Optimizing Chemotherapy in an HIV Model

by Sky McKinley

Abstract
The general purpose of this paper is to examine a differential equation published in a journal article.  For my paper, I used an article published by the Electronic Journal of Differential Equations, Vol. 1998(1998), No. 32, pp. 1-12.  The link for this web published journal follows this paper.  The article proposed to optimize the amount of chemotherapy necessary to rid a human host of the HIV virus, given certain growth parameters and restraints on the viral population.

Introduction

    The particular differential equation that I selected from the article is the basic model for the quantity of CD4+T cells, a type of of white blood cells, in an infected subject .

These CD4+T cells, were they not infected with the virus, would be the ones to defend the body from invasion.  The onset of AIDS is usually determined by the severe depletion of these cells.

A Little History

    In order to understand some of the terms designated in this differential equation, a cursory explanation of some of the players is necessary.

    The CD4+T cells are a type of white blood cell, the natural defense force of the body.  When HIV infects a human host, these are the white blood cells which are attacked and invaded by the virus.  This greatly weakens the host because the CD4+T cells are the cells necessary to fight off the HIV infection.

    CD4+T , or simply T cells, cells are produced by the thymus gland, hence the "T".  The CD4 is a protein marker which determines the type of the white blood cell. Immature T glands are produced by the bone marrow, after which they make their way to the thymus gland where they mature in to active T cells.

    Once the HIV virus infects a host, the viruses begin to infect healthy T cells.  The T cells are "colonized" by the virus, which grows inside the cell until the cell bursts, creating more free viral agents to infest the host.

Discussion of Variables

    In the above mentioned equation, the concentration of CD4+T cells in the infected host is represented by T.  The change in the concentration of these cells, , is a dependent upon the rate of production of new CD4+T cells, the natural death rate of CD4+T cells, the relative quantities of uninfected, latently infected, and actively infected CD4+T cells, and the quantity of free virus in the host.
    The production rate of immature T cells is represented by .  The value for s is determined by the host's natural antibody production rate, in proportion to the number of free virus in the body, V.  Values for s are assumed to be on the interval , but this was not explicitly stated in the article.  One would assume that that the rate of  T cell production would increase as the concentration of the viral population increased, and for this assumption to hold, the value for s would need to be between zero and one.

    The natural death rate of the T cells is represented by .  This death rate can be adjusted to accommodate hostile environments in the host, but is generally assumed to remain constant.

    The growth rate of the mature, active T cells is represented by the logistic growth term .  It is a relatively simple logistic growth term, with r being the growth coefficient of the T cells and is the maximum concentration of the T cells.  Where this term differs from basic logistic growth terms is the () term.  T has already been defined to be the concentration of healthy, uninfected T cells.  The T* is the concentration of latently infected cells, and T** is the concentration of actively infected cells.  This ensures that the concentration of T cells does not exceed the maximum concentration value.  It is unfortunate, however, for the infected host.  As the number of latently infected and actively infected cells increases, the maximum concentration for healthy T cells is decreased, as their combined concentrations cannot exceed .

    Lastly, the term represents the rate at which the virus infects the healthy T cells.

References

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