Sliding motion and global dynamics of a Filippov fire-blight model with economic thresholds

Cutting off infected branches has always been an effective method for removing fire-blight infection in an orchard. We introduce a Filippov fire-blight model with a threshold policy: cutting off infected branches and replanting susceptible trees. The dynamics of the proposed piecewise smooth model are described by differential equations with discontinuous right-hand sides. For each susceptible threshold value S _T , we investigate the global dynamical behaviour of the Filippov system, including the existence of all the possible equilibria, their stability and sliding-mode dynamics, as we vary the infected threshold level I _T . Our results show that model solutions ultimately approach the equilibrium that lies in the region above I _T or below I _T or on I=I _T , or the equilibrium E _T =(S _T ,I _T ) on the surface of discontinuity. Furthermore, control strategies should be taken when the solution of this system approaches the equilibrium that lies in the region above I _T . The findings indicate that proper choice of susceptible and infected threshold levels can either preclude an outbreak of fire blight or lead the number of infected trees to a desired level.