Hazard curvature makes within-host variability costly for survival

Tumors, pathogens, and immune responses are increasingly modeled as ecological and evolutionary systems inside hosts. Yet the clinically meaningful endpoint is usually not the trajectory of those internal populations but the fate of the host. In this article a framework is developed that links mechanistic within-host dynamics to outcomes through a hazard map. The framework gives rise to a curvature principle: If the instantaneous hazard is a convex function of a harmful host state, variability in that state is costly: That is, among trajectories with the same temporal mean, the path with least variance minimizes cumulative hazard, and any mean-preserving spread increases it. A local expansion shows that the penalty is set by hazard curvature and temporal variance. We extend the framework to proportional-hazards joint models, where the standard exponential link is convex by construction. This yields a curvature penalty, ½γ 2 ·Var w (z), such that for sinusoidal trajectories under a constant baseline hazard, an exact Bessel-function expression, I 0 (γA), for cumulative-hazard inflation is described. Both quantities can be derived from fitted joint models and quantify when the variability in the trajectory is likely to matter. The same framework highlights an endpoint mismatch: Strategies that improve burden-based control metrics, such as time to threshold, can worsen survival by repeatedly visiting high-risk states. The theory is illustrated with competitive tumor dynamics, a pathogen–immune–damage model and an example using parameters from a published joint model of SARS-CoV-2 viral kinetics and mortality. Hazard curvature links within-host dynamics, time-to-event outcomes, and treatment design.