Is Earlier Better for AZT Therapy in HIV Infection? A Mathematical Model

Abstract

A mathematical model is proposed to determine the optimal level of the CD4-lymphocyte count at which to begin intervention therapy in Human Immunodeficiency Virus (HIV) infection. In the deterministic formulation of the model, the CD4 count for an untreated HIV-infected individual is assumed to decline linearly over time during the asymptomatic phase of the infection. Treatment with an antiviral such as zidovudine (AZT) is assumed to delay further decline in CD4 counts for a time period which may be constant or which functionally may depend upon the attained CD4 level. The optimal CD4 level for starting therapy is defined to be that which maximizes an individual’s sojourn time above a predetermined critical level of CD4, below which serious health consequences are likely to occur. It is shown that earlier is better provided the duration of the effective therapeutic period increases with the starting CD4 level at a rate that is greater than or equal to the difference in the inverse CD4 slopes before and after treatment. The stochastic-model formulation of the CD4 count is defined as a time series which is the sum of the deterministic function and random errors which are independent and identically distributed at successive time points. An application is made to the specific case where the parameters of the deterministic model and of the (normal) error distribution were estimated from CD4 data obtained from intravenous drug users in New York City. The implications of the stochastic model are generally similar to the deterministic model. Whenever the deterministic model indicates that earlier is better, so does the stochastic model. For a logistic treatment-duration function ranging from a minimum of 12 months (for low starting levels) to a maximum of 30 months (for high starting levels), the model implies that an early start to AZT therapy is a good choice, unless the after-treatment slope is more than double the before-treatment slope.