Applied Mathematics, 2011, 2, 131-140
doi:10.4236/am.2011.21015 Published Online January 2011 (http://www.SciRP.org/journal/am)
Mathematical Modeling and Analysis of Tumor Therapy
with Oncolytic Virus
Manju Agarwal, Archana S. Bhadauria
Department of Mathematics and Astronomy, Lucknow University, Lucknow, India
Received October 1, 2010; revised December 2, 2010; accepted November 5, 2010
Abstract
In this paper, we have proposed and analyzed a nonlinear mathematical model for the study of interaction
between tumor cells and oncolytic viruses. The model is analyzed using stability theory of differential equa-
tions. Positive equilibrium points of the system are investigated and their stability analysis is carried out.
Moreover, the numerical simulation of the proposed model is also performed by using fourth order Runge-
Kutta method which supports the theoretical findings. It is found that both infected and uninfected tumor
cells and hence tumor load can be eliminated with time, and complete recovery is possible because of virus
therapy, if certain conditions are satisfied. It is further found that the system appears to exhibit periodic limit
cycles and chaotic attractors for some ranges of the system parameters.
Keywords: Tumor, Oncolytic Virus, Limit Cycles, Chaotic Behavior, Attractors, Asymptotic Stability
1. Introduction
A connection between cancer and viruses has long been
theorized and case reports of cancer regression (cervical
cancer, Burkitt lymphoma, Hodgkin lymphoma) after
immunization or infection with an unrelated virus ap-
peared at the beginn ing of the 20th cen tury [1 ]. Efforts to
treat cancer through immunization or deliberate infection
with a virus began in the mid 20th century [1,2]. As the
technology for creating a custom virus did not exist, all
early efforts focused on finding natural oncolytic viruses.
During the 1960s, promising research involved in using
poliovirus [3], adenovirus [1], Coxsackie virus [4], and
others [2]. The early complications were occasional cas-
es of uncontrolled infection, resulting in significant mor-
bidity and mortality; the very frequent development of an
immune response, while harmless to the patient [1], de-
stroyed the virus and thus prevented it from destroying
the cancer [3]. In a number of cases, cancer cells exposed
to viruses have experienced widespread necrosis, which
cannot be entirely accounted for by viral replication
alone. Cytotoxic T-cell responses directed against virus-
infected cells have been identified as an important factor
in tumor necrosis. However, since viruses are normal
human pathogens, they induce an immune response,
which reduces the effectiveness of viruses. For example,
increased antibody could deactivate viruses before the
tumor has been destroyed. This can be overcome by us-
ing parental viruses that are not normal human pathogens,
thereby avoiding any pre-existing immunity. However,
this does not avoid subsequent antibody generation. Al-
ternatively, the viral vector can be coated with a polymer
such as po lyethylene glycol, sh ielding it from antib odies,
but this also preven ts viral coat proteins adhering to host
cells. Deactivation of the immune syste m is not desirable,
since it has a positive effect on tumor necrosis. Even
when a response was seen, these responses were neither
complete nor durable [1].
With the later development of advanced genetic engi-
neering techniques, researchers gained the ability to de-
liberately modify existing viruses, or to create new ones.
All modern research on oncolytic viruses involves vi-
ruses that have been modified to be less susceptible to
immune suppression, to more specifically target particu-
lar classes of cancer cells, or to express desired cancer-
suppressing genes. The first Oncolytic virus to be ap-
proved by a regulatory agency was a genetically modi-
fied adenovirus named H101 by Shanghai Sunway Bio-
tech. It gained re gulatory approval in 2005 fro m China’s
State Food and Drug Administration (SFDA) for the
treatment of head and neck cancer [5,6 ]. Sunway’s H101
and the very similar Onyx-15 have been engineered to
remove a viral defense mechanism that interacts with a

