Digital Signal Processing Reference
In-Depth Information
Fig. 8.10. Schematic models for
the dc motor with transfer
function
H
′
(
s
).
T
d
−
k
m
+
1
+
+
+
V
a
(
t
)
applied
voltage
w
(
t
)
output
angular velocity
sL
a
+
R
a
sJ
+
r
−
armature
load
V
emf
(
t
)
k
f
8.4 Immune system in humans
We now apply the Laplace transform to model a more natural system, such as
the human immune system. The human immune system is non-linear but, with
some assumptions, it can be modeled as a linear time-invariant (LTI) system.
Below we provide the biological working of the human immune system, which
is followed by an explanation of its linearized model.
Human blood consists of a suspension of specialized cells in a liquid, referred
to as plasma. In addition to the commonly known erythrocytes (red blood
cells) and leukocytes (white blood cells), blood contains a variety of other
cells, including lymphocytes. The lymphocytes are the main constituents of
the immune system, which provides a natural defense against the attack of
pathogenic microorganisms such as viruses, bacteria, fungi, and protista. These
pathogenic microorganisms are referred to as antigens. When the lymphocytes
come into contact with the foreign antigens, they yield antibodies and arrange
the antibodies on their membrane. The antibody is a molecule that binds itself
to antigens and destroys them in the process. When sufficient numbers of anti-
bodies are produced, the destruction of the antigens occurs at a higher rate than
their creation, resulting in the suppression of the disease or infection. Based on
this simplified explanation of the human immune system, we now develop the
system equations.
8.4.1 Mathematical model
The following notation is used to develop a mathematical model for the human
immune system:
g
(
t
)
=
number of antigens entering the human body;
a
(
t
)
=
number of antigens already existing within the human body;
l
(
t
)
=
number of active lymphocytes;
p
(
t
)
=
number of plasma cells;
b
(
t
)
=
number of antibodies.
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