A Mathematical Model for the Vector Transmission and Control of Banana Xanthomonas Wilt

Banana {\it {Xanthomonas}} wilt is currently wrecking havoc in East and Central Africa. In this paper, a novel theoretical model for the transmission of banana {\it {Xanthomonas}} wilt by insect vectors is formulated and analyzed. The model incorporates roguing of infected plants and replanting using healthy suckers. The model is analyzed for the existence and stability of the equilibrium points. The global stability of the disease-free equilibrium point was determined by using a Lyapunov function and LaSalle's invariance principle. For the global stability of the endemic equilibrium point, the theory of competitive systems, compound matrices and stability of periodic orbits were used. It was established that if the basic reproduction number satisfies $R_0 \leq 1$, the disease-free equilibrium point is globally stable and the disease will be wiped out and if $R_0 > 1,$ the endemic equilibrium is stable and the disease persists. A numerical simulation of the model was also carried out. It was found out that at appropriate roguing and replanting, the disease can be contained.