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(Pandey and Stauffer 1990; Sieburg, McCutchan, Clay, Caballero, Ostlund, and
James 1990; Mosier and Sieburg 1994; Zorzenon dos Santos and Coutinho 2001).
Other possible approaches to study different aspects of the immune system dynamics
in a discrete space/time framework can be found in Atlan and Cohen (1989).
Hereafter, we introduce specific models of the HIV infection. The focus is on
C-ImmSim that can be seen as a synthesis of a number of different approaches.
8.3 Discrete Models of HIV Infection
As an extension of the already mentioned work with Stauffer (Pandey and Stauffer
1990; Pandey 1991), Pandey proposed a model that views a whole organism (e.g., a
person) as a three-dimensional cellular automaton. The entities represented are the
helper T cells, the killer T cells, macrophages, and virions (cells infected by HIV or
free virus particles).
The evolution of the automaton is determined by two sets of rules corresponding
to different infection modes (fast replication followed by rupture of the cell and
slower reproduction). The rules take the form of logical statements using the Boolean
operators on the binary codes of the entities. The simulation of the model produced
nontrivial results, but it did not show the characteristic “three-phase'' dynamics of
HIV (Pantaleo, Graziosi, and Fauci 1993). An interesting variation of that model
allowed the authors to study the viral load as a function of viral growth factor and
mutation rate (Ruskin, Pandey, and Liu 2002).
Perelson and Nelson presented various models of HIV infection (Perelson and
Nelson 1999) based on sets of simple ordinary differential equations (ODEs). By
means of clinical data, the value of the parameters is estimated and a classic stability
analysis is carried out for the different phases of the HIV infection. The authors
described the impact of drug therapies on the HIV dynamics by means of appropriate
changes in the equations. In a model extension, they also consider the role of macro-
phages and write the corresponding ODE. However, the model does not include
B cells. This choice is very common (see other models below) and it may appear like
a major omission in any description of the immune response.
Hershberg, Louzoun, Atlan, and Solomon (2001) proposed a discrete model that
acts in the Perelson “generalized shape space.” The mutations of HIV are represented
by propagation of the virions in the infinite-dimensional shape space. There is nei-
ther a spatial structure nor a distinction of the immune system components. The
status of each site of the (discrete) shape space is represented simply by the number
of virions existing with that shape and by the number of immune system cells that
recognize the same shape.
The virions of any shape proliferate exponentially killing immune system cells at
random until an immune system cell with that shape starts to proliferate and kill the
virions in turn. In the absence of virus diffusion in shape space (i.e., if there were no
mutations), this would stop the dynamics of the system, that is, the disease would be
defeated. The diffusion of the virus in the shape space is responsible for the con-
tinuation of the infection.
