A Stochastic Model for Early HIV-1 Population Dynamics

Abstract

A simple stochastic mathematical model is developed and investigated for early human immunodeficiency virus type-1 (HIV-1) population dynamics. The model, which is a multi-dimensional diffusion process, includes activated uninfected CD4+T cells, latently and actively infected CD4+T cells and free virions occurring in plasma. Stochastic effects are assumed to arise in the process of infection of CD4+T cells and transitions may occur from uninfected to latently or actively infected cells by chance mechanisms. Using the best currently available parameter values, the intrinsic variability in response to a given initial infection is examined by solving the stochastic system numerically. We estimate the statistical distributions of the time of occurrence and the magnitude of the early peak in viral concentration. The maximum of the viral load has a value in the experimental range and its time of occurrence has a 95% confidence interval from 19.4 to 25.1 days. The stochastic nature of the growth of viral density is extremely pronounced in the first few days after initial infection. Threshold effects are noted at virion levels of about 3–5×10−5mm−3. In addition to modeling the intrinsic variability in HIV-1 growth, we have explored the effects of perturbations in the parameter values in order to assess the additional stochastic effects of between-patient variability. We found that changes in the initial number of virions or dose size, the rate at which latently infected CD4+T cells are converted to the actively infected form and the fraction of latent cells has only minor effects on the size, speed and variability of the response. In contrast, decreased speed and magnitude but greater variability in response were obtained when the death rate of uninfected CD4+T cells, the death rate of actively infected cells and the clearance rate of the virus were increased or when the appearance rate of uninfected CD4+T cells, the number of virions produced by infected cells, the infection rate of CD4+T cells and the initial number of uninfected activated CD4+T cells were decreased. We also determined the distribution of the time to reach a given virion density. From this distribution the probability of detection of the virus as a function of time can be estimated. The numerical results obtained are in the range of experimental values and are discussed in relation to recently proposed detection and testing procedures.