We developed an electrothermal model of NbN superconducting nanowire avalanche photodetectors (SNAPs) on sapphire substrates. SNAPs are single-photon detectors consisting of the parallel connection of N superconducting nanowires. We extrapolated the physical constants of the model from experimental data and we simulated the time evolution of the device resistance, temperature and current by solving two coupled electrical and thermal differential equations describing the nanowires. The predictions of the model were in good quantitative agreement with the experimental results.