Biomedical Engineering Reference
In-Depth Information
S(t)I(t)
E(t)
=− μ+σ
(
)E(t)
+β
(2)
N
I(t)
=− μ+γ
(
)I(t)
+σ
E(t)
(3)
R(t)
=− μ+ω
(
)R(t)
+γ
I(t)
+μ
NV(t)
(4)
subject to initial conditions S(0)
≥
0
, E(0)
≥
0
, I(0)
≥
0
and R(0)
≥
0
under a vac-
[
)
cination function
V:
, with
→
. In the above SEIR model,
0,
∞∩
0
+
0
+
0
+
N0
>
is the total population at any time instant
t
∈
,
is the rate of deaths
μ >
0
0
+
and births from causes unrelated to the infection,
ω≥
0
is the rate of losing immuni-
ty,
is the transmission constant (with the total number of infections per unity of
time at time t being
β >
0
−
σ>
1
−
1
are, respectively, the
average durations of the latent and infective periods. The total population dynamics
can be obtained by summing-up (1)-(4) yielding:
β
S(t)I(t) N
) and,
0
and
γ
>
0
N(t)
=+ ++ =
S(t)
E(t)
I(t)
R(t)
0
(5)
so that the total population
N(t)
. Then, this model is
suitable for epidemic diseases with very small mortality incidence caused by infection
and for populations with equal birth and death rates so that the total population may
be considered constant for all time.
is constant
∀∈
t
=
N(0)
=
N
0
+
3
Vaccination Strategy
An ideal control objective is that the removed-by-immunity population asymptotically
tracks the whole population
. In this way, the joint infected plus infectious population
asymptotically tends to zero as t
, so the infection is eradicated from the popula-
tion. A vaccination control law based on a static-state feedback linearization strategy
is developed for achieving such a control objective. This technique requires a nonli-
near coordinate transformation, based on the Lie derivatives Theory [14], in the sys-
tem representation.
The dynamics equations (1)-(3) of the SEIR model can be equivalently written as
the following nonlinear control affine system:
(
→∞
)
(
)
(
)
x(t)
=
f
x(t)
+
g x(t) u(t) ; y(t)
=
h x(t)
(6)
[
]
T
where
are,
respectively, considered as the output signal, the input signal and the state vector of
the system
y(t)
=∈
I(t)
,
u(t)
=∈
V(t)
and
x(t)
=
I(t) E(t) S(t)
∈
3
0
0
+
0
+
+
∀∈
t
and R(t)
=−
N
S(t)
−
E(t)
−
I(t)
has been used, with:
0
+
−μ+γ
(
)I(t)
+σ
E(t)
(
)
3
f x(t)
=
− μ +σ
(
)E(t)
+β
I(t)S(t)
∈
1
(
)
(
)
(7)
−ω
I(t)
+
E(t)
+ μ + ω
(
) N
−
S(t)
−β
I(t)S(t)
1
(
)
[
]
T
(
)
3
0
g x(t)
=
0 0
−μ
N
∈
; h x(t)
=
I(t)
∈
−
0
+
Search WWH ::
Custom Search