Spatial Aspects of HIV Infection - Mathematical Methods and Models in Biomedicine - page 11

, β , δ , ,

( d c ) is the same on each site of the grid, the dynamics differed from the one

observed in the basic ODE model, especially early in the infection where the spatial
model predicts a rapid increase of the virus load followed by a slower increase as
target cells are depleted in some regions of the grid whereas the nonspatial model
only shows a one-phase increase (see Fig. 2 in [ 34 ]). The authors suggest that due
to this deviation parameters estimated using the basic ODE model might contain
systematic errors [ 34 ].
On a heterogeneous grid, where the rate constants for each grid site are sampled
randomly from a uniform distribution, spatial coupling between the different sites
can change the equilibrium properties of target cells compared to the basic model
of viral dynamics. However, increased spatial coupling between the sites leads to a
more averaged outcome similar to the mean-field approximation given by a model
with a well-mixed assumption for the different populations: If spatial coupling is
high, the transport rate m V for virions between the different sites has to be high,
indicating that the average virus load in the neighborhood of a grid site has a higher
influence on the equilibrium viral load at this site than more distant sites. Increased
spatial coupling will lead to a smoothed viral load between the different sites ([ 34 ]
and Fig. 4 ).
In addition to their basic model, the authors extended their model by considering
an immune response targeted against infected cells. Similar to the virions, immune
cells were assumed to move among the different sites of the spatial grid. Their
spatially explicit model equilibrated much faster, with more damped oscillations,
compared to a version of the basic model of infection dynamics that incorporated an
immune response [ 91 ]. The risk that the infection persists during the invasion phase
was markedly increased even for parameter regimes where the nonspatial models
would predict extinction.
Similar to the consideration of spatial aspects for viral replication, viral clear-
ance, and the dynamics of target cell populations, spatial heterogeneity can also
be included when modeling the dynamics of specific immune responses. Similar to
Funk et al. [ 34 ], Louzon et al. [ 68 ] studied the influence of spatial heterogeneity in
a model capturing the activation and proliferation of lymphocytes. They defined a
simple deterministic model considering two different populations: antigen, A ,which
enters the system at rate

p

and is cleared at rate d A , and lymphocytes, L ,which
proliferate upon activation by antigen with a constant rate r and die with death rate
d L [Eq. ( 3 )].

λ

d L
d t =

rAL − d L L ,

d A
d t = λ −

d A A . (3)

The authors showed that according to this ODE model the lymphocyte population
will either grow exponentially or decrease to zero if the antigen concentration is
either above or below a certain threshold, respectively. In a second step, the authors