Biomedical Engineering Reference
In-Depth Information
described by the SEIR mathematical model (1)-(4) with
μ=
−
1
70 years
=
25550 days
,
β=
1.66 per day
,
σ=γ=
−
1
−
1
2.2 days
and
ω=
−
1
15 days
. A total population of
N1000 bo s
=
is considered with the initial condition given by S(0)
=
800 boys
,
. Two sets of simulation results
are presented to compare the evolution of the SEIR model populations in two differ-
ent situations, namely: when no vaccination control actions are applied and if a con-
trol action based on the feedback input-output linearization approach is applied.
E(0)
=
100 boys
, I(0)
=
60 boys
and R(0)
=
40 boys
4.1
Epidemic Evolution without Vaccination
The time evolution of the respective populations is displayed in Fig. 1. The model
tends to its endemic equilibrium point as time tends to infinity. There are susceptible,
infected and infectious populations at such an equilibrium point. As a consequence, a
vaccination control action has to be applied in order to eradicate the epidemics.
800
700
600
R(t)
500
400
S(t)
300
200
I(t)
100
E(t)
0
0
10
20
30
40
50
60
Fig. 1.
Time evolution of the individual populations without vaccination
Time (days)
4.2
Epidemic Evolution with a Feedback Control Law
The control law given by (20)-(21) is applied with
δ=
0.001
and the free-design
{
}
controller parameters
, being chosen so that the roots of the cha-
racteristic polynomial P(s) associated with the closed-loop dynamics (13) are
r
λ
, for
i , , 2
∈
i
are obtained from
(38). The time evolution of the respective populations is displayed in Fig. 2 and the
vaccination function in Fig. 3.
The vaccination control action achieves the control objectives as it is seen in Fig. 2.
The infection is eradicated from the population since both infectious and infected
populations converge rapidly to zero. Also, the susceptible population converges to
zero while the removed-by-immunity population tracks asymptotically the whole
population as time tends to infinity. Such a result is coherent with the result proved in
Theorem 3. Moreover, the positivity of the system is maintained for all time as it can
−=−γ
,
−=−μ + γ
r
(
)
and
−=−μ + γ
r
(2)
. Such values for
λ
i
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