Biomedical Engineering Reference
In-Depth Information
Since
h
(
t
)
<
0forall
t
>
0, by the LaSalle invariance principle [
7
] it follows that
x
0
h
(
t
)
→
0as
t
→
∞
. This easily leads to the conclusion that
x
(
t
)
→
as
t
→
∞
,so
that
x
0
is a globally asymptotically stable equilibrium.
A function of the form Eq. (
7
) is a Liapounov function if
x
∗
)
, F
x
0
)
V
=
V
(
(
x
)
≤F
(
i
i
i
i
for all
x
∗
=(
m
+
,
x
0
=(
x
∗
,
x
m
+
1
,...,
x
n
)
,
x
1
,...,
x
m
)
∈
R
and
m
i
=
1
V
ij
λ
i
>
0
.
Similarly one can show, under somewhat different conditions, that if
R
0
>
m
j
=
1
x
j
∂
∂
x
0
)
−V
x
∗
)
≤
x
0
)
−V
x
∗
))
,
F
(
(
x
j
(
F
(
(
i
i
i
j
1then
there is a function
h
(
t
)
of the form Eq. (
7
) which satisfies:
h
(
t
)
≥
δ
h
(
t
)(
δ
>
0
)
provided
∑
i
=
j
x
i
(
t
)
<
η
where
η
is sufficiently small and
t
is sufficiently large
depending on
h
. This instability result of the DFE can be used, with the aid of
Horn
's
l
e
mma [
5
](seealso[
4
]) to deduce the existence of an (endemic) equilibrium
point
x
,
x
(
0
)
=
0, provided the system (
5a
)and(
5b
) has a compact convex invariant set
with
x
0
in its interior.
3
The Basic Reproduction Number for Nonautonomous
Systems
Consider now a nonautonomous system of the same structure as in Eqs. (
5a
)
and (
5b
),
d
x
i
d
t
=
F
i
(
t
,
x
)
−V
i
(
t
,
x
)
(
1
≤
i
≤
m
)
,
(8a)
d
x
j
d
t
=
g
j
(
t
,
x
)
(
m
+
1
≤
j
≤
n
)
.
(8b)
Following [
12
](seealso[
2
,
13
]) we assume:
n
+
F
V
R
×
R
(B1)
i
,
i
and
g
j
and their first
x
-derivatives are continuous functions in
,
for all
i
,
j
.
n
0
+
,
F
=
V
=
R
×
R
(B2)
0on
,forall
i
.
i
i
F
≥
(B3)
0, for all
i
.
i
Search WWH ::
Custom Search