Flavescence dor\'ee (FD) is a disease of grapevine transmitted by an insect
vector, Scaphoideustitanus Ball. At present, no prophylaxis exists, so
mandatory control procedures (e.g. removal of infected plants, and insecticidal
sprays to avoid transmission) are in place in Italy and other European
countries. We propose a model of the epidemiology of FD by taking into account
the different aspects involved into the transmission process (acquisition of
the disease, latency and expression of symptoms, recovery rate, removal and
replacement of infected plants, insecticidal treatments, and the effect of
hotbeds). The model was constructed as a system of first order nonlinear ODEs
in four compartment variables. We perform a bifurcation analysis of the
equilibria of the model using the severity of the hotbeds as the control
parameter. Depending on the non-dimensional grapevine density of the vineyard
we find either a single family of equilibria in which the health of the
vineyard gradually deteriorates for progressively more severe hotbeds, or
multiple equilibria that give rise to sudden transitions from a nearly healthy
vineyard to a severely deteriorated one when the severity of the hotbeds
crosses a critical value. These results suggest some lines of intervention for
limiting the spread of the disease
