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The VISTA Variables in the Via Lactea (VVV) survey obtained near-infrared photometry toward the Galactic bulge and the southern disc plane for a decade (2010 - 2019). We designed a modified Lomb-Scargle method to search for large-amplitude ($Δ$Ks > 1.5 mag) mid to long-term periodic variables (P > 10 d) in the 2nd version of VVV Infrared Astrometric Catalogue (VIRAC2-$β$). In total, 1520 periodic sources were discovered, including 59 candidate periodic outbursting young stellar objects (YSOs), based on the unique morphology of the phase-folded light curves, proximity to Galactic HII regions and mid-infrared colours. Five sources are spectroscopically confirmed as accreting YSOs. Both fast-rise/slow-decay and slow-rise/fast-decay periodic outbursts were found, but fast-rise/slow-decay outbursts predominate at the highest amplitudes. The multi-wavelength colour variations are consistent with a variable mass accretion process, as opposed to variable extinction. The cycles are likely to be caused by dynamical perturbations from stellar or planetary companions within the circumstellar disc. An additional search for periodic variability amongst YSO candidates in published Spitzer-based catalogues yielded a further 71 candidate periodic accretors, mostly with lower amplitudes. These resemble cases of pulsed accretion but with unusually long periods and greater regularity. The majority of other long-period variables are pulsating dusty Miras with smooth and symmetric light curves. We find that some Miras have redder $W3 - W4$ colours than previously thought, most likely due to their surface chemical compositions.