Stochastic Field Theory of HIV Latency: Instanton Dynamics and the Path to Viral Rebound
The transition from clinical latency to active HIV infection is a stochastic process. Traditional deterministic models based on ordinary differential equations (ODEs) fail to capture the extinction boundary and the subsequent rebound of the virus because they neglect demographic noise in the small-population limit. We map the resulting Master Equation onto a coherent-state path integral using the Doi-Peliti second-quantization formalism and verify that the stochastic Hamiltonian reproduces the correct mean-field ODEs on the classical manifold $\barϕ_i = 1$. In the semiclassical (large system-size) limit we derive eight coupled Hamilton-Jacobi equations, identify the virus-free and endemic fixed points, and obtain the basic reproduction number $R_0$ for the four-species network. For the analytically tractable case of single-virion bursting ($n=1$) we show that the stochastic Hamiltonian \emph{factorizes} into two bilinear terms, exposing a non-trivial zero-energy surface that constitutes the instanton trajectory. A quasi-steady-state reduction then yields a first-order linear ODE whose closed-form solution gives the instanton action $S_{\rm inst}$ in terms of biological parameters. Calibrating to clinical data---infected-cell half-life $δ_I^{-1}\approx 2\,\rm d$, viral burst size $N\approx 10^3$, clearance rate $δ_V\approx 23\,\rm d^{-1}$, reactivation rate $η\approx 10^{-3}\,\rm d^{-1}$, and latent reservoir size $L_0\approx 10^6$ cells---we obtain a mean first-passage time (MFPT) to viral rebound of $\approx 6$ months for a typical patient, with a range of days to years depending on reservoir size. These predictions are qualitatively consistent with observed post-interruption rebound timescales.