  Fire blight is a bacterial plant disease that affects apple and pear trees.
We present a mathematical model for its spread in an orchard during bloom. This
is a PDE-ODE coupled system, consisting of two semilinear PDEs for the
pathogen, coupled to a system of three ODEs for the stationary hosts.
Exploratory numerical simulations suggest the existence of travelling waves,
which we subsequently prove, under some conditions on parameters, using the
method of upper and lower bounds and Schauder's fixed point theorem. Our
results are likely not optimal in the sense that our constraints on parameters,
which can be interpreted biologically, are sufficient for the existence of
travelling waves, but probably not necessary. Possible implications for fire
blight biology and management are discussed.
