Biomedical Engineering Reference
In-Depth Information
{
}
{
}
(a)
0r
<<μ +
Min ,
σ
γ
,
r
= μ + γ and
r
> μ +
ax
σ
,
γ
, and
1
(b)
r and
r satisfy the inequalities:
2
r
+≥μ +σ+γ + β −ω
r
2
; r r
≥ μ +σ
(
)(r
+
r )+(
γ −σ
)(2
μ +σ+γ − μ + γ
)
(
) ;
.
1
3
1 3
1
3
(r
−
r )(r
−
− γ
)
≥ β
3
1
3
Then:
(i)
the application of the control law (12) to the SEIR model guarantees that the epi-
demics is asymptotically eradicated from the population while I(t)
≥
0
, E(t)
≥
0
and
, and
(ii)
the application of the control law (20) guarantees the epidemics eradication after a
finite time
S(t)
≥
0
∀∈
t
0
+
[
)
t , the positivity of the controlled SEIR epidemic model
∀∈
t
, t
and
f
[
)
that u(t)
=
V(t)
≥
1
∀∈
t
, t
so that u(t)
≥
0
∀∈
t
,
f
0
+
{
}
provided that the controller tuning parameters
λ
,
i , , 2
∈
, are chosen so that
i
{
}
, be the roots of the characteristic polynomial P(s) associated with
the closed loop dynamics (13).
Proof
.
(i)
On one hand, the epidemics asymptotic eradication is proved by following the
same reasoning that in Theorem 2. On the other hand, the dynamics (13) of the con-
trolled SEIR model can be written in the state space defined by
T
(r)
−
,
j
∈
, , 3
j
as in (16)-(17). From such a realization and taking into
account the first equation in (8) and that
x(t)
=
I(t)
E(t) S(t)
{
}
(r)
−
for
j
∈
, , 3
are the eigenvalues of
j
A , it follows that:
− −
=== +
rt
r t
c e
−
rt
I(t)
I (t)
y(t)
c e
c e
+
(22)
1
2
3
1
2
3
{
}
∀∈
t
for some constants
c for
j
∈
, , 3
being dependent on the initial con-
0
+
ditions y(0) , y(0
and y(0)
. In turn, such initial conditions are related to the initial
conditions of the SEIR model in its original realization, i.e., in the state space defined
by
[
]
T
{
}
x(t)
=
I(t) E(t) S(t)
via (8). The constants
c for
j
∈
, , 3
can be ob-
tained by solving the following set of linear equations:
I
(0)
=
y(0)
=
c
+
c
+
c
=
I(0)
1
2
3
E
(0)
=
y(0)
= −
(c r
+
c r
+
c r )
= −
(
+ γ
)I(0)
+ σ
E(0)
(23)
11
2 2
3 3
2
2
2
2
S(0)
=
y(0)
=
c r
+
c r
+
c r
=
(
+ γ
) I(0)
− σ
(2
+
+ γ
)E(0)
+
β
I(0)S(0)
11
2 2
3 3
1
where (8) and (22) have been used. Such equations can be compactly written as
p
RKM
⋅
= where:
111
c
I(0)
1
R
=
r
r
r
; K
=
c
; M
=
(
μ + γ
)I(0)
−σ
E(0)
(24)
p
1
2
3
2
222
2
r
r
r
c
(
μ+γ
) I(0)
−σ
(2
μ+σ+γ
)E(0)
+σβ
I(0)S(0)
123
3
1
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