Biomedical Engineering Reference
In-Depth Information
(
)
12
(
)
12
where (25), (26),
and the constraints in (a) and
(b) have been used. In particular, the coefficient multiplying to I(0) in (29) is non-
negative if
Gr,r
=−μ − γ <
r
0
,
Fr,r
=
0
1
r and
r satisfy the third inequality of the constraints (b) by taking into
account
σβ
1
S(0)
=σβ
S(0) N
≤σβ
and S(0)
≤
N
. This later inequality is directly
implied by N
=+ + +
I(0)
E(0)
S(0)
R(0)
with I(0)
≥
0
, E(0)
≥
0
, S(0)
≥
0
and
R(0)
≥
0
. Finally, for the case (iv), i.e., if
2
c0
<
and
3
c0
<
, it follows that:
(
)
(
)
[
]
−
rt
−
r t
−
rt
−
rt
−
r t
−
rt
−
rt
−
rt
I(t)
=−−
I(0)
c
c
e
+ + =
c e
c e
I(0)e
+
c
e
−+
e
c
e
−≥
e
0
(30)
1
2
3
1
2
1
3
1
2
3
2
3
2
3
−
rt
−
−≤
rt
−
rt
−
−≤
rt
∀∈
t
, where
e
e
0
and
e
e
0
, since
1
rr r
<<
, and
2
1
3
1
0
+
2
3
I(0)
=++ ≥
c
c
c
0
have been taken into account. In summary, I(t)
≥
0
∀∈
t
1
2
3
0
+
if all partial populations are initially non-negative and the roots
(r)
−
, for
j
{
}
, of the closed-loop characteristic polynomial satisfy the constraints in (a)
and (b). On the other hand, one obtains from (22) and the reverse coordinate trans-
formation to (8) that:
j
∈
, , 3
1
1
3
[
]
−
rt
E(t)
=
E(t)
+ μ+γ
(
) I (t)
=
c (
μ+γ−
r )e
j
j
j
σ
σ
j1
=
S(t)
+ μ+σ μ+γ
(
)(
)I
(
t)
+
(2
μ+σ+γ
)E(t)
S(t)
=
σβ
I(t)
(31)
1
3
2
−
rt
c
r
−
(2
μ+σ+γ
)r
+ μ+σ μ+γ
(
)(
) e
j
j
j
j
j1
=
=
σβ
I(t)
1
from the facts that E(t)
=
I(t)
and S(t)
=
I (t)
. If one fixes
r
= μ + γ
then:
1
−
rt
−
rt
E(t)
=
c (
μ + γ −
r )e
+
c (
μ + γ −
r )e
1
3
1
1
3
3
σ
c
2
11
r
−
(2
μ+σ+γ
)r
+ μ+σ μ+γ
(
)(
) e
−
rt
1
1
S(t)
=
(32)
σβ
I(t)
1
2
33
−
rt
c
r
−μ+σ+γ +μ+σμ+γ
(2
)r
(
)(
) e
3
+
3
σβ
I(t)
1
where the function H :
defined as:
+
→
2
H(v)
v
−
(2
μ +σ+γ
)v
+ μ +σ μ + γ
(
)(
)
(33)
is zero for
vr
==μ + γ
has been used. From the first equation in (32), it follows that
2
c(
μ + γ −=σ −μ + γ −
r)
E(0) c(
r)
and then:
3
3
1
1
(
)
−
rt
−
rt
−
rt
c(
μ+γ−
r) e
−
e
+σ
E(0)e
1
3
3
(34)
E(t)
=
1
1
≥
0
σ
∀∈
t
by applying such a relation between
c and
c in (32) and by taking into
0
+
−
rt
account that
c(
μ + γ −>
r)
0
, E(0)
≥
0
and
e
−
rt
−≥
e
0
∀∈
t
since
rr
<
.
1
3
1
1
0
+
1
3
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