Using the disc instability model for dwarf novae and soft X-ray transients, we investigate the stability of accretion discs in long-period binary systems. We simulate outbursts resulting from this thermal-viscous instability for two symbiotic systems, RS Ophiuchi and Z Andromedae. The outburst properties deduced from our simulations suggest that, although the recurrent nova events observed in RS Oph are caused by thermonuclear runaway at the white dwarf surface, these runaways are triggered by accretion disc instabilities. In quiescence, the disc builds up its mass, and it is only during the disc-instability outburst that mass is accreted on to the white dwarf at rates comparable to or larger than the mass-transfer rate. For a mass-transfer rate in the range 10^−8 to |$10^{-7} \, {\rm M_{\odot}}$| yr^−1, the accretion rate and the mass accreted are sufficient to lead to a thermonuclear runaway during one of a series of a few dwarf nova outbursts, which are barely visible in the optical but easily detectable in the X-rays. In the case of Z And, persistent irradiation of the disc by the very hot white dwarf surface strongly modifies the dwarf nova outburst properties, making them significant only for very high mass-transfer rates, of the order of |$10^{-6} \, {\rm M_{\odot}}$| yr^−1, much higher than the expected secular mean in this system. It is thus likely that the so-called ‘combination nova’ outburst observed in the years 2000 to 2002 was triggered not by a dwarf-nova instability but by a mass-transfer enhancement from the giant companion, leading to an increase in nuclear burning at the accreting white dwarf surface.