Biomedical Engineering Reference
In-Depth Information
n
+
(B4)
V
i
≤
0on
R
×
R
if
x
i
=
0;
i
is any number 1
,
2
,...,
m
.
(B5)
j
.
(B6) There exists a disease-free periodic solution
x
0
F
i
,
V
i
,
g
j
are
ω
-periodic in
t
,forall
i
,
x
m
+
1
(
(
t
)=(
0
,
0
,...,
0
,
t
)
,...,
x
n
(
t
of Eqs. (
8a
)and(
8b
).
We introduce the Jacobian matrices
)
∂
x
0
g
i
(
t
,
(
t
))
(
)=
n
,
G
t
∂
x
j
m
+
1
≤
i
,
j
≤
∂
F
i
(
∂
V
i
(
x
0
x
0
t
,
(
t
))
t
,
(
t
))
F
(
t
)=
m
,
V
(
t
)=
m
.
∂
x
j
∂
x
j
1
≤
i
,
j
≤
1
≤
i
,
j
≤
Let
Y
(
t
,
s
)
denote the solution of
d
Y
d
t
=
−
V
(
t
)
Y
,
Y
(
s
,
s
)=
I
,
where
I
is the unit matrix, and set
Y
(
ω
)=
Y
(
ω
,
0
)
.
We assume that
x
0
(
t
)
is linearly asymptotically stable by imposing the
condition
(B7)
1.
We also assume that the internal evolution in the infectious disease
compartments due to death is dissipative, that is,
(B8) In
ρ
(
G
(
ω
))
<
ρ
(
Y
(
ω
))
<
1.
m
Denote by
C
ω
the Banach space of
ω
-periodic functions from
R
into
R
equipped with the maximum norm, and denote by
C
ω
the subspace of functions
m
+
with values in
R
. We introduce the bounded linear operator
∞
(
L
φ
)(
t
)=
Y
(
t
,
t
−
a
)
F
(
t
−
a
)
φ
(
t
−
a
)
d
a
,
t
∈
R
0
C
ω
. The basic reproduction number
R
0
is defined as the spectral radius of
the operator
L
,
for
φ
∈
R
0
=
ρ
(
L
)
.
(9)
This definition coincides with the definition of
R
0
given in Sect.
2
when the
system (
8a
)and(
8b
) is autonomous [
12
].
Theorem 3.1 ([
12
]).
Under the assumptions (B1)-(B8), x
0
(
t
)
is asymptotically
stable if R
0
<
1
, and unstable if R
0
>
1
.
In the sequel we consider nonautonomous systems (
8a
)and(
8b
)wherethe
dependence on
t
is in terms of parameters
(
)
,...,
γ
k
(
)
. We associate with Eqs.
(
8a
)and(
8b
) the autonomous system (
5a
)and(
5b
) with parameters
γ
t
t
1
,...,
γ
k
and
γ
1
DFE
x
0
independent of
t
.
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