Biomedical Engineering Reference
In-Depth Information
{
}
from (36). The controller tuning parameter
λ
for
i
∈
, , 2
are related to the roots
i
{
}
(r)
−
, for
j
∈
, , 3
, of the closed-loop characteristic polynomial P(s) , see Re-
j
mark 1, by:
λ=
rr r ;
λ=
rr
+
rr +r r ;
λ= + +
r
r
r
.
(38)
0
1 2 3
1
1 2
1 3
2 3
2
1
2
3
{
}
Then, the assignment of r for
j
∈
, , 3
such that the constraints (a) and (b) are
fulfilled implies that:
λ+ω−μ −σ− γ − β ≥
μ + γ
3
2
0
2
2
(
)
+
(2
μ +σ+γμ+σ +λ −λ
)(
)
(2
μ +σ+γ
)
≥
0
.
(39)
1
2
μσβ +λ −λ μ + γ +λ
(
)
(
μ + γ
)
2
− μ + γ
(
)
3
= μσβ ≥
0
0
1
2
[
)
Then,
R(t )
≥
0
from (39) and (37). The facts that R(t)
≥
0
∀∈
t
, t
,
R(t )
=
0
0
0
0
[
]
and
via complete induction.
On the other hand, from (19) and (20), it follows:
3
R(t )
≥
0
imply that R(t)
≥
0
∀∈
t
, t
0
f
2
μσβ− μ+γ
(
)
+λ −λ
(
μ+γ +λ
)
(
μ+γ
)
ω
(3
μ+σ+ γ−ω−λ
2
)
u(t)
=
0
1
2
+
R(t)
−
2
S(t)
μσβ
μ
N
μ
N
(40)
2
σ
E(t)S(t)
β
(
μ+γ
)
+
(2
μ+σ+γ
)(
μ+σ +λ −λ
)
(2
μ+σ+γ
)
E(t)
+
−
I(t)S(t)
+
1
2
2
μ
N
I(t)
μ
N
μβ
I(t)
[
]
∀∈
t
, t
by taking into account that S(t)
+++ =
E(t)
I(t)
R(t)
N
. Moreover:
f
μσβ− μ+γ
(
)
3
+λ −λ μ+γ +λ
(
)
(
μ+γ
)
2
λ +ω− μ−σ− γ−β
3
2
u(t)
≥
0
1
2
+
2
S(t)
μσβ
μ
N
(41)
2
(
μ+γ
)
+
(2
μ+σ+γ μ+σ +λ −λ
)(
)
(2
μ+σ+γ
)
E(t)
[
]
+
1
2
∀ ∈
t
0, t
f
μβ
I(t)
[
]
where the facts that 0
<δ≤
I(t)
≤
N
, E(t)
≥
0
, S(t)
≥
0
and R(t)
≥
0
∀∈
t
, t
f
have been used. If the roots of the polynomial P(s) satisfy the conditions in (a) and
(b), it follows from (41) that:
λ+ω−μ−σ−γ−β
≥+
3
2
u(t)
1
2
S(t)
μ
N
(42)
(
μ+γ
)
2
+
(2
μ+σ+γ μ+σ +λ −λ
)(
)
(2
μ+σ+γ
)
E(t)
1
2
+
≥
1
μβ
I(t)
[
]
∀∈
t
, t
by taking into account (39) and the non-negativity of S(t) , E(t) and
f
[
]
from (20) and (42). ***
+
I(t)
∀∈
t
, t
. Finally, it follows that u(t)
≥
0
∀∈
t
f
0
4
Simulation Results
An example based on an outbreak of influenza in a British boarding school in early
1978 [2] is used to illustrate the presented theoretical results. Such an epidemic can be
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