Backward bifurcation of a plant virus dynamics model with nonlinear continuous and impulsive control

Roguing and elimination of vectors are the most commonly seen biological control strategies regarding the spread of plant viruses. It is practically significant to establish the mathematical models of plant virus transmission and regard the effect of removing infected plants as well as eliminating vector strategies on plant virus eradication. We proposed the mathematical models of plant virus transmission with nonlinear continuous and pulse removal of infected plants and vectors. In terms of the nonlinear continuous control strategy, the threshold values of the existence and stability of multiple equilibria have been provided. Moreover, the conditions for the occurrence of backward bifurcation were also provided. Regarding the nonlinear impulsive control strategy, the stability of the disease-free periodic solution and the threshold of the persistence of the disease were given. With the application of the fixed point theory, the conditions for the existence of forward and backward bifurcations of the model were presented. Our results demonstrated that there was a backward bifurcation phenomenon in continuous systems, and there was also a backward bifurcation phenomenon in impulsive control systems. Moreover, we found that removing healthy plants increased the threshold $ R_{1}. $ Finally, numerical simulation was employed to verify our conclusions.