Biomedical Engineering Reference
In-Depth Information
Using Theorem
4.2
we shall also prove:
Theorem 4.3.
If
[
R
0
]
>
1
, then there exists an
ω
-periodic solution of Eqs. (
17a
)-
(
17d
) with i
(
t
)
>
0
,w
(
t
)
>
0
for all t
>
0
.
5
Proof of Theorem
4.1
Since
s
(
t
)
≤
1, we obtain from Eqs. (
17a
)-(
17d
)
i
w
i
w
d
d
t
≤
A
(
t
)
T
T
where we have used the notation
(
x
,
y
)
≤
(
x
1
,
y
1
)
if
x
≤
x
1
,
y
≤
y
1
.
T
We introduce the solution
z
=(
z
1
,
z
2
)
of
d
z
d
t
=(
T
A
(
t
)+
δ
I
)
z
,
z
(
0
)=(
i
(
0
)+
δ
,
w
(
0
)+
δ
)
,
(20)
T
for any
δ
>
0. Then
z
>
(
i
,
w
)
for small
t
. We claim that this inequality holds for
all
t
0.
Indeed, otherwise there is a smallest
t
s
uch t
h
at at least
o
ne of the strict
inequalities is vi
o
lated
a
t
t
>
=
t
. Sup
p
ose
z
1
(
t
)=
i
(
t
)
.Thend
z
1
(
t
)
/
d
t
≤
d
i
(
t
)
/
d
t
.
We a l s o h ave
z
2
(
t
)
≥
w
(
t
)
. Hence, at
t
,
d
z
1
d
t
≤
d
i
d
t
≤
β
2
z
2
+(
β
1
−
γ
−
μ
)
z
1
,
which contradicts the fir
s
t equat
i
on in Eq. (
20
). By a similar argument one derives a
contradiction in case
z
2
(
t
)=
w
(
t
)
.
Taking
δ
→
0 we conclude that
i
(
t
)
≤
z
1
(
t
)
,
w
(
t
)
≤
z
2
(
t
)
for all
t
>
0
,
(21)
T
where
z
=(
z
1
,
z
2
)
is the solution of
d
z
d
t
=
T
A
(
t
)
z
,
z
|
t
=
0
=(
i
,
w
)
|
t
=
0
.
(22)
By integration and use of the
ω
-periodicity of
A
(
t
)
we obtain
⎛
⎞
n
ω
+
τ
e
n
ω
A
exp
⎝
⎠
z
z
(
t
)=
A
(
s
)
d
s
(
0
)
if
t
=
n
ω
+
τ
,
n
ω
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