Abstract
We present a simple SIR-Ross-MacDonald like model of the infestation of a honeybee (Apis mellifera) colony by the Acute Bee Paralysis Virus (ABPV), which is transmitted by parasitic varroa mites (Varroa destructor) as vector. This is a four dimensional system of nonlinear ordinary differential equations for the dependent variables healthy and virus infected bees, number of mites in the colony overall and number of mites that carry the virus. In the autonomous case we study the model with analytical techniques deriving conditions under which the bee colony can fight off a ABPV epidemic. These results are then used to design and discuss numerical simulations of the more realistic case with periodic coefficient functions that mimic the seasonal changes in bee colonies. keywords: honeybees, varroa destructor, acute bee paralysis virus, mathematical model MSC: 92D25, 92D30