First passage time to detection in stochastic population dynamical models for HIV-1

The detection of HIV-1 levels in human hosts is cast as a first exit time problem for a multidimensional diffusion process. We consider a four-component model for early HIV-1 dynamics including uninfected CD4+ T-cells, latently infected cells, actively infected cells, and HIV-1 virions. An analytical framework is presented for the distribution of the time at which a given virion level is attained. A one-dimensional diffusion approximation for a branching process leads to an estimate for the distribution of the virion density and an expression for the mean detection time for any given detection threshold.