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Global dynamic analysis of a vector-borne plant disease model. (English) Zbl 1344.92180
Summary: An epidemic model which describes vector-borne plant diseases is proposed with the aim to investigate the effect of insect vectors on the spread of plant diseases. Firstly, the analytical formula for the basic reproduction number $$R_0$$ is obtained by using the next generation matrix method, and then the existence of disease-free equilibrium and endemic equilibrium is discussed. Secondly, by constructing a suitable Lyapunov function and employing the theory of additive compound matrices, the threshold for the dynamics is obtained. If $$R_0\geq 1$$, then the disease-free equilibrium is globally asymptotically stable, which means that the plant disease will disappear eventually; if $$R_0> 1$$, then the endemic equilibrium is globally asymptotically stable, which indicates that the plant disease will persist for all time. Finally some numerical investigations are provided to verify our theoretical results, and the biological implications of the main results are briefly discussed in the last section.

##### MSC:
 92D30 Epidemiology 34D20 Stability of solutions to ordinary differential equations
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