<p>This article investigates the spreading properties of yellows viruses within sugar beet agro-ecosystems using reaction-diffusion systems, with spreading speeds and traveling wave solutions serving as key analytical tools. In these systems, each unknown function corresponds to a distinct ecological variable: infected hosts (i.e., sugar beets), infected vectors, susceptible vectors, and vector predators. In the absence of predators, we analyze the spreading characteristics of infected hosts and infected vectors, with susceptible vectors treated as the native population. The spreading speed of yellows viruses is given, which equals the minimal wave speed of monotonic traveling wave solutions modeling disease spreading and prevalence. When predators are introduced for biological control, the existence and nonexistence of traveling wave solutions starting from the disease-free and predator-free steady state are studied. For the corresponding Cauchy problem, the invasion speed of predators is established. Specifically, this speed is derived under the assumption that vectors are native species, and it remains independent of disease prevalence. Subsequently, we numerically compare the observed viral prevalence under the presence of predators with different expansion capabilities. We then present two distinct scenarios focusing on the spread or extinction of yellows viruses. In the context of virus prevention and control, these results deepen our understanding of the importance of the predation rate, predator mobility, and biting rate.</p>