Spatial Aspects of HIV Infection - Mathematical Methods and Models in Biomedicine - page 18

a lower effective infection rate than if infection were initiated with single infected
cells where all the neighbors are infectable. However, the effective infection rate can
be kept high if one allows for occasional jumps of the infection to areas where most
of the cells are still uninfected. With the same argument, the type of replacement of
dead cells influences the infection dynamics. A global regeneration rule in which
a dead cell is replaced by an uninfected cell placed randomly on the grid, as
assumed in the basic ODE model [Eq. ( 1 )], would allow for the appearance of
new uninfected cells in completely infected areas which would increase the number
of infected cells. On the other hand, if the replacement of dead cells depends on
the proliferation of uninfected cells in the direct neighborhood of a dead cell, the
infection propagates in thin circular waves as this regeneration rule limits the growth
of the infection by starving it of target cells. In this scenario, the replacement of dead
cells left behind by the propagating wave of infection can only occur if immune cells
detect the infection and breach the infection wave. Beauchemin [ 8 ] could show that
the fraction of dead cells over time generated by this model closely agreed with
experimentally observed curves [ 8 , 14 ] in contrast to the result obtained by an ODE
model where the replacement only depends on the number of uninfected cells and
not on their location.

5.2 Agent-Based Models

Cellular automata represent a specific type of agent-based model where cells,
which can change their status, are positioned at spatially fixed sites. However, the
simulation of cells does not need to be constrained on a regular grid with cells of
regular shape. Rather, each cell can be followed as an individual agent. As advanced
experimental techniques, such as two-photon microscopy, provide additional in-
sights into the dynamics of infection processes in vivo and computational power
increases, the simulation environments described above have become more and
more detailed. Bogle and Dunbar [ 15 ] used a simulation environment where cells
move on a grid to simulate T-cell motility in a lymph node. The authors show that
this simulation environment was able to reproduce experimental observations. Graw
and Regoes [ 37 ] used a similar approach to study how the killing of infected cells
by cytotoxic CD8 + T cells in lymphoid organs is influenced by spatial aspects and
what type of mathematical models should be used to analyze experimental data. In
the following, we present several novel types of agent-based models that have been
applied to the study of HIV infection dynamics and other aspects in immunology.

Agent-Based Models for HIV

Lin and Shuai [ 65 ] extended the modeling approach originally presented by
Zorzenon dos Santos and Coutinho [ 130 ] to study the interplay between HIV disease
dynamics and the immune response. In their simulations, they include HIV virus