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
)
0 , t =
σ ) .
1
γ (
1
1
In
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 (
0
,
1 (respectively, F N , t (
) >
) <
>
0
0
1), is equivalent to
1
0
,
t
<
(respectively,
1).
The threshold parameter (basic reproduction number),
0
,
t
T
t = 0 F N , t ( 0 ) ,
1
0 =
determines the long-term behavior of the disease in Model ( 5 ) , where the total
population is asymptotically constant. That is, we obtain that
0 <
1 implies disease
extinction whereas
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
0 <
1
,
then lim
t
I
(
t
)=
0 . That is, the disease goes extinct.
(b) If
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.
 
Search WWH ::




Custom Search