M. AGARWAL ET AL. 132
normal human gene p53, which is very frequently dere-
gulated in cancer cells [6]. Onyx-015 is a adenovirus th at
was developed in 1987 with the function of the E1B gene
knocked out, meaning cells infected with Onyx-015 are
incapable of blocking p53’s function. If Onyx-015 in-
fects a normal cell, with a functioning p53 gene, it will
be prevented from multiplying by the action of the p53
transcription factor. However if Onyx-015 infects a p53
deficient cell it should be able to survive and replicate,
resulting in selective destruction of cancer cells [7,8].
The interaction between the growing tumor and the
replicating oncolytic virus are highly complex and non-
linear. Thus to precisely define the conditions that are
required for successful therapy by this approach, mathe-
matical models are needed. Several mathematical models
that describe the evolution of tumors under viral injection
were recently developed [9-11]. Other mathematical mo-
dels for tumor-virus dynamics are, mainly, spatially ex-
plicit models, described by systems of partial differen-
tial equations (which is an obvious and necessary ex-
tension of ordinary differential equations models in as
much as most solid tumors have distinct spatial structure),
the local dynamics is usually modeled by systems of or-
dinary differential equations that bear close resemb-
lance to a basic model of virus dynamics [12]. Wu et al.
modeled and compared the evolution of a tumor under
different initial cond itions [13]. Friedman and Tao (2003)
presented a rigorous mathematical analysis of somewhat
different model [14]. Our model builds upon the model
of Artem S. Novozhilov [9] with a modified functional
response between the cells. Novozhilov presented a ma-
thematical model that describes the interaction between
two types of tumor cells (the cells that are not infected
but are susceptible so far as they have the cancer pheno-
type) with ratio dependent functional response between
them. We consider a more realistic type of functional re-
sponse with saturation effect of virus–cell interaction as
even when a virus is oncolytic and it punches a hole in a
tumor, the immune respon se of the ind ividu al to the viru s
occurs so fast that the effects are quickly wiped out and
the tumor continues to grow. We discuss the linear sta-
bility analysis of the biologically feasible equilibrium
states of this model. The ranges of the system parameters
for which the system has chaotic behavior is found.
Some authors discussed the problem of chaos and stabil-
ity analysis of some biological models such as cancer
and tumor model, genital herpes epid emic, stochastic lat-
tice gas prey–predator model and many other models, see,
for example, [15-19]. The problem of optimal control of
the unstable equilibrium states of cancer self-remission
and tumor system using a feedback control approach is
studied by Sarkar and Banerjee and El-Gohary [15,19].
The objective of present work is to study the interaction
between growing tumor and the replicating Oncolytic
virus with a functional response with saturation effect.
The saturation effect accounts for the fact that the num ber
of contacts an individual cell reaches some maxi- mal
value as our immune system evolves to stop virus just as
the virus evolves to enter cells and replicate. Our model
exhibits that complete elimination of tumor is possible
with the help of oncolytic virus therapy in the treatment of
tumor.
This paper is organized as follows: In Section 2, we
outline our model. In Section 3, Boundedness of solu-
tions of the system is studied. Section 4, gives a review
of equilibrium points of the system. Sections 5, deals
with stability analysis of equilib rium points. In Section 6,
we have determined conditions for global stability of in-
terior equilibrium point. It is further followed by nume-
rical simulation in Section 7. Lastly, a short discussion is
represented in Secti on 8.
2. Mathematical Model
The model considers two types of tumor cells
x
and
y growing in logistic fashion.
x
is the size of the un-
infected tumor cell population andyis the size of in-
fected tumor cell population. It is assumed that onco lytic
viruses enter tumor cells and replicate. These tu mor cells,
infected with oncolytic viruses, fu rther cause infection in
other tumor cells. Oncolytic virus preferentially infects
and lysis cancer cells both by direct destruction of the
tumor cells, and, if modified, as vectors enabling genes
expressing anticancer proteins to be delivered specifically
to the tumor site. Based on these assumption model tak es
the following form:
1
2
1,
1,
dxx ybxy
rx
dtKxy a
dyx ybxy
ry y
dtKxy a



 




 


(2.1)
With initial conditions:

0
00xx and


0y

00y.
Here 1 r and 2 r are the maximum per capita growth
rates of uninfected and infected cells correspondingly;
K
is the carrying capacity , b is the transmission rate
(this parameter also includes the replication rate of the
viruses) ; The expression by
x
ya, displays a satura-
tion effect accounting for the fact that the number of
contacts an individual cell reaches some maximal value
as our immune system evolves to stop virus just as the
viruses evolve to enter cells and replicate. a is the
measure of the immune response of the individual to the
viruses which prevents it from destroying the cancer and

is the rate of infected cell killing by the viruses (cy-

M. AGARWAL ET AL. 133
totoxicity). All the parameters of the model are supposed
to be nonnegative.
3. Boundedness of Solutions
Boundedness may be interpreted as a natural restriction
to grow because of limited resources. To establish the
biological validity of the model system, we have to show
that the solutions of system (2.1) are bounded for this we
find the region of attraction in the following lemma.
Lemma 1.: All the solutions of (2.1) starting in the
positive orthant 
2
0
R either approaches, enter or re-
main in the subset of 
2
0
R defined by



