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discrete models of the immune system can be found in Perelson and Weisbuch
(1997), Zorzenon dos Santos (1999), and Forrest and Hofmeyr (2000).
The discrete models of the immune system in use at this time (Zorzenon dos
Santos 1999) can be classified, from the technical viewpoint, as Boolean networks,
cellular automata, and lattice gases. We now describe briefly some of these models.
Kaufman, Urbain, and Thomas (1985) proposed one of the first applications of dis-
crete automata in the study of the adaptive immune response. The original model
( KUT model ) considered five types of cells and molecules but, for the sake of sim-
plicity, we describe here a simplified version (Fig. 1). This “submodel” considers
antibodies (Ab), helper cells (T H ), lymphocytes, B cells (B), and antigens (Ag). Each
entity is represented by a two-valued variable denoting high/low concentration. The
rules governing the network of interdependencies/interactions among these variables
are expressed by logical operations. The application of the rules is iterated over dis-
crete steps and the dynamics is observed. The rules are:
Ab t+1 = Ag t AND B t AND T H t
B t+1 = T H t AND (Ag t OR B t )
(4)
Ag t+1 = Ag t AND NOT Ab t
T H t+1 = T H t OR Ag t
where AND, OR, and NOT are the standard logical operators. These rules should be read
as follows: Abs are produced at time step t+1 by B cells if (at time t ) the antigen is pre-
sent together with stimulating lymphocytes and T H cells; the B cells grow if T H cells and
either antigens or other B cells are present; the antigen proliferates in the absence of
Fig 1. Simplied KUT Model with antibodies (Ab), T helper cells (T), lymphocytes, (B) cells,
and antigens (Ag).
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