Biomedical Engineering Reference
In-Depth Information
The first example is an SEIR tuberculosis model [
8
]
d
E
d
t
=(
S
N
1
−
q
)
β
(
t
)
I
−
(
μ
+
k
(
t
))
E
,
(45a)
d
I
d
t
=
S
N
q
β
(
t
)
I
−
(
μ
+
d
+
r
)
I
,
(45b)
d
S
d
t
=
μ
−
β
(
S
N
t
)
I
−
μ
S
,
(45c)
d
R
d
t
=
rI
−
μ
R
,
(45d)
where
N
=
S
+
E
+
I
+
R
.Here
x
1
=
E
,
x
2
=
I
,
x
3
=
S
,
x
4
=
R
and
m
=
2,
n
=
4
in the notation of Sect.
3
.Since
S
≤
N
, we can apply the proof of Theorem
4.1
to
deduce that if
[
R
0
]
<
1 then the DFE,
(
1
,
0
,
0
,
0
)
, is globally asymptotically stable,
so that
R
0
<
1. However, as shown in [
8
], for some choices of
β
(
t
)
,
k
(
t
)
there holds:
[
R
0
, so that Theorem
4.2
does not hold.
The second example is a model of staged progression in disease transmission of
HIV [
6
]:
R
0
]
>
1
>
d
I
1
d
t
=
β
1
(
S
N
S
N
t
)
I
1
+
β
2
(
t
)
I
2
−
(
ν
1
+
d
1
)
I
1
,
(46a)
d
I
2
d
t
=
ν
−
(
ν
+
)
,
1
I
1
d
2
I
2
(46b)
2
d
I
3
d
t
=
ν
2
I
2
−
d
3
I
3
,
(46c)
d
S
d
t
=
μ
−
β
S
N
S
N
(
)
−
β
(
)
−
μ
,
t
I
1
t
I
2
S
(46d)
1
2
with
DF E
=(
1
,
0
,
0
,
0
)
;here
N
=
S
+
I
1
+
I
2
+
I
3
.Asshownin[
12
], for some
choice of
β
1
(
t
)
,
β
2
(
t
)
and the parameters
ν
i
d
j
there holds:
R
0
>
1
>
[
R
0
]
,sothat
Theorem
4.1
does not hold.
In order to prove Theorem
4.2
for the system (
46a
)-(
46d
) we proceed with steps
(i), (ii) and (iii) described above. The first step is to show that if
I
1
(
t
)
≤
η
for 0
<
t
and positive constants
c
i
.
This follows from the differential equations for
I
2
and
I
3
, using Lemma
6.1
(i). The
second step, (ii), follows by the same arguments as in the proof of Lemma
6.6
.
We next note that if
I
1
(
<
Λ
then
I
2
(
t
)
≤
c
1
η
and
I
3
(
t
)
≤
c
2
η
for
T
0
<
t
<
Λ
t
)
>
η
for 0
<
t
<
Λ
then
I
2
(
t
)
≥
c
3
η
and
I
3
(
t
)
≥
c
4
η
for
t
0
<
t
<
Λ
and positive constants
c
3
,
c
4
. Also, if
I
2
(
t
)
≥
η
then
I
1
(
t
)
≥
c
5
η
for
t
0
<
0; indeed the proof is similar to the proof of
Lemma
6.4
. First we estimate
S
analogously to Eq. (
32
), and then use the estimate
t
<
Λ
with another
t
0
,and
c
5
>
S
N
c
N
≥
c
η
β
(
t
)
I
2
≥
2
N
(
0
)
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