2
0
,:0
x
yR xyK

  
where 
2
0
R denote the non-negative cone of 2 R in-
cluding its lower dimensional faces.
Proof: From system (1) we get:

12
1xy
x
tyt rxryy
K



 



 
1
x
y
xtytx yK







where

12
max ,rr  
then by usual comparison theorem, we get the following
expression as ,t
 
limsup
t
x
tytK


Thus, it suffices to consider solution s in the region

.
Solutions of the initial valu e problem starting in

and
defined by (2.1) exist and are unique on a maximal in-
terval [20]. Since solutions remain bounded in the posi-
tively invariant region , the maximal interval is well
posed both mathematically and epidemiologically.
4. Existence and Uniqueness of Equilibrium
Points
An equilibrium point is a point at which variables of a
system remain unchanged over time. System (2.1) pos-
sesses the following equ ilibria:

00,0E,

1,0EK ,

20,Ey
and

3,Exy

,
where

2
2
K
yr
r


and

22
2
4
2
MM rN
yfx
r
 

where
 
22 2
2MxrarKr

 and
  

2
222 2
.Nrxxr aKrKbKar

  
Existence of


00,0E. The existence of trivial equi-
librium point


00,0Eis obv iou s. This eq uilibriu m point
implies the complete elimination of tumor. Biologically,
it means that both infected and uninfected tumor cells
can be eliminated with time, and complete recovery is
possible because of the virus therapy.
Existence of


1,0EK . The existence of equilibrium
point


1,0EK is obvious. This equilibrium implies the
failure of virus therapy. Biologically, it means that both
infected and uninfected tumor cells tend to the same state,
as they would have been reached without virus admini-
stration.
Existence of


20,Ey. It can be checked out that the
equilibrium point


20,Ey exists if 2
r

. This equi-
librium implies complete infection of tumor cells and
stabilization of tumor load to a finite minimal size
y


2
2
.
Kr
r

 Biologically, it gives a real life situation in
which tumor load can be reduced to lower size if tumor
is detected at initial stage.
Existence of


3,Exy

. To see the existence of in-
terior equilibrium point


3,Exy

we note that
,
x
y

 are positive solutions of the system of alge-
braic equations given below:
110
xy by
rKxya






(4.1)
210
xy bx
rKxya 





 (4.2)
Now substituting the value of
y
 in Equation (4.1),
we get



 

 
10rK xfxxfxaKbfx

 (4.3)
To show existence of
x

, it suffices to show that Eq-
uation (4.3) has a unique positive solution.
Taking





  


1
GxrKxf xxf xaKbf x
We note that





 

 
1
00000GrKf faKbf


provided
12 2
0 and 0.rrb r



 (4.4)
or
12
1
rr
b 
 (4.5)
and
 




10GKfK rKfKabK


provided
.
bK
K
a
 

(4.6)

M. AGARWAL ET AL. 134
Thus, there exist a
x
 in the interval 0
x
K



such that

0Gx

For
x
 to be unique, we must have
    
1122 0,
dG rfxxfxaKKbfx
dx 

(4.7)
Corresponding value of
y
 is given by


yfx

.
Thus if all the three conditions (4.5)-(4.7) hold an uni-
que interior equilibrium always exists. However, these
conditions depend upon the parameter values so they do
not always hold. We will show further that if Equation
(4.5) and Equation (4.6) does not hold other equilibrium
points of the system become locally asymptotically sta-
ble.
5. Local Stability Analysis of Equilibrium
Points
An equilibrium point is locally asymptotically stable if
all solutions of the system approaches it as t. To
discuss the local stability of equilibrium points we com-
pute the variational matrix of system (2.1). 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 differ-
entiating the right hand side of system (2.1) with respect
to ,
x
y. The matrix is given by

A
B
VE CD



where


1
12
1rx
x ybybxy
Ar KKxya
x
ya


 
 
 


