Relapse timing and the endemic basin: how the shape of the relapse hazard moves the fold

Susceptible–Addicted–Reformed models with nonlinear relapse admit subthreshold endemic states: an established addiction epidemic persists below a basic reproduction number of one, down to a saddle-node fold. Such models give every reformed individual the same relapse hazard regardless of how long they have abstained, yet empirically that hazard is sharply front-loaded, with most relapses occurring within weeks to months of quitting.

Resolving the reformed class by time-since-reversion, this paper shows that the entire age-structured mechanism re-enters the bifurcation through a single functional, the reformed pool. Its size sets the fold, yielding two comparison principles: the fold decreases in the cumulative relapse hazard, and — at fixed total relapse feedback — in the earliness of relapse. Its derivative governs stability: the characteristic equation of the age-structured linearization collapses to a single transform, giving an explicit stability criterion which, together with a Nyquist count on the same kernel, rules out oscillations on the whole upper branch. The same depletion of the reformed pool that lowers the fold is what keeps the endemic state quiet.

At fixed relapse intensity, hazard shapes fitted to published relapse-survival data move the fold from 0.863 to roughly 0.09–0.13, and the shape parameter tracks treatment context rather than substance. Eradicability is governed less by how much relapse occurs than by when.

Code archived at DOI 10.5281/zenodo.21863449.