Spatial Aspects of HIV Infection - Mathematical Methods and Models in Biomedicine - page 9

Tabl e 1 Quantification of parameters to describe disease dynamics in HIV or SIV based on
models relying on ordinary differential equations
Parameter

Unit Value Reference

day − 1

.

Viral clearance rate, c 5 5 [ 110 ]

23 [ 109 ]

300 [ 128 ] SIV

10 2

Viral production rate, p RNA copies ≈ [ 29 , 44 ]

cell − 1 day − 1 10 3 0 . 7 − 3 . 4 × [ 110 ]

10 4 2 − 5 × [ 21 ] SIV

Source rate of CD4 + T cells, l − 1 day − 1

λ μ 8 [ 125 ]

Death rate of CD4 + T cells, d day − 1 ∼ 0.01 [ 111 ]

day − 1

Death rate of productively infected cells, δ 1 [ 69 , 104 ]

Values are taken from the corresponding publications

3.2 Shortcomings of ODE Models in Spatial Situations

As suggested by the estimates above, one can expect infection dynamics to vary
among different anatomical compartments [ 27 ]. However, most of the infection
occurs in solid tissue, such as in lymph nodes, the spleen or other organs in which
target cells are plentiful [ 7 , 9 ]. In solid tissues, local effects such as the focal
release of virions from infected cells, as observed in vivo [ 44 , 76 ],mayplayan
important role in the infection dynamics. Once virus is released it diffuses and will
preferentially infect nearby cells. In general, ODE models are not able to capture
this type of spatial aspect.
In ecology and epidemiology, there is a large body of literature studying
the influence of spatial structure on dynamics [ 28 , 39 , 58 , 121 ]. The space being
analyzed need not be Euclidean space but can also be a tree or more general
graph representing interactions in a social network [ 59 , 74 , 88 ] or meta-populations
[ 39 , 58 , 95 ]. Levin and Durrett [ 64 ] and later Webb et al. [ 124 ] analyzed the
circumstances under which the results of a spatial model would differ from a
mean-field model, such as that given by a simple ODE model of population
dynamics which assumes a well-mixed population. For their spatial model, Webb
et al. [ 124 ] examined an undirected regular network of sites which can be either
empty or occupied by a susceptible, infected, or resistant individual. They found
that dependent on the degree of spatial connectivity and the underlying ecological
situation the prediction of the spatial and mean-field models can differ substantially
given similar parameter ranges. Similar studies have been made in related fields
such as statistical mechanics or genetics (reviewed in [ 30 ]).
In the following sections, we will review several methods for including spatial
aspects into the modeling of infection dynamics within a host with a specific
emphasis on HIV viral dynamics.