# Perelson, Kirschner, de Boer, 1993

#### Model Status

This model has been built with the differential expressions in Perelson, Kirschner and de Boer's 1993 paper for the 4th model (perelson_1993d); the quasi-steady-state approximation to the [previous model] dynamics. This file is known to run in OpenCell and COR, and is set to the parameters in Table 1 and figure 13 of the paper. The condition t=t_min in the paper was replaced with t less than/equal to t_min because otherwise the model would be incorrect. By changing the the parameters, the model will replicate all panels in figures 13. Note that in the paper, some figure diagrams are scaled logarithmically.

#### Model Structure

ABSTRACT: We examine a model for the interaction of HIV with CD4+ T cells that considers four populations: uninfected T cells, latently infected T cells, actively infected T cells, and free virus. Using this model we show that many of the puzzling quantitative features of HIV infection can be explained simply. We also consider effects of AZT on viral growth and T-cell population dynamics. The model exhibits two steady states, an uninfected state in which no virus is present and an endemically infected state, in which virus and infected T cells are present. We show that if N, the number of infectious virions produced per actively infected T cell, is less a critical value, Ncrit, then the uninfected state is the only steady state in the nonnegative orthant, and this state is stable. For N > Ncrit, the uninfected state is unstable, and the endemically infected state can be either stable, or unstable and surrounded by a stable limit cycle. Using numerical bifurcation techniques we map out the parameter regimes of these various behaviors. oscillatory behavior seems to lie outside the region of biologically realistic parameter values. When the endemically infected state is stable, it is characterized by a reduced number of T cells compared with the uninfected state. Thus T-cell depletion occurs through the establishment of a new steady state. The dynamics of the establishment of this new steady state are examined both numerically and via the quasi-steady-state approximation. We develop approximations for the dynamics at early times in which the free virus rapidly binds to T cells, during an intermediate time scale in which the virus grows exponentially, and a third time scale on which viral growth slows and the endemically infected steady state is approached. Using the quasi-steady-state approximation the model can be simplified to two ordinary differential equations the summarize much of the dynamical behavior. We compute the level of T cells in the endemically infected state and show how that level varies with the parameters in the model. The model predicts that different viral strains, characterized by generating differing numbers of infective virions within infected T cells, can cause different amounts of T-cell depletion and generate depletion at different rates. Two versions of the model are studied. In one the source of T cells from precursors is constant, whereas in the other the source of T cells decreases with viral load, mimicking the infection and killing of T-cell precursors. The latter gives more realistic predictions than the model with a constant source.

Schematic diagram of the model. |

The original paper reference is cited below:

Dynamics of HIV infection of CD4+ T cells, Perelson, Kirschner, Boer, *Mathematical Biosciences*, 114, 81-125. PubMed ID: 8096155