Spatial Aspects of HIV Infection - Mathematical Methods and Models in Biomedicine - page 15

transmission [ 117 ]. In biology the pair approximation has so far only been applied
to epidemiological and ecological problems [ 54 , 64 ], and in these applications
gives consistently better results than the mean-field approximation in describing
the dynamics of a stochastic system.
The pair approximation can be thought of as being in between the mean-field
approximation of a dynamical system and its stochastic and individual simulation
as done by agent-based models, which are introduced below.

5 Cellular Automata and Agent-Based Models

Several stochastic in silico simulation tools have been applied to the study of viral
dynamics within a host [ 7 , 9 , 18 , 120 , 130 ]. In silico simulations tools, such as
cellular automata and agent-based models, allow one to model individual units of
investigation, e.g., viruses, cells, patients, or animals, in a spatially explicit way.
These types of simulation tools have been used to study an epidemic caused by a
pathogen inside a population or on an agricultural farm (see, e.g., [ 57 ] and references
therein).
Cellular automata and other agent-based models have three main advantages
in describing complex biological systems compared to ODE or PDE models:
First, the behavior of different elements in a complex biological system can be
sometimes more easily described in terms of rules of interaction rather than
by specific rate constants that have to be determined and that usually average
the individual behavior over the whole population. Second, due to these rules,
these types of models easily incorporate nonhomogeneous spatial information.
Lastly, due to their implementation, cellular automata and agent-based models
usually include stochasticity. In ODE and PDE-models, stochasticity needs to be
explicitly implemented by assuming specific probability distributions for certain rate
constants. However, the ODE and PDE models will still only represent the average
dynamics of the modeled system, while in agent-based models the same starting
conditions can lead to different outcomes. Agent-based models add an additional
layer of complexity to the mean- field approaches by trying to mimic the underlying
biological system in more detail.

5.1 Cellular Automata

Cellular automata can be considered as a specific type of agent-based model. In a
cellular automaton, cells are organized in a 2D or 3D grid. Each site on the grid
represents a cell or a comparable predefined unit. The position of the cell is fixed
with a well-defined distance to its neighboring cells (see also Fig. 4 ). According to
environmental conditions and predefined rules, the status of a cell changes during
a simulation run. Cellular automata were first introduced by von Neumann and