Spatial Aspects of HIV Infection - Mathematical Methods and Models in Biomedicine - page 14

showed that the speed of HIV progression depends on the initial viral distribution
in the considered grid. Therapy has to inhibit viral accumulation in local regions in
order to be effective.
The models proposed by Funk et al. [ 34 ] and Graziano et al. [ 38 ] aim to include
spatial information into the analysis of HIV infection. However, they still rely on
ordinary or PDE. PDEs are more computationally challenging than ODEs, making it
more difficult to determine general behaviors of the system. Furthermore, similar to
ODE models, they only provide an average or mean-field description of the system
behavior [ 7 ]. Because of this, there are only a limited number of studies applying
PDEs to within-host dynamics. In virus dynamics, PDE models have also been used
to develop age-structured models of infection that keep track of the time a cell has
been infected, i.e., its age of infection [ 2 , 61 , 86 ]. These models add realism in that
they can account for the fact that when a cell is first infected no virus is produced
until a number of steps in the viral life cycle have been completed. Thus they allow
the rate of virus production to depend on the age of an infected cell. The models can
also allow the death rate of an infected cell to depend on its age.

4.3 Pair Approximation

Ordinary and PDE are so-called mean-field approximations of the dynamical system
that both kinds of model systems try to capture. In short, the solutions of both model
systems are based on the following assumption: If our representation of space is a
lattice with a total of S sites, where the status of each site i

∈ S is denoted by σ i ,then

the probability that the two neighboring sites i , j ∈ S with status σ i σ j change their

σ i σ j is given by status into

r σ ( σ i σ j → σ i σ j )= r σ ( σ i → σ i ) r σ ( σ j → σ j ) . (5)

This assumption, used in mean-field models, means that each site is considered
to change its status independently of the others. However, as local interactions at
different sites of the lattice might occur, this approximation might be too crude
in some circumstances. The so-called pair approximation approach represents an
improvement to the mean-field approximation by considering local interactions
between neighboring sites. In this case the probability that the two neighboring sites
i

σ i σ j depends on the status of all
pairs of neighboring sites in the direct neighborhood of i and j , E ij :

, j ∈ S with status σ i σ j change their status into

r σ ( σ i σ j → σ i σ j )= r σ ( σ i σ j → σ i σ j | σ E ij ) . (6)

The application of this kind of modeling approach to HIV infection dynamics
might be relevant especially if we consider each grid site to represent a spe-
cific target cell and also include the fact that HIV can spread by cell-to-cell