Both the linear (at low doses)-no-threshold (LNT) and the threshold models (S- shapes) dose-response lead to no benefit from low exposure. We propose three new models that allow and include, but do not require – unlike LNT and S-shaped models — this strong assumption. We also provide the means to calculate benefits associated with bi-phasic biological behaviors, when they occur and propose:

1. Three hormetic (phasic) models: the J-shaped, inverse J-shaped, the min-max, and

2. Method for calculating the direct benefits associated with the J and inverse J- shaped models.

The J-shaped and min-max models for mutagens and carcinogenic agents include an experimentally justified repair stage for toxic and carcinogenic damage. We link these to stochastic transition models for cancer and show how abrupt transitions in cancer hazard rates, as functions of exposure concentrations and durations, can emerge naturally in large cell populations even when the rates of cell-level events increase smoothly (e.g., proportionally) with concentration. In this very general family of models, J-shaped dose- response curves emerge. These results are universal, i.e., independent of specific biological details represented by the stochastic transition networks. Thus, using them suggests a more complete and realistic way to assess risks at low doses or dose-rates.