Qualitative analysis of a mathematical model for Xylella fastidiosa epidemics

In Southern Italy, since 2013, there has been an ongoing Olive Quick Decline Syndrome (OQDS) outbreak, due to the bacterium Xylella fastidiosa. In a couple of previous papers, the authors have proposed a mathematical approach for identifying possible control strategies for eliminating or at least reduce the economic impact of such event. The main players involved in OQDS are represented by the insect vector, Philaenus spumarius, its host plants (olive trees and weeds) and the bacterium, X. fastidiosa. A basic mathematical model has been expressed in terms of a system of ordinary differential equations; a preliminary analysis already provided interesting results about possible control strategies within an integrated pest management framework, not requiring the removal of the productive resource represented by the olive trees. The same conjectures have been later confirmed by analyzing the impact of possible spatial heterogeneities on controlling a X. fastidiosa epidemic. These encouraging facts have stimulated a more detailed and rigorous mathematical analysis of the same system, as presented in this paper. A clear picture of the possible steady states (equilibria) and their stability properties has been outlined, within a variety of different parameter scenarios, for the original spatially homogeneous ecosystem. The results obtained here confirm, in a mathematically rigorous way, what had been conjectured in the previous papers, i.e. that the removal of a suitable amount of weed biomass (reservoir of the juvenile stages of the insect vector of X. fastidiosa from olive orchards and surrounding areas is the most acceptable strategy to control the spread of the OQDS. In addition, as expected, the adoption of more resistant olive tree cultivars has been shown to be a good strategy, though less cost-effective, in controlling the pathogen.