Biomedical Engineering Reference
In-Depth Information
We d e fi n e
n
,
0
+
=
{
n
+
,
R
x
∈
R
x
i
=
0for
i
=
1
,...,
m
}
and make the following assumptions:
F
i
,V
i
and
g
j
and their first
x
-derivatives are continuous functions in
R
n
(A1)
,for
+
all
i
,
j
.
n
,
0
(A2)
F
i
=
V
i
=
0on
R
,forall
i
.
+
(A3)
F
i
≥
0, for all
i
.
(A4)
V
i
≤
0 whenever
x
i
=
0;
i
is any number 1
,
2
,...,
m
.
m
∑
(A5)
i
=
1
V
i
≥
0.
d
x
j
d
t
=
(A6) The disease free system
g
j
(
0
,...,
0
,
x
m
+
1
,...,
x
n
)
has a unique equi-
librium point
x
0
x
m
+
1
,...,
x
n
)
=(
0
,...,
0
,
which is globally asymptotically
stable.
We refer to
x
0
as the DFE. We introduce the Jacobian matrices
∂
F
i
(
∂
V
i
(
x
0
x
0
)
)
F
=(
F
ij
)=
1
≤
i
,
j
≤
m
,
V
=(
V
ij
)=
∂
x
j
∂
x
j
1
≤
i
,
j
≤
m
and assume that
(A7)
V
is nonsingular.
Then
V
−
1
and
FV
−
1
are nonnegative matrices [
11
].
We denote by
the spectral radius of a bounded linear operator
A
;if
A
is a
matrix, viewed as a linear operator in
ρ
(
A
)
n
,then
R
ρ
(
A
)
is the maximum absolute value
of the eigenvalues of
A
.
Theorem 2.1 ([
11
]).
Under the assumptions (A1)-(A7) there holds:
FV
−
1
=
ρ
(
)
.
R
0
Since
R
0
is the principal eigenvalue of the nonnegative matrix
FV
−
1
and thus also
of its transpose, there exists an eigenvector
m
+
FV
−
1
T
λ
=(
λ
,...,
λ
)
R
(
)
λ
=
in
of
1
m
R
0
λ
,sothat
m
i
=
1
F
ij
λ
i
=
R
0
m
i
=
1
V
ij
λ
i
.
(6)
1then
x
0
is
globally
asymptotically stable for the system Eqs. (
5a
)and(
5b
). The proof uses a
Liapounov function of the form
Using Eq. (
6
) one can show, under some additional conditions, that if
R
0
<
m
i
=
1
(
λ
i
+
εμ
i
)
x
i
(
t
)
h
(
t
)=
(7)
for appropriate parameters
μ
i
and
ε
positive and sufficiently small.
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