12
rx bx bxy
BKxya
x
ya
 
 


22
ry by bxy
CKxya
x
ya
 
 


2
22
1ry
x ybxbxy
Dr KKxya
xya



 
 
 
We denote the variational matrix corresponding to i E
by

i
VE , 0,1,2,3i
5.1. Local Stability Analysis of

00,0E
To explore local stability o f trivial equilibrium point, we
compute variational matrix of 0
E. The variational ma-
trix of equilibrium point 0 E is given by

1
02
0
0
r
VE r








 (5.1)
Eigenvalues of


0
VE are given by,r

 2.r




0
E is an unstable equilibrium point since both the ei-
genvalues of the matrix are positive.
5.2. Local Stability Analysis of

1,0EK
Now, to study the stability behavior of 1 E, we compute
the variational matrix


1
VE corresponding to 1 E as
follows:

11
10
bK
rr
K
a
VE bK
Ka


















(5.2)
From Equation (5.2) we observe that eigenvalues of
the matrix


1
VE are given by 1 r

 and
bK
Ka




. Thus,


1
VE has negative eigenvalues
and 1 E is stable equilibrium point if bK
K
a
 

, con-
sequently 1 E is a saddle point if bK
K
a
 .
Biological interpretation: Inequality bK
K
a
 

im-
plies that equilibrium point 1
E is locally asymptotically
stable if death rate of infected cells due to the virus at-
tack is larger than the rate of transmission of infection
from virus to the uninfected cells if measure of the im-
mune response of the individual to the viruses is very
less as compared to carrying capacity of the cells. Stabil-
ity of this equilibrium point suggests that infected cells
would die without having time to infect other cells and
tumor grows unaffectedly.
Remark: 1
E is stable if interior equilibrium point
does not exist .
5.3. Local Stability Analysis of

20,
E
y
The variational matrix of equilibrium point


20,Ey
where

2
2
K
yr
r


 is given by




 
2
1
22 2
2
2
22
22
0
bK r
r
rKr ar
MbK r
rr
Kr ar






















 





(5.3)

M. AGARWAL ET AL. 135
From Equation (5.3) we observe that eigenvalues of
the matrix

1
VE are given by


2
1
222
bK r
r
rKr ar




 
and

2.r

   Thus

2
VE has negative eigenvalues
and 2
E is stable equilibrium point if


2
1
22 2
bK r
r
rKr ar




 or


221
12
Kr
abrr
rr





consequently 2
E is a saddle point if


221
12
Kr
abrr
rr




 .
Biological interpretation: Local asymptotic stability
of 2
E implies that 21 br r

, Since a is a positive
parameter and it cannot be less than a negative quantity,
so 2
E is locally asymptotically stable point if 21
rr
b


together with 21
r

 or 21 max 1
rr
,b




. Hence 2 E is
locally asymptotically stable under any of the two situa-
tions:
1) If net growth rate of uninfected cells is less than the
rate of transmission of virus infection than growth rate of
infected cells must be greater than the death rate of in-
fected cells caused by the viruses.
2) If net growth rate of uninfected cells is more than
the rate of transmission of virus infection then the ratio
of net growth rate of uninfected cells to the death rate of
infected cells caused by the v iruses is more than the ratio
of net growth rate of uninfected cells to rate at which
they become infective.
Remark: 2 E is stable if interior equilibrium point
does not exist .
5.4. Local Stability Analysis of


,

3
E
xy
Variational matrix of

3,Exy

is given by

3
A
B
VE CD





(5.4)
where


1
1
2
1
rx xy by
Ar KK
x
ya
bx y
xya
  






 
 




12
rx bxbx y
BKxya
x
ya


 
 
 


22
ry bybx y
CKxya
x
ya


 
 
 


2
2
2
1
ry xy bx
Dr KK
x
ya
bx y
xya


 






 
 



From variational matrix

3
VE , we find that eigen-
values are


where



12
22
12
12 3
2
14
2
rx ry
K
bx yrr
rx ryab x y
KxyaK xya




 
 






 




 

The signs of the real parts of

 and

 are nega-
tive. This implies that 3
E is always locally asymptoti-
cally stable if it exists.
6. Global Stability of Interior Equilibrium
Point
An equilibrium point is globally asymptotically stable if
system always approaches it regard less of its initial posi-
tion. We construct Lyapunov functions that enable us to
find biologically realistic conditions sufficient to ensure
existence and uniqueness of a globally asymptotically
stable equilibrium state. Global stability of the interior
equilibrium point of system (2.1) is determined in the
theorem 6 given below:
Theorem 6.: If the following inequality holds




