Biomedical Engineering Reference
In-Depth Information
(c)
β
2
φ
β
N
∞
φ
β
t
I
N
∞
2
β
t
I
N
∞
t
N
∞
t
F
N
∞
,
t
(
I
)=
γ
−
(
N
−
I
)+
.
φ
<
φ
≥
Since
0and
0on
[
0
,
∞
)
,
we have
F
N
∞
,
t
(
I
)
<
0on
[
0
,
N
∞
]
.
F
N
∞
,
t
(
(d)
F
N
∞
,
t
(
0
)=
0 implies that
y
=
0
)
I
is the tangent line to the graph of
F
N
∞
,
t
(
I
)
at
I
=
0
.
Since
F
N
∞
,
t
is concave down on
[
0
,
N
∞
]
,
its graph is below the tangent
line at the origin on
[
0
,
N
∞
]
.
Hence,
F
N
∞
,
t
(
(
)
<
)
(
,
N
∞
]
.
F
N
∞
,
t
I
0
I
on
0
Since
F
N
∞
,
t
(
(
N
∞
)=
γ
((
−
σ
)
N
∞
)
<
N
∞
.
)
>
,
(
)
(e)
F
N
∞
,
t
1
0
1
the graph of
F
N
∞
,
t
I
starts
=
N
∞
.
out higher than the diagonal and must cross it before
I
The concavity
(
)
property of
F
N
∞
,
t
I
(see (c)) implies that there is a unique positive fixed
point.
Let
−
γβ
t
φ
(
0
)
ℜ
0
,
t
=
−
σ
)
.
1
−
γ
(
1
1
In
ℜ
0
,
t
,
is the product of the average death adjusted infectious period in
(
1
−
γ
(
1
−
σ
))
generations;
is the proportion that can be invaded by the disease (survival first then
infection) and at time
t
,
γ
φ
(
−
β
)
is the maximum rate of infection of new recruits
and susceptible individuals per infective. Thus, at time
t
,
0
t
t
gives the average
number of secondary infections due to small initial infective individuals over their
life-time.
We note that
F
N
∞
,
t
(
ℜ
0
,
1 (respectively,
F
N
∞
,
t
(
)
>
)
<
>
0
0
1), is equivalent to
ℜ
1
0
,
t
<
(respectively,
1).
The threshold parameter (basic reproduction number),
ℜ
0
,
t
T
t
=
0
F
N
∞
,
t
(
0
)
,
−
1
ℜ
0
=
determines the long-term behavior of the disease in Model (
5
) , where the total
population is asymptotically constant. That is, we obtain that
ℜ
0
<
1 implies disease
extinction whereas
ℜ
0
>
1 implies disease persistence. We collect these results in
Theorem
4.1
.
Theorem 4.1.
In Model
(
5
)
,letN
(
t
)=
N
∞
and N
∞
≥
I
(
0
)
>
0
.
(a) If
ℜ
0
<
1
,
then
lim
t
→
∞
I
(
t
)=
0
. That is, the disease goes extinct.
(b) If
ℜ
0
>
1
,
then
lim
t
F
N
∞
,
t
◦···◦
F
N
∞
,
1
◦
F
N
∞
,
0
(
I
)
≥
η
>
0forsome
η
>
0
. That is,
→
∞
the infected population is uniformly persistent.
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