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and continuous models is in the choice of the numerical space chosen to represent,
for instance, the time scale. It is pretty different to let time change continuously, and
to consider quanta of time in which a number of events may happen simultaneously.
The choice of the mathematical space determines the mathematical techniques one is
able to use, the kind of solution that can be derived and, in general, the difficulty in
carrying on a nontrivial analysis. A deterministic process is completely predictable
provided an exact knowledge is available about the initial conditions whereas a sto-
chastic process, by definition, can be described only by means of the methods of
statistics and probability theory. Stochastic components are often included to take
into account that we do not have perfect knowledge either of the initial conditions or
of the process that a system follows (or both). Therefore, stochastic models seem
more suitable to describe biology but more difficult to analyse.
All existing models of the immune system derive from either the
clonal selection
theory
or the
idiotypic network theory
. Nowadays immunologists consider these as
two independent and perhaps complementary theories (Zorzenon dos Santos 1999).
However, while clonal selection theory is believed to be the fundamental one in the
present knowledge of the immune system, the idiotypic network theory is believed
correct as far as the existence of anti-idiotypic reactions but probably not relevant in
determining the immune response (Anachini and Mortarini 1999).
Both immunological theories inspired a number of continuous models (Perelson
1988a; Perelson 1988b) whereas most of the discrete models are based on Jerne's
(
idiotypic network
) theory (Jerne 1973; Jerne 1974). On the other hand, the Celada-
Seiden model, which
may
include both theories, rests its foundation on the clonal
selection theory (Celada and Seiden 1992).
Basically, continuous mathematical models try to represent the dynamics of cer-
tain quantities, like the number of cells or the concentration of a molecule in a com-
partment of the immune system, as a function of the birth and death rate and of other
variables. A simple example is the so-called AB model (Segel and Perelson 1991)
used in theoretical studies of the immune network. It describes the dynamics of the
i
-
th clone (
i
=1,…,M) of B-lympocytes (
B
i
(t)
) and antibodies (
A
i
(t)
) by means of the
following two equations:
dB
i
/d
t
=
m
+
B
i
(pf(h
i
)
-
dB)
(1)
dA
i
/
d
t
=
sB
i
f(h
i
)
-
d
C
h
i
A
i
-
d
A
A
i
(2)
Here
m
is the rate of maturation of B cells from the bone marrow,
p
is the division
rate of activated lymphocytes,
d
characterizes the death rate,
s
is the secretion rate of
antibodies, and
d
C
is the elimination rate of antibody-antigen complexes. Idiotypic
interactions are mediated through a field
h
i
which is determined by the concentration
of antibodies
A
j
and by the affinity
J
ij
:
M
∑
=
h
=
J
A
(3)
i
ij
i
j
1
