Spatial Aspects of HIV Infection - Mathematical Methods and Models in Biomedicine - page 10

4 Getting More Spatial: From ODE to PDE

4.1 Spatial Discrete ODE Models

One way to include spatial aspects into ODE models for HIV infection dynamics
on a macroscopic scale is to use compartment models. These models distinguish
between different anatomical compartments in which HIV can infect cells and
replicate. The published models of this type developed by De Boer and coworkers
consider different sites of HIV replication and clearance, e.g., lymphoid tissue,
blood, plasma and other organs such as the liver or the lung, and the interchange
of viral particles among these compartments [ 27 , 85 ]. Applying these models to
experimental data, De Boer et al. found that the clearance rate of free virus in
lymphoid organs should be rapid and that the previously estimated clearance rates
measured in the blood most likely correspond to the efflux of virions from the blood
to other organs [ 27 ]. By considering different compartments for viral replication
and clearance, De Boer et al. were able to explain previously estimated differences
in viral clearance rates as found in [ 109 , 127 , 128 ].
Another way to combine an ODE model with spatial aspects was introduced by
Funk et al. [ 34 ]. In this case, the authors did not specifically consider different com-
partments and their physiological interdependence. Instead, the authors assumed
a well-defined two-dimensional grid with n

n grid sites for a predefined total
volume. Each grid site could represent different anatomical sites inside the host or
different spatially adjoining compartments inside a specific organ, such as a lymph
node. At each of the different grid sites

×

n , the concentration of
target cells, T i , j , infected cells, I i , j , and the viral load, V i , j , were described by ODE
according to the basic model of viral dynamics in Eq. ( 1 ). Target cells and infected
cells were assumed to be stationary, while virions were allowed to migrate from one
grid site to a neighboring site. Virion movement is accounted for by an additional
term in the virus equation [Eq. ( 2 )], where m V denotes the diffusion rate of free
virions from site

( i , j ) , i , j = 1 ,...,

( i , j ) to the eight nearest neighboring sites [ 34 ].

d T i , j
d t = λ −

dT i , j − β V i , j T i , j ,

d I i , j
d t

= β − δ ,

V i , j T i , j I i , j

1
∑
i 0 =

i + 1

∑
j 0 =

j +

V i , j −

V i 0 , j 0 .

d V i , j
d t =

m V
8

pI i , j − cV i , j − (2)

i − 1 j − 1

Varying the degree of spatial coupling between the different sites by changing
the value of the parameter m V , the authors compared the outcome of this model to
the basic population dynamics model defined by Eq. ( 1 ). They showed that given a
homogeneous spatial environment, meaning that each of the different rate constants