  We propose a honeybee-mite-virus model that incorporates (1) parasitic
interactions between honeybees and the Varroa mites; (2) five virus
transmission terms between honeybees and mites at different stages of Varroa
mites: from honeybees to honeybees, from adult honeybees to phoretic mites,
from honeybee brood to reproductive mites, from reproductive mites to honeybee
brood, and from honeybees to phoretic mites; and (3) Allee effects in the
honeybee population generated by its internal organization such as division of
labor. We provide completed local and global analysis for the full system and
its subsystems. Our analytical and numerical results allow us have a better
understanding of the synergistic effects of parasitism and virus infections on
honeybee population dynamics and its persistence. Interesting findings from our
work include: (a) Due to Allee effects experienced by the honeybee population,
initial conditions are essential for the survival of the colony. (b) Low adult
honeybee to brood ratios have destabilizing effects on the system, generate
fluctuated dynamics, and potentially lead to a \emph{catastrophic event} where
both honeybees and mites suddenly become extinct. This catastrophic event could
be potentially linked to Colony Collapse Disorder (CCD) of honeybee colonies.
(c) Virus infections may have stabilizing effects on the system, and could make
disease more persistent in the presence of parasitic mites. Our model
illustrates how the synergy between the parasitic mites and virus infections
consequently generates rich dynamics including multiple attractors where all
species can coexist or go extinct depending on initial conditions. Our findings
may provide important insights on honeybee diseases and parasites and how to
best control them.
