Biomedical Engineering Reference
In-Depth Information
values of time. Here, different paths can be proposed. We can adjust the external
source of effector cells
s
1
(
t
)
every time
S
(
t
)
starts to decrease or, alternatively, the
functions
r
2
(
t
)
,
a
(
t
)
can be made
E
-dependent in a way that the stability condition
S
S
0
holds until complete eradication of the tumor is achieved. Below we
propose conditions which do not involve effector cells,
E
(
t
)
>
explicitly. In the next
theorem,
T
0
plays a key role: if
T
falls below
T
0
, the cancer is assumed cleared.
Theorem 4.2 (Main Stability Theorem).
Let the following condition be satisfied
for all t
(
t
)
b
−
1
≥
t
0
with some constants t
0
≥
0
,
ε
>
0
,
σ
>
0
,
β
>
0
and T
0
∈
(
0
,
)
⎧
⎨
2
μ
2
(
ε
+
θ
r
2
(
t
))
+(
μ
2
f
−
p
3
−
s
1
(
t
)
−
c
(
t
)
T
0
)(
ε
+
θ
r
2
(
t
))
a
(
t
)
a
2
−
(
s
1
(
t
)+
fc
(
t
)
T
0
−
β
)
(
t
)
<
0
ε
+
θ
r
2
(
t
)
(15)
⎩
is a nonincreasing function of time
(
)
a
t
fc
(
t
)
T
0
+
s
1
(
t
)
>
σ
>
0
where
2
g
2
+
(
1
−
bg
2
)
Case a:
θ
=
4
b
or
b
−
1
Case b:
θ
=
g
2
and g
2
>
The following statements hold:
Case I
(partial clearance).
For every solution
(
T
(
t
)
,
E
(
t
))
of Eq. (
5
)givenby
initial condition
(
T
(
t
1
)
,
E
(
t
1
))
,
t
1
≥
t
0
with
T
(
t
1
)
>
T
0
the function
T
(
t
)
will
reach in finite time the value
T
0
.
Case II
(complete clearance).
If condition (
15
) is satisfied with
T
0
=
0, then for
all solutions of the system (
5
)
lim
T
(
t
)=
0
.
(16)
t
→
+
∞
Proof.
We write the first equation of the system (
5
) as follows
K
(
E
)
E
=
f
,
(17)
E
+
where
K
is quadratic polynomial with respect to
E
given by
˜
E
2
K
=
α
˜
+
β
E
+
γ
˜
(18)
˜
and ˜
.
The conditions (
7
)and(
8
) can be all expressed, for suitable real positive number
α
=
−
μ
2
,
β
=
c
(
t
)
T
(
t
)+
p
3
+
s
1
(
t
)
−
μ
2
f
, ˜
γ
=
fc
(
t
)
T
(
t
)+
fs
1
(
t
)
θ
in the following form
E
(
t
)
−
h
(
t
)
>
0
(19)
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