The Prevention Theorem: Time-Dependent Constraints on Post-Exposure Prophylaxis for HIV

Antiretroviral agents for HIV prevention are typically evaluated in terms of trial efficacyand programmatic coverage, but rarely in terms of whether they admit a true mathematicalsolution to prevention. Here we introduce the Prevention Theorem, which formalizesprevention for a given exposure e as the condition R0(e) = 0, meaning that the probability of establishing a productive, transmissible infection is exactly zero. Within this framework,post-exposure prophylaxis (PEP) is not delayed treatment but a time-dependent operatoracting on within-host infection establishment dynamics. Using a mechanistic model of reservoirseeding and proviral integration, we derive the PEP Window Corollary: PEP can enforce R0(e) = 0 only when initiated within a finite biological window prior to irreversible integrationand initial reservoir establishment. Beyond this window, all reachable system statessatisfy R0(e) > 0 and are irreducible by post-exposure intervention. Parameterization usingvirological data indicates that this window extends to approximately 72 hours for mucosalexposures but is compressed to roughly 12–24 hours for parenteral exposures due to bypass ofearly immune bottlenecks. As an applied example, we show that structural access delays inhigh-risk populations—such as people who inject drugs—frequently exceed this compressedparenteral window. Consequently, for such exposures the condition R0(e) = 0 is mathematicallyand biologically unreachable before access is even attempted, rendering the failure ofpost-exposure prevention a consequence of violated biological boundary conditions ratherthan pharmacological efficacy.