Abstract
Diseases, in particular, vector-borne diseases are very important issues in crop protection. However, despite their impact on food safety, very few mathematical models have been developed in order to improve control strategies. Motivated by existing literature, we begin by considering a temporal model of vector-borne diseases in (annual) crops. Using appropriate mathematical methods, we show the existence of threshold parameters and discuss several control strategies based on the model. Then, the model is modified to include a spatial component in order to take into account that vectors are moving. We study both theoretically and numerically the related system and other subsystems that are easier to handle using the theory of monotone dynamical systems. We show that traveling wave solutions may exist, with traveling wave speed dependent on the virus lifespan and the contact rate between the pest and the crop. Finally, we discuss the consequences in terms of control strategies.