Stochastic First-Passage Theory of HIV Viral Rebound Following Latent Reservoir Reactivation

Abstract

In our earlier work, we modeled the stochastic initiation of HIV rebound by treating latent-cell reactivation as a Poisson-driven process during antiretroviral-therapy (ART) washout, immune modulation, and therapeutic perturbation~\cite{Taye2025CM}. That framework characterized activation survival, cumulative hazards, waiting-time laws, and expected viral-load trajectories. However, the endpoint observed in analytical treatment interruption (ATI) studies is not the hidden time of first successful reactivation. It is the first time at which plasma virus exceeds an assay-defined detection threshold. Here we reformulate post-treatment rebound as a stochastic first-passage problem, with $T_{\rm reb}=\inf\{t\ge t_w:V(t)\ge V_{\rm det}\}$. Successful reactivation events arrive with a time-dependent intensity, and each event seeds an exponentially expanding viral lineage. The total plasma viral load is therefore a Poisson shot-noise process, and rebound corresponds to its first threshold crossing. In the rare-reactivation regime, this crossing is dominated by the earliest successful lineage. Rebound timing then separates into two components: a stochastic waiting time for reservoir reactivation and a deterministic growth delay to detectability. This separation gives a shifted-hazard survival law and yields closed-form rebound-time distributions for constant activation, ART-washout-dependent activation, immune-periodic activation, Cox-process activation, and heterogeneous-reservoir activation. The same formulation also provides a likelihood suitable for the interval-censored sampling structure of ATI trials.