Biomedical Engineering Reference
In-Depth Information
{
}
appropriate choice of the free-design controller parameters
.
Then, the control law (12) guarantees the exponential stability of the controlled SEIR
model (6)-(11) while achieving the eradication of the infection from the population as
t →∞. Moreover, the SEIR model (1)-(4) has the properties: E(t) , I(t) , S(t)I(t)
and
λ>
0
, for
i , , 2
∈
i
[
]
S(t)
+
R(t)
=
N
−
E(t)
+
I(t)
are bounded for all time, E(t)
→
0
, I(t)
→
0
,
(
)
S(t)
+
R(t)
→ and S(t)I(t)
N
→
0
exponentially as t →∞, and
I(t)
=
o 1 S(t)
.
Proof.
The dynamics of the controlled SEIR model (13) can be equivale
nt
ly rewritten
with the state equation (16)-(17) and the output equation y(t)
=
Cx(t)
, where
[
]
C100
=
, by taking into account that y(t)
=
I(t)
, y(t)
=
E(t)
and y(t)
=
S(t)
.
T
The initial condition
in such a realization is bounded
since it is related to x(0) via the coordinate transformation (8) and x(0) is assumed
to be bounded. The controlled SEIR model is exponentially stable since the eigenva-
lues of the matrix A are the roots
x(0)
=
I (0)
E(0)
S(0)
{
}
of P(s) which are as-
sumed to be in the open left-half plane. Then, the state vector x(t) exponentially
converges to zero as t →∞ while being bounded for all time. Moreover, I(t) and
E(t) are also bounded and converge exponentially to zero as t →∞ from the boun-
dedness and exponential convergence to zero of x(t) as t →∞ according to the
coordinate transformation (8). Then, the infection is eradicated from the host popula-
tion. Furthermore, the boundedness of S(t) R(t+ follows from that of E(t) and
I(t) , and the fact that the total population is constant for all time. Also, the exponen-
tially convergence of S(t)
−< for
j
r0
j
∈
, , 3
+
R(t)
to the total population as t →∞ is derived from the
fact that S(t)
and the exponential convergence to
zero of I(t) and E(t) as t →∞. Finally, from the third equation of (8), it follows
that S(t)I(t) is bounded and it converges exponentially to zero as t →∞ from the
boundedness and convergence to zero of I(t) , E(t) and x(t) as t →∞. The facts
that I(t)
+++ =
E(t)
I(t)
R(t)
N
∀∈
t
0
+
(
)
→
0
and S(t)I(t)
→
0
as t →∞ imply directly that
I(t)
=
o 1 S(t)
. ***
Remark 4
. Theorem 2 implies the existence of a finite time instant t after which the
infectious disease is eradicated if the vaccination control law (20) is used instead of
(12). Concretely, such an existence derives from the fact that I(t)
→
0
as t →∞ via
the application of the control law (12).
***
Theorem 3.
Assume an initial condition for the SEIR model satisfying R(0)
≥
0
,
x(0)
∈
, i.e., I(0)
3
0
≥
0
,
E(0)
≥
0
and S(0)
≥
0
, and the constraint
+
S(0)
+++ =. Assume also that some strictly positive real numbers
r for
E(0)
I(0)
R(0)
N
{
}
j
∈
, , 3
are chosen such that:
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