Abstract
A mathematical model for the honeybee-varroa mite-ABPV system is proposed in terms of four differential equations for the: infected and uninfected bees in the colony, number of mites overall, and of mites carrying the virus. To account for seasonal variability, all parameters are time periodic. We obtain linearized stability conditions for the disease-free periodic solutions. Numerically, we illustrate that, for appropriate parameters, mites can establish themselves in colonies that are not treated with varroacides, leading to colonies with slightly reduced number of bees. If some of these mites carry the virus, however, the colony might fail suddenly after several years without a noticeable sign of stress leading up to the failure. The immediate cause of failure is that at the end of fall, colonies are not strong enough to survive the winter in viable numbers. We investigate the effect of the initial disease infestation on collapse time, and how varroacide treatment affects long-term behavior. We find that to control the virus epidemic, the mites as disease vector should be controlled.