Models are presented to study the damages that the ectoparasitic mite Varroa destructor produces on the honey bee Apis mellifera colonies. The mathematical systems differ by either assuming a constant mite population size or by allowing it to be bee-dependent, [1, 2]. The negative influence of mites on beehives is not just related to their parasitic action, namely its sucking hemolymph from the host. Mainly it is the fact that the mite becomes a vector which increases the transmission of several viral diseases, among which the worst are acute paralysis and deformed wing viruses. There represent the main causes of Colony Collapse Disorder. The models possess only four possible realizable stable equilibria. The point with just the thriving healthy bees, i.e. the healthy beehive. An alternative is the mite-free but endemic equilibrium, with the disease among the thriving bee population. Infected bees can also coexist with the invading Varroa mites; here the disease affects all the bees, driving the healthy bees to extinction. Finally there is coexistence, with both populations of bees and mites thriving and with a disease endemically affecting both species. The analysis is in line with some observed facts in natural honey bee colonies. Namely, these diseases are endemic. Further, if the mite population is present, necessarily the viral infection occurs. The findings of this study indicate the at a low horizontal transmission rate of the virus among honey bees in beehives will help in protecting bee colonies from Varroa infestation and viral epidemics.

References

[1] S. Bernardi, E. Venturino. Viral epidemiology of the adult Apis Mellifera in fested by the Varroa destructor mite, to appear in HELIYON (2016).

[2] Sara Bernardi, Ezio Venturino, An epidemiological model of viral infections in a Varroa-infested bee colony: the case of a bee-dependent mite population size, Biomat 16, R. P. Mondaini Editor, 165-186, World Scientific, 2017.