Abstract
In this paper, a plant disease model with continuous cultural control strategy and time delay is formulated. Then, how the time delay affects the overall disease progression and, mathematically, how the delay affects the dynamics of the model are investigated. By analyzing the transendental characteristic equation, stability conditions related to the time delay are derived for the disease-free equilibrium. Specially, whenR0=1, the Jacobi matrix of the model at the disease-free equilibrium always has a simple zero eigenvalue for allτ≥0. The center manifold reduction and the normal form theory are used to discuss the stability and the steady-state bifurcations of the model near the nonhyperbolic disease-free equilibrium. Then, the sensitivity analysis of the threshold parameterR0and the positive equilibriumE*is carried out in order to determine the relative importance of different factors responsible for disease transmission. Finally, numerical simulations are employed to support the qualitative results.
Highlights
Viral disease is a key constraint on the production of staple food crop in the lesser developed countries
Diseases caused by plant viruses in cassava (Manihot esculenta), sweet potato (Ipomoea batatas) and plantain are among the main constraints on sustainable production of these vegetatively propagated staple food crops, see Rybicki and Pietersen [1], Dahal et al [2], Gibson and Aritua [3], and Thresh and Cooter [4]
A cultural control strategy including replanting or roguing diseased plants is a widely accepted treatment for plant epidemics which involves periodic inspections followed by removal of the infected plants
Summary
Viral disease is a key constraint on the production of staple food crop in the lesser developed countries. They found a threshold parameter R0 and proved that (4) owns a unique positive equilibrium when R0 > 1, and its stability was numerically checked They have shown that the development of new and improved disease control methods for viral diseases of vegetatively propagated staple food crops ought to take evolutionary responses of the virus into consideration. Delay differential equations are difficult to analyze; see