Abstract
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 R0 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 R0 ≤ 1, then the disease-free equilibrium is globally asymptotically stable, which means that the plant disease will disappear eventually; if R0 > 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.
Highlights
In the natural world, plants are very important, since they are the survival foundation for all kinds of creatures, including human being, animals, and even microbes
The research of plant diseases is attractive to epidemiologists. They need to establish a simple plausible mechanism to protect susceptible hosts, allowing coexistence of pathogens and hosts, which is consistent with empirical studies of diseases in plant populations. The dynamics of these host-pathogen systems are routinely modeled by compartmental susceptible-infected-removed (SIR) epidemic models
We will use the results for the three dimensional competitive systems that live in convex sets [ – ] and a powerful theory of additive compound matrix to prove asymptotic orbital stability of periodic solutions [, ]
Summary
Plants are very important, since they are the survival foundation for all kinds of creatures, including human being, animals, and even microbes. An antagonist is included in their model to control plant diseases They obtained invasion criteria for all three species: host, pathogen and antagonist. The research of plant diseases is attractive to epidemiologists They need to establish a simple plausible mechanism to protect susceptible hosts, allowing coexistence of pathogens and hosts, which is consistent with empirical studies of diseases in plant populations. The dynamics of these host-pathogen systems are routinely modeled by compartmental susceptible-infected-removed (SIR) epidemic models. Early investigations of the epidemics caused by plant pathogens seldom included the demographics of the host population, replenishment of susceptible hosts is common in those models [ , , , ]. The rate at which new infections are created is determined by the matrix F, and the rates of transfer into and out of the class of infected states are represented by the matrix V ; these are given by
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