Seismicity models are probabilistic forecasts of earthquake rates to support seismic hazard assessment. Physics-based models allow extrapolating previously unsampled parameter ranges and enable conclusions on underlying tectonic or human-induced processes. The Coulomb Failure (CF) and the rate-and-state (RS) models are two widely-used physics-based seismicity models both assuming pre-existing populations of faults responding to Coulomb stress changes. The CF model depends on the absolute Coulomb stress and assumes instantaneous triggering if stress exceeds a threshold, while the RS model only depends on stress changes. Both models can predict background earthquake rates and time-dependent stress effects, but the RS model with its three independent parameters can additionally explain delayed aftershock triggering. This study introduces a modified CF model where the instantaneous triggering is replaced by a mean time-to-failure depending on the absolute stress value. For the specific choice of an exponential dependence on stress and a stationary initial seismicity rate, we show that the model leads to identical results as the RS model and reproduces the Omori-Utsu relation for aftershock decays as well stress-shadowing effects. Thus, both CF and RS models can be seen as special cases of the new model. However, the new stress response model can also account for subcritical initial stress conditions and alternative functions of the mean time-to-failure depending on the problem and fracture mode.