  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 ($\Delta$Ks > 1.5 mag) mid to long-term periodic variables (P >
10 d) in the 2nd version of VVV Infrared Astrometric Catalogue
(VIRAC2-$\beta$). 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.
