Bifurcation and optimal control analysis of delayed models for huanglongbing

In this paper, a delayed differential model of citrus Huanglongbing infection is analyzed, in which the latencies of the citrus tree and Asian citrus psyllid are considered as two time delay factors. We compute the equilibrium points and the basic reproductive numbers with and without time delays, i.e. R-0 and (R) over tilde (0), and then show that (R) over tilde (0) completely determines the local stability of the disease-free equilibrium. Moreover, the conditions for the existence of transcritical bifurcation are derived from Sotomayor's Theorem. The stability of the endemic equilibrium and the existence of Hopf bifurcation are investigated in four cases: (1) tau(1) = tau(2) = 0, (2) tau(1) > 0, tau(2) = 0, (3) tau(1) = 0, tau(2) > 0 and (4) > tau(1) > 0, tau(2) > 0. Optimal control theory is then applied to the model to study two timedependent treatment efforts and minimize the infection in citrus and psyllids, while keeping the implementation cost at a minimum. Numerical simulations of the overall SN Si em S are implemented in MatLab for demonstration of the theoretical results.