Biomedical Engineering Reference
In-Depth Information
Remark 2 . The control (12) may be rewritten as:
3
2
(
μ+ω σβ− μ+γ
)
(
)
+λ −λ μ+γ +λ
(
)
(
μ+γ
)
ω
[
]
0
1
2
u(t)
=
I(t)
+
E(t)
μσβ
μ
N
(3
μ+σ+ γ−λ
2
)
σ
E(t)S(t)
β
2
S(t)
+
I(t)S(t)
(19)
2
μ
N
μ
N
I(t)
μ
N
2
(
μ+γ
)
+
(2
μ+σ+γ μ+σ +λ −λ
)(
)
(2
μ+σ+γ
) E(t)
+
1
2
μβ
I(t)
by using (8) and (15).
***
Remark 3 . The control law (12) is well-defined for all
x
except in the surface
3
0
+
. However, the infection may be considered eradicated from the population once
the infectious population strictly exceeds zero while it is smaller than one individual,
so the vaccination strategy may be switched off when 0
I0
=
< <. This fact im-
plies that the singularity in the control law is not reached. i.e., such a control law is
well-defined by the nature of the system. In this sense, the control law
I(t)
1
u(t) for 0
t ≤≤
t
t
f
u(t)
=
(20)
p
0 for t
>
f
may be used instead of (12) in a practical situation. The signal u(t) in (20) is given
by the linearizing control law (12) while t denotes the eventual time instant after
which the infection propagation may be assumed ended. Formally, such a time instant
is defined as:
{
}
+
t
Mint
I(t )
< δ
f r ome 0
< δ <
1
.
(21)
f
0
f
Then, the control action is maintained active while the infection persists within the
population and it is switched off once the epidemics is eradicated.
***
3.1
Control Parameters Choice
The application of the control law (12), obtained from the exact input-output lineari-
zation strategy, makes the closed-loop dynamics of the infectious population be given
by (13). Such a dynamics depends on the control parameters
{
}
.
Such parameters have to be appropriately chosen in order to guarantee the following
suitable properties: (i) the stability and positivity of the controlled SEIR model and
(ii) the eradication of the infection, i.e., the asymptotic convergence of I(t) and E(t)
to zero as time tends to infinity. The following theorems related to the choice of the
controller tuning parameter in order to meet such properties are proven.
Theorem 2. Assume that the initial condition
λ
, for
i , , 2
i
[
]
T
3
0
x(0)
=
I(0)
E(0) S(0)
is
+
{
}
bounded and all roots
of the characteristic polynomial P(s)
associated with the closed-loop dynamics (13) are of strictly negative real part via an
(r)
for
j
, , 3
j
 
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