Modelling the effects of cutting off infected branches and replanting on fire-blight transmission using Filippov systems

Fire blight is one of the most devastating plant diseases in the world. This paper proposes a Filippov fire-blight model incorporating cutting off infected branches and replanting susceptible trees. The Filippov-type model is formulated by considering that no control strategy is taken if the number of infected trees is less than an infected threshold level I _c ; further, we cut off infected branches once the number of infected trees exceeds I _c ; meanwhile, we replant trees if the number of susceptible trees is less than a susceptible threshold level S _c . The global dynamical behaviour of the Filippov system is investigated. It is shown that model solutions ultimately converge to the positive equilibrium that lies in the region above I _c , or below I _c , or on I=I _c , as we vary the susceptible and infected threshold values S _c and I _c . Our results indicate that proper combinations of the susceptible and infected threshold values based on the threshold policy can lead the number of infected trees to an acceptable level, when complete eradication is not economically desirable.