Digital Signal Processing Reference
In-Depth Information
at any time
t
is given by
d
b
d
t
=
p
(
t
)
− σ
a
(
t
)
− λ
b
(
t
)
.
(8.52)
8.4.2 Transfer function
Equations (8.49)-(8.52) describe the linearized model used to analyze the
human immune system. To develop the transfer function, we take the Laplace
transform of Eqs. (8.49)-(8.52). The resulting expressions can be expressed as
follows:
1
(
s
− α
)
[
G
(
s
)
− η
B
(
s
)];
number of antigens
A
(
s
)
=
(8.53)
number of lymphocytes
L
(
s
)
= β
A
(
s
);
(8.54)
−τ
s
(
s
+ γ
)
L
(
s
);
e
number of antigens
P
(
s
)
=
(8.55)
1
(
s
+ λ
)
[
P
(
s
)
− σ
A
(
s
)]
.
number of antibodies
B
(
s
)
=
(8.56)
In Eqs. (8.53)-(8.56), variables
A
(
s
),
G
(
s
),
L
(
s
),
P
(
s
), and
B
(
s
) are, respec-
tively, the Laplace transforms of the number of antigens
a
(
t
) present within the
human body, the number of antigens
g
(
t
) entering the human body, the number
of lymphocytes
l
(
t
) within the blood, the total number of antigens
p
(
t
) within
the human body, and the number of antibodies
b
(
t
) in the blood. Assuming the
number of antigens
g
(
t
) entering the human body to be the input and the number
of antibodies
b
(
t
) produced to be the output, the human immune system can be
modeled by the schematic diagram shown in Fig. 8.11(a). Figure 8.11(b) is the
simplified version of Fig. 8.11(a), which yields the following transfer function
for the human immune system:
−τ
s
− σ
(
s
+ γ
)
(
s
− α
)(
s
+ λ
)(
s
+ γ
)
+ η
[
β
e
M
(
s
)
(1
+ η
M
(
s
))
β
e
T
(
s
)
=
=
−τ
s
− σ
(
s
+ γ
)]
.
(8.57)
8.4.3 System simulations
The simplified model of the human immune system is still a fairly complex
system to be analyzed analytically. The characteristic equation of the human
immune system is not a polynomial of
s
, therefore evaluation of its poles is
difficult. In this section, we simulate the human immune system using the
simulink toolbox available in M
ATLAB
.
8.4.3.1 Simulation 1
In simulink, a system is simulated using a block diagram where the subblocks
represent different subsystems. Figure 8.12 shows the simulink representation of
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