Biomedical Engineering Reference
In-Depth Information
4
Stability Conditions for Nonautonomous Gene
Therapy Model
In this section we derive conditions for global stability of the cancer free state
(
T
0) for the Gene Therapy model (
5
). First, we investigate the second equation
of system (
5
) independently from the first equation. Thus, we consider
r
2
(
=
t
)
,
a
(
t
)
and
e
(
t
)=
E
(
t
)
as
arbitrary
positive data functions:
e
(
t
)
T
T
=
r
2
(
t
)
T
(
1
−
bT
)
−
a
(
t
)
g
2
.
(6)
T
+
b
−
1
The only biological plausible solutions should satisfy the condition
T
(
t
)
∈
[
0
,
]
.
b
−
1
Moreover, as easily seen from the second equation of system (
5
), the interval
]
is dynamically invariant under the flow. Our first aim is to derive conditions on the
functions
r
2
(
[
0
,
, which would imply asymptotical global stability of
the cancer free equilibrium state
T
t
)
,
a
(
t
)
and
e
(
t
)
=
0.
Theorem 4.1.
Let one of the following two conditions holds
Condition 1:
There exist t
0
>
0
and
ε
>
0
such that
2
a
(
t
)
e
(
t
)
g
2
+
(
1
−
bg
2
)
ε
r
2
(
>
+
)
, ∀
t
≥
t
0
(7)
r
2
(
t
)
4
b
t
or
Condition 2:
b
−
1
and there exist t
0
>
g
2
>
0
and
ε
>
0
such that
a
(
t
)
e
(
t
)
ε
r
2
(
>
g
2
+
)
, ∀
t
≥
t
0
.
(8)
r
2
(
t
)
t
Then every solution of Eq.
(
6
)
satisfies
lim
T
(
t
)=
0
with exponential conver-
t
→
+
∞
gence.
Proof.
We write Eq. (
6
) in the form:
r
2
(
t
)
T
T
=
g
2
V
(
T
)
,
(9)
T
+
where
V
is a quadratic polynomial with respect to
T
given by:
a
(
t
)
e
(
t
)
bT
2
V
(
T
)=
−
+(
1
−
bg
2
)
T
+
g
2
−
,
(10)
r
2
(
t
)
with discriminant
D
as follows:
4
b
g
2
−
a
(
t
)
e
(
t
)
2
D
=(
1
−
bg
2
)
+
.
(11)
r
2
(
t
)
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