cells decrease automatically.
10
9
8
x(t) for b = 0.03
y(t) for b = 0.03
x(t) for b = 0.06
y(t) for b = 0.06
7
x(t),y(t)
6
5
4
3
2
1
0
0
100
200
300
400
500
Time (t)
Fig.2, Variation of x(t ), y(t ) with time for different value of b and other values of
parameters are same as (6.10.1)
Fig.3, Variation of x(t ), y(t ) with time for different value of  and other values of
parameters are same as (6.10.1)
Fig. 3 displays variation of cancer cells with time for different cytotoxicity
rate,  , of oncolytic virus. We observe that as  increase infected cancer cells
134
decrease on the other hand uninfected cancer cells increase. Equilibrium level of
infected cancer cells decrease because oncolytic virus kill cancer cells by infecting
them. But it is not able to annihilate the proliferation of new cancer cells. Therefore,
number of uninfected cancer cells keep on proliferating in the human body.
In addition to the values of parameters given in (6.10.1), we choose following
more parameters for the model with virus specific immune response.
  0.91, d  0.1, c  1,   1.2,   0.01.
(6.10.2)
The interior equilibrium point for this system is
x   9.9616, y  0.0382, z   0.0294. It is found that all the conditions for local
asymptotic stability of interior equilibrium point are satisfied for above parameter
values given in (6.10.1) and (6.10.2). The eigenvalues for the system are as follows,
- 1.4536, - 0.9424, - 0.0001. Negative eigenvalues predict the local asymptotic
stability of the system. It is observed that presence of virus specific immune cells
results into considerable decrease in the number of infected cancer cells. It
demonstrates the fact that patient’s own immune system destroys the virus and
removes them from the body and prohibits virus to annihilate cancer cells
proliferation. It implies that modified oncolytic virus that are less susceptible to
immune suppression decrease cancer cells in the human body.
In Figs. 4 and 5, variation of infected cells with time for different replication
or infection transmission rate and different cytotoxicity rate of oncolytic virus is
determined. It is observed that infected cells increase with the increase replication rate
of oncolytic virus and decrease with the increase in cytotoxicity rate of oncolytic
virus. If we compare this result of two models (6.1.1) and (6.6.1), it is observed that in
the presence of virus specific immune cells quantitative values of cancer cells infected
135
with oncolytic virus differ considerably but qualitative behavior remains similar. It is
inferred that presence of virus specific immune cells causes significant suppression in
the density of infected cancer cells.
Fig.4, Variation of y (t ) with time for different value of b and other values of
parameters are same as (6.10.1) and (6.10.2)
Fig.5, Variation of y (t ) with time for different value of  and other values of
parameters are same as (6.10.1) and (6.10.2)
Our main objective is to explore the conditions for decreasing cancer load in
the body. For this, we increase value of replication rate of oncolytic virus from
136
b  0.03 to b  0.8 in Fig. 6. We found that cancer load starts decreasing for
b  0.8 and then on further increasing b , cancer load keeps on decreasing until
b  6 . Further increase in value of b does not make any significant change in the
equilibrium level of cancer load. Thus, biologically, we interpret that for high
replication rate of virus, cancer load decreases. In fact, there is a range 0.8  b  6
between which replication of oncolytic virus produces significant effect on cancer
cells and decreases cancer load in the patient’s body.
Fig.6, Variation of cancer load x(t )  y(t ) with time for different value of b and other values of
parameters are same as (6.10.1) and (6.10.2)
Further, for a higher value of replication rate of oncolytic virus, that is for b  4 , we
increased cytotoxicity rate of oncolytic virus (Fig. 7) and observed that cancer load
increase with the increase in cytotoxicity rate  . In this case, all the cancer cells are
killed by virus so fast that virus do not get time to replicate and spread in other cancer
cells and virus therapy fails.
137
Fig.7, Variation of cancer load x(t )  y(t ) with time for b  4 and different value of  and other
values of parameters are same as (6.10.1) and (6.10.2)
For b  6 , we increased value of  (as shown in Fig. 8) and observed that
cancer load decreased. Cancer load decreased on increasing  from   0.003 to
  0.3 and then from   1, it again starts increasing. Thus, we observe that cancer
load decrease for high replication rate of oncolytic virus and cytotoxicity rate of
cancer cells lying in within the range 0.003    1 .
Fig.8, Variation of cancer load x(t )  y(t ) with time for b  6 and different value of  and other
values of parameters are same as (6.10.1) and (6.10.2)
138
6.11 CONCLUSION
In this chapter, we have studied two nonlinear models to examine the
interaction of oncolytic virus with cancer cells in the absence and presence of virus
specific immune cells. The positive equilibrium points of the system are investigated.
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, which supports the theoretical findings.
The first model has two differential equations dealing with the interaction of cancer
cells with oncolytic virus without considering the presence of immune response in the
body. It is observed that replication rate of oncolytic virus within cancer cells and
cytotoxicity rate of cancer cells by oncolytic virus play important role in dynamics of
the system.
The second model has three differential equations and takes into consideration
the effect of immune response on oncolytic virus. It is observed from numerical
analysis that in the presence of virus specific immune cells, quantitative values of
cancer cells infected with oncolytic virus decreases considerably but its qualitative
behavior remains same as in model without immune response. We have found that
cancer load decreases for high replication rate of oncolytic virus. However, for high
replication rate, cancer load decreases for a large value of cytotoxicity rate of cancer
cells by oncolytic virus in a given range and beyond this range cancer load again
starts rising.
139