2
12
4
by
axy a
rr by bx
KK
axy aKaxya



 






 
 

 
 
 
then 3
E is globally asymptotically stable with respect
to the solutions initiating in the interior of the positive
orthant

.
Proof: Consider the following positive definite func-
tion abou t 3
E
ln ln
x
y
Wxxxyy y
x
y
 

 .
Computing the derivative of W with respect to t
and after some algebraic manipulations, we get

M. AGARWAL ET AL. 136









2
1
12
2
2
()
r dWby xx
dtK xyaxya
rr
K
x
xyy
by x
xyaxya
rbx yy
Kxyaxya









 

 




 








 



 

 

We note that dW
dt can be made negative definite inside
 if







2
12
1
2
4
by x
rr
Kxyaxya
rby
Kxyaxya
rbx
Kxyaxya













 







 





 

After maximizing left hand side and minimizing right
hand side of above in equality we note that dW
dt will be
negative definite if following hold:




2
12
4
by
axy a
rr by bx
KK
axy aKaxya



 





 
 
 
 
 
 
,
This completes the proof of the theorem (6).
7. Numerical Simulation
To substantiate the above analytical findings, the model
is studied numerically. The system of differential equa-
tion is integrated using fourth order Runge-Kutta method
under the following set of compatible (hypothetical) pa-
rameters, which satisfy the stability conditions.
12
40, 100, r2, 0.05, 0.02, 0.003.rKab

  
(7.1)
The equilibrium points for this set of parameters are:

00,0E,

1100,0E


20,99.85E and

310.5295, 89.4258E
It is found that all the conditions for local asymptotic
stability of interior equilibrium point are satisfied for
above parameter values. Further, to illustrate the global
stability of the equilibrium point graphically, numerical
simulation is performed for different initial starts an d the
result is displayed in Figure 1. It is found from the gr aph
that all the trajectories starting from different initial starts,
reach the endemic equilibrium 3 E.
In Figure 2 densities of tumor cells for parameter val-
ues (7.1) is shown. It is observed from the figure that in-
fected tumor cell population first rise and then attain a
constant equilibrium value whereas uninfected tumor cell
population first rise abruptly and then decrease due to
virus infection and cytotoxicity to attain its equilibrium
value.
In Figure 3, density of tumor cells is drawn for
Figure 1. Variation of infected tumor cells with uninfected
tumor cells for different initial starts.
Figure 2. The densities of tumor cells for all the parameter
values given in Equation (7.1).

M. AGARWAL ET AL. 137
0.3

and other parameters remaining same as in Eq-
uation (7.1). This figure implies that the system con-
verges to equilibrium point

1100,0E when death rate
of infected tumor cells due to the viruses increases from
0.003

 to 0.3

. Figure 4 shows the density of
tumor cells for 06.0b and other parameters remain-
ing same as in Equation (7.1). It demonstrates the con-
vergence of the system to equilibrium point


20,99.85E
for higher infectivity of the oncolytic virus.
Numerically it is found that tumor load decreases
when per capita growth rate of infected cells (2 r) is less
than or equal to the death rate of infected cells due to
virus (

) for high replication rate of viruses in the cells.
Figure 5 shows variation of tumor load with time for
different transmission rate of virus infection (b). It is
Figure 3. Convergence of the system to the equilibrium
point


1100,0E, for 0.3

and other parameters re-
maining same as in Equation (7.1).
Figure 4. Convergence of the system to the equilibrium
point


20,99.85Efor 0.06band other parameters re-
maining same as in Equation (7.1).
found from the graph that for 2 r2

, tumor load
decreases with the increase in b for all 1b and van-
ishes for 50b, other parameters remaining same as in
Equation (7.1). Figure 6 display the formation of limit
cycle in the system. It is observed from numerical simu-
lation that stable limit cycles are formed for 1035b


and for 35b no limit cycles are formed. However,
when 2 r


i.e. for parameter values2 r1,

2


and other parameters remaining same as (7.1), we have
found that stable limit cycles are formed for 1040b

.
Figure 7 shows stable limit cycle for 30band keep-
ing other parameters fixed as above. However, when
40b, tumor load decreases and strange chaotic attrac-
tors are obtained. Figure 8 display chaotic attractor for
this range.

Figure 5. Variation of tumor load with time for different
transmission rate of infection from virus and parameters
140r, 100K

, 2 r2

, 0.05a, 2

.

Figure 6. Stable limit cycles for parameter values, 140r,
100K

, 2 r2

, 0.05a

, 2

, 20b initial values




01,y01x

.

M. AGARWAL ET AL. 138
Figure 7. Stable limit cycles for parameter values, 140r,
100,K2 r2,0.05,a2,

30b and initial values
 
01,y01x .
Figure 9 shows that for a particular value of rate of
transmission of infection, 50b

tumor load vanishes
when per capita growth rate of infected cells (2
r) is less
than or equal to the death rate of infected cells due to
virus (

) but tumor load remains at its maximal size if
2
r

.
Now keeping all the parameters fixed at 140r

,
100,K2 r2,0.05,a50band varying ,

we ob-
serve that tumor load increases with increase in alpha
and acquires maximum possible load for


50b  
which implies that tumor grows unaffectedly when death
rate of infected cells due to the virus attack exceeds the
rate of transmission of infection from virus to the unin-
fected cells. However, tumor load decreases giving peri-
odic oscillations for b

. Figure 10 shows variation
of tumor load with time for different values of

. Fig-
ure 11 display three dimensional attractors obtained for
the range 310

 and other parameters fixed as
above. Figure 12 shows table limit cycle for the range
10 30

 . Further, it is found that no limit cycles exist
for 30

.
8. Conclusions
It is well known that the cancer is one of the greatest
killers in the world and the control of tumor growth re-
quires great attention. Various efforts have been made
for its treatment. Equally extensive efforts have been
dedicated over many years to mathematical modeling of
cancer development. Here we address a complex process
that involves both virus-cell interaction and tumor
growth. The positive equilibrium points of the system are
investigated. The stability and instability of the equilib-
Figure 8. Tumor three dimensional attractors for the value
of the system parameters 140,r

100,K2 r1,0.05a

,
60b

, 50


and


01,y01x

.
Figure 9. Variation of tumor load with time for different
growth rate of infected tumor population relative to their
mortality rate due to virus with parameters 140,r
100K

, 0.05a

, 50b

.
rium 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 sys-
tem of differential equation is integrated using fourth
order Runge-Kutta method, which supports the theoretic-
cal findings.
It is found that virus therapy fails if death rate of in-
fected cells due to virus attack exceeds the transmission
rate of infection from virus to the uninfected tumor cells
for small immune response of the individual to the virus
action. It is observed from our analysis that tumor load
can be reduced to a lower value under any one of the two
conditions: If net growth rate of uninfected cells is less
than the rate of transmission of virus infection than

M. AGARWAL ET AL. 139
Figure 10. Variation of tumor load with time for different
values of

and parameters 140,r100,K2 r1,

0.05a, 50b.
Figure 11. Tumor three dimensional attractors for the value
of the system parameters 140,r100,K2
r2,0.05,a

50b, 5

and initial values

01x,


y0 1

.
growth rate of infected cells must be greater than the
death rate of infected cells caused by the viruses and if
net growth rate of uninfected cells is more than the rate
of transmission of virus infection then the ratio of net
growth rate of uninfected cells to the death rate of in-
fected cells caused by the viruses must exceed the ratio
of net growth rate of uninfected cells to rate at which
they become infective. Further it is found from our anal-
ysis that interior equilibrium point of the system exists
under certain condition which is always locally asympto-
tically stable if we assume that rate of growth of unin-
fected tumor cells is more than the growth rate of in-
fected tumor cells.
From numerical simulation it is found that tumor load
decreases when per capita growth rate of infected cells is
Figure 12. Stable limit cycle for the set of parameters:
140r

, 100K

, 2 r2

, 0.05a, 50b, 20


and




01,y01x

.
less than or equal to the death rate of infected cells due to
virus for high replication rate of viruses in the cells.
However, tumor grows unaffectedly when death rate of
infected cells due to the virus attack exceeds the rate of
transmission of infection from virus to the uninfected
cells. The system appears to exhibit different attractors
and stable limit cycles for some ranges of the system
parameters.
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