Biomedical Engineering Reference
In-Depth Information
(
)
We denote by
ω
the common period of the functions
γ
t
,andset
j
ω
1
ω
=
(
)
z
z
t
d
t
0
for any
ω
-periodic function
z
(
t
)
. Suppose
R
0
for Eqs. (
5a
)and(
5b
) is given as a
function
R
0
=
R
0
(
γ
1
,...,
γ
k
)
(10)
and set
R
0
(
t
)=
R
0
(
γ
1
(
t
)
,...,
γ
k
(
t
))
,
[
R
0
]=
R
0
(
γ
1
,...,
γ
k
)
.
(11)
If
F
=
diag
(
F
1
,...,
F
m
)
,
V
=
diag
(
V
1
,...,
V
m
)
,
then
(
)
F
i
t
R
(
t
)=
max
1
)
.
V
i
(
t
≤
i
≤
m
As shown in [
12
], in this case
m
F
i
R
0
=
max
1
V
i
,
≤
i
≤
so that
R
0
=[
R
0
]
;
(12)
but, in general,
R
0
=[
R
0
]
. For example, in a model of tuberculosis it was shown,
in [
8
], that
R
0
<
[
when both vary in an interval containing 1,
as shown in Fig.
1
a, and in a model of Dengue fever it was shown, in [
12
], that
R
0
>
[
R
0
]
for
R
0
and
[
R
0
]
in an interval containing 1; see Fig.
1
b.
Since it is much harder to compute
R
0
than to compute
R
0
]
for
R
0
and
[
R
0
]
[
R
0
]
, the question arises
whether one can use
[
R
0
]
to estimate
R
0
. We would especially want to derive such
estimates when
or
R
0
are equal to (or near to) the value 1, a value associated
with stability/instability of the DFE. If we can show, for example, that
[
R
0
]
[
R
0
]
<
1 implies that
R
0
<
1
(13)
and
[
R
0
]
>
1 implies that
R
0
>
1
,
(14)
then it would follow that Eq. (
12
) holds when either side of the equation is equal
to 1. This situation is illustrated in Fig.
2
.
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