A Model for the CD4 Cell Counts in an HIV/AIDS Patient and its Application in Treatment Interventions

The number of CD4 white blood cells has been established as an important clinical marker of disease progression to acquired immunodeficiency syndrome (AIDS) for persons infected with human immunodeficiency virus (HIV). The number of CD4 cells per unit volume is expected to decrease with time since infection by the virus. However on introduction of treatment interventions, the process is expected to reverse with the counts increasing to return to the normal level. In this study we deduce that the count per unit volume of blood of an HIV/AIDS patient has a linear relationship with the time since infection during the short period of time immediately treatment usage begins. We show one application of the model in treatment selection strategy.


INTRODUCTION
An understanding of typical course of the infection within an individual and the quantitative information about latent period, infectious period or in some cases incubation period is essential in designing effective treatment strategies. In a given population, it is considered desirable to classify individuals (further stratified by spatial locality, age or gender) for modeling purposes into the categories that are: * Immunized (protected from the infection by maternally derived factors or through vaccination) * Susceptible (not yet infected but may get infected if exposed to the disease) * Infected (and in the various stages of infection) * Permanently removed (AIDS or dead) In order to describe the various stages of HIV infection one can make use of the corresponding differential equations showing the rates of changes in the number of individuals at the respective stages with the help of relevant parameters [1,2] . Simwa and Pokhariyal [3] constructed a dynamical model for stagespecific HIV incidences, using two systems of ordinary differential equations that are coupled through a delay in one of the systems. Further by incorporating possible vaccine interventions in the dynamical stage specific model simulations of the HIV pandemic show the vaccine effect on the stage specific projections [4] .
Other forms of interventions include use of antiretroviral (ARV) drugs by the HIV infected patients. These drugs include AZT, among others. In this study the effect of such drugs to a patient is considered. The CD4 cells forms a group of white cells that provide immune capability to ones body. The HIV virus is known to attack and destroy these cells once an individual is infected. The depletion of the cells leads to loss of body immunity and thus AIDS. In fact the absolute number of CD4 cells has been established as a significant predicator of disease progression in the patient [5] . However application of the anti-retroviral drugs may lead to reduced rate of destruction of these cells by the virus such that the cell count per unit volume of blood of the patient starts increasing. We model the CD4 cells counts per unit volume of blood in an HIV patient as from time of treatment application, as a function of time since infection. Theories from survival and actuarial studies are adapted to analyse the distribution of the future lifetime of an arbitrary newly formed CD4 cell in an HIV infected patient. An expression for the counts per unit volume (milliliter) as a linear function of the hazard rate of the distribution of the CD4 cell lifetime on commencement of treatment is derived in the next section. From these results we deduce strategies that can be used in the analysis of treatment interventions, with the analysis based on Simwa [6] .
is the expected value of N(t), then the estimate for N for a healthy person has been approximated to be 1200 per mm 3 [5] . For an HIV patient it has been noted that this number reduces to, say (2.6) Proof: A CD4 cell in an HIV patient is exposed to extra risk of dying due to the attack by the HIV viruses and hence the average lifetime of the cells is less than the case before the infection. Hence we may adopt the theories on Exposed-to-Risk to estimate ( ) can also be taken as the ratio of the expected total exposure time (expected total lifetime) of CD4 cells to the risk of dying at time t with and without infection, of a given constant number of these cells, say k (t) cells.
For a healthy HIV non infected individual at time t, is the lifetime of the th j CD4 cell (j = 1, 2, …, k(t)). On the other hand, for an HIV infected patient (infectedτ years ago), lim E T t, j  From theories on survival analysis [7] , note that that the hazard function is independent of the 'age' of the CD4 cell (as sometimes assumed in demography) [8,9] , but depends on time since HIV infection of the individual. Then where, ω is the maximum (limiting) 'age' of a CD4 cell [7] . Thus, Let X denote the incubation period of AIDS with ( ) x h as the hazard function of the distribution of the incubation period. Kiuchi et al. [5] note that in the absence of treatment as x increases ( ) decreases. For a treatment that has been developed to improve the immune system of an HIV patient we expect that once the patient takes up the treatment, say The Weibull distribution has been assumed to approximate the distribution of X [10,11] , such that ( )  form. Fig. 1 will be used to describe the procedure.
HIV infection leads to a gradual reduction in the number of CD4 cells while treatment by the ARV drugs is expected to lead to an initial abrupt increase in this number. If 1 N and 2 N are the average numbers before and after the infection at time 0 and ( ) * 2 τ = t years respectively since the infection, treatments type X and type Y may lead to the curves sketched in the figure. Treatment type X requires duration of length , the corresponding length with respect to treatment type Y, to get the CD4 cells counts back to the normal level, 1 N ("healing phase" period). Thus, the magnitude (and sign) of the slope of either curve during time interval ( ) provides a relative measure of effectiveness of the treatments. Therefore, the following steps could lead to classifying various types of treatment forms that may be available for the "healing phase" of an HIV infected patient under treatment. Step 1: For each patient, once treatment begins, record time of treatment and the corresponding CD4 cells count per millilitre for several treatment sessions and treatment forms. A patient is assumed to undergo one type of treatment.
Step 2: Assume linearity for a specified short time interval and determine the slopes of the curve. The approximate duration of the interval can be estimated by observing the plot of the relevant data ( x t and y t in Fig. 1).
Step 3: Test whether there is marked and significant relative treatment effects basing on the slope parameter. Then select the predominant treatment form, if any. For instance in Fig. 1, we note that the slope parameter estimates are such that y b x b> and hence treatment type X is better than treatment type Y. However standard statistical tests may be used for more rigorous analysis.