Biomedical Engineering Reference
In-Depth Information
(
)
Proof.
a
Since
T
t
=
0
F
N
∞
,
t
(
0
)
<
1
,
−
1
=
ℜ
0
Lemma
4.1
gives
F
N
∞
,
0
(
I
(
1
)=
F
N
∞
,
0
(
I
(
0
))
≤
0
)
I
(
0
)
.
Thus,
F
N
∞
,
1
(
F
N
∞
,
1
(
F
N
∞
,
0
(
(
)=
(
(
))
≤
)
(
)
≤
)
)
(
)
I
2
F
N
∞
,
1
I
1
0
I
1
0
0
I
0
and inductively
t
−
1
t
=
0
F
N
∞
,
t
(
0
)
I
j
−
1
t
=
0
F
N
∞
,
t
(
0
)
I
)
≤
(
ℜ
0
)
t
/
T
I
(
t
)
≤
(
0
max
j
∈{
1
,
2
,···,
T
−
1
}
(
0
)
.
Since the sequence
{
I
(
t
)
}
is dominated by the decreasing sequence
j
−
1
t
=
0
F
N
∞
,
t
(
0
)
I
(
ℜ
0
)
t
/
T
max
(
0
)
,
j
∈{
1
,
2
,···,
T
−
1
}
it converges to zero. Hence lim
t
→
∞
I
0. The proof of Theorem
4.1
(b) is
similar to that of Theorem 4.6 in [
18
] and is omitted.
(
t
)=
In Model (
5
), the demographic dynamics does not drive the disease dynamics.
Now, we use constant recruitment and Theorem
4.1
in Example
4.1
to illustrate
an infective population on a 2-cycle attractor, where the demographic dynamics is
noncyclic.
Example 4.1.
In Model (
5
), let
β
t
I
N
k
(
1
−
γ
)
γ
e
−
β
t
N
f
(
N
)=
and
φ
=
,
where
t
k
=
5
,
β
t
=
a
+
b
×
(
1
+(
−
1
)
)
,
a
=
0
.
1
,
b
∈
[
42
,
50
]
,
γ
=
0
.
1and
σ
=
0
.
01.
5, the
interaction between susceptible and infected is modeled as a Poisson process, and
disease transmission is 2
In Example
4.1
, the total population is asymptotically constant,
N
∞
=
−
periodic
(
β
t
+
2
=
β
t
). When
b
=
42, then
β
0
=
84
.
1,
β
1
=
0
.
1,
ℜ
0
=
0
.
927
<
1 and the disease goes extinct (Theorem
4.1
and Fig.
1
).
However, when
b
1 and the infective population
persists on a positive 2-cycle attractor (Theorem
4.1
and Fig.
1
). That is, increasing
the periodic infection rate in Example
4.1
shifts the system from disease extinction
phase to disease persistence on a 2-cycle attractor, where the total population
persists on a fixed point attractor.
=
45
.
38 then
ℜ
0
=
1
.
000 1
>
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