Digital Signal Processing Reference
In-Depth Information
2000
g
(
t
)
3000
b
(
t
)
1600
2000
1200
800
1000
p
(
t
)
400
a
(
t
)
0
0
0 0.5
2
4
6
8
10
12
1 0
1 2
t
(s)
t
(s)
(a)
(b)
4000
a
(
t
)
3000
Fig. 8.13. Results of Simulations
1 and 2. (a) Number of antigens
g
(
t
) entering the human body.
(b) Time evolution of the
number of antigens
a
(
t
), plasma
cells
p
(
t
), and antibodies
b
(
t
)in
Simulation 1. (c) Same as (b) for
Simulation 2.
2000
p
(
t
)
b
(
t
)
1000
0
−1000
1 0
1 2
t
(s)
(c)
that the number of antigens increases linearly for the initial duration of 0.5 s.
Since the human body generates lymphocytes with a delay
τ
, which is 0.2 s in
our simulation, the number of plasma cells
p
(
t
) starts rising with a delay of 0.2 s.
After 0.5 s, no external antigens enter the human body. However, new antigens
are being reproduced by the already existing antigens present inside the human
body. As a result, the number of antigens
a
(
t
) keeps rising, even after 0.5 s.
After roughly 3 s, the respective strengths of lymphocytes and plasma cells is
high enough to impact the overall population of the antigens. The number of
antigens
a
(
t
) starts decreasing after 3 s. After 5.3 s, all antigens in the body
are destroyed. At this time, the body stops producing any further plasma cells.
After this stage, the number of plasma cells
p
(
t
) starts decreasing, as some of
these cells die naturally due to aging. As the number of plasma cells decreases,
the number of antibodies
b
(
t
) also decreases such that after 10 s no antibodies
are present in the simulation.
8.4.3.2 Simulation 2
Simulation 1 models successful eradication of the antigens. Let us now consider
the other extreme, where the antigens are lethal such that the human immune
system is unable to terminate the infection. The proportionality constants
β
and
related to the antibodies have lower values than those specified in Simulation
1. Also, the delay
τ
between the instant when the antigens are detected to the
instant when antibodies are produced is increased to 1 s. The simulated values
of the constants are given by
α =
0
.
1
,β=
0
.
1
,γ=
0
.
1
,=
0
.
3
,τ=
1
,λ=
0
.
1
,σ=
0.1,
and
η =
0
.
5
.
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