Biomedical Engineering Reference
In-Depth Information
=
μ
−
β
(
)
−
β
(
)
−
μ
,
s
t
si
t
sw
s
(17a)
1
2
i
=
β
(
)
+
β
(
)
−
γ
−
μ
,
t
si
t
sw
i
i
(17b)
1
2
=
ξ
(
)(
−
)
,
w
t
i
w
(17c)
=
γ
−
μ
,
r
i
r
(17d)
where
β
1
(
t
)
,
β
2
(
t
)
,
ξ
(
t
)
are continuous
ω
-periodic functions, with initial conditions.
(
s
(
0
)
,
i
(
0
)
,
w
(
0
)
,
r
(
0
))
∈
Ω
.
(18)
It easily follows that the solution
(
s
(
t
)
,
i
(
t
)
,
w
(
t
)
,
r
(
t
))
remains in
Ω
for all
t
>
0.
The DFE for both systems (
15a
)-(
15d
)and(
17a
)-(
17d
)is
(
s
,
i
,
w
,
r
)=(
1
,
0
,
0
,
0
)
.
The basic reproduction number for Eqs. (
15a
)-(
15d
) is easily computed to be [
8
]
)=
β
1
+
β
2
γ
+
μ
≡
(
β
,
β
,
R
0
R
0
1
2
so that
)=
β
1
+
β
2
γ
+
μ
[
R
0
]=
R
0
(
β
,
β
.
(19)
1
2
Note that
R
0
<
1 if and only if all the eigenvalues of the matrix
β
1
−
γ
−
μβ
2
ξ
A
=
−
ξ
have negative real parts. Similarly, setting
β
β
(
)
−
γ
−
μβ
(
)
−
γ
−
μ
β
t
t
1
2
1
2
A
(
t
)=
,
A
=
,
ξ
(
t
)
−
ξ
(
t
)
ξ
−
ξ
[
have negative real parts.
In the following sections we shall prove the assertions (
13
)and(
14
)forthe
system (
17a
)-(
17d
). This will be a consequence of the following theorems.
R
0
]
<
1 if and only if all the eigenvalues of
A
Theorem 4.1.
If
[
R
0
]
<
1
, then the DFE for Eqs. (
17a
)-(
17d
) is globally asymptot-
ically stable.
[
]
>
Theorem 4.2.
If
R
0
1
, then there exists a positive number
δ
such that for any
(
)
>
<
(
)
≤
(
)
initial values with s
0
0
, and
0
w
0
i
0
, the solution of Eqs. (
17a
)-(
17d
)
satisfies:
(
)
>
δ
(
)
>
δ
.
i
t
and
w
t
if
t
is sufficiently large
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