Modeling and Analysis of Vector-borne Plant Disease with Two Delays

Abstract

In this paper, a non-linear mathematical model is formulated and analyzed to understand the dynamics of plant disease in presence of predators. Here predators act as biological control agent that reduce the vectors carrying the disease pathogen. The mathematical model is formulated using system of delay differential equations by considering two delays. The delays are time taken for the plant to become infected after contagion and time taken for the insect to become infected after contagion. The existence and stability of different equilibria of the model are discussed in detail. The basic reproduction number R 0 is computed. The Routh-Hurwitz criteria is used to investigate the stability of the disease-free equilibrium and the endemic equilibrium in absence of delay. The stability of endemic equilibrium is also investigated in presence of delay. The occurrence of Hopf-bifurcation is demonstrated by considering the delay as the bifurcation parameter. The critical values of the delays are obtained which preserve the stability of the endemic equilibrium point. The model system shows an oscillatory behavior beyond this critical values of delays. Numerical simulation is also performed to support the analytical results. It is found that biological control has positive impact on reducing the transmission of plant disease.