Real-time assessment of debris-flow hazard is fundamental for developing warning systems that can mitigate risk. A convenient method to assess the possible occurrence of a debris flow is to compare measured and forecasted rainfalls to critical rainfall threshold (CRT) curves. Empirical derivation of the CRT from the analysis of past events' rainfall characteristics is not possible when the database of observed debris flows is poor or when the environment changes with time. For debris flows and mud flows triggered by shallow landslides or debris avalanches, the above limitations may be overcome through the methodology presented. In this work the CRT curves are derived from mathematical and numerical simulations, based on the infinite-slope stability model in which slope instability is governed by the increase in groundwater pressure due to rainfall. The effect of rainfall infiltration on landside occurrence is modelled through a reduced form of the Richards equation. The range of rainfall durations for which the method can be correctly employed is investigated and an equation is derived for the lower limit of the range. A large number of calculations are performed combining different values of rainfall characteristics (intensity and duration of event rainfall and intensity of antecedent rainfall). For each combination of rainfall characteristics, the percentage of the basin that is unstable is computed. The obtained database is opportunely elaborated to derive CRT curves. The methodology is implemented and tested in a small basin of the Amalfi Coast (South Italy). The comparison among the obtained CRT curves and the observed rainfall amounts, in a playback period, gives a good agreement. Simulations are performed with different degree of detail in the soil parameters characterization. The comparison shows that the lack of knowledge about the spatial variability of the parameters may greatly affect the results. This problem is partially mitigated by the use of a Monte Carlo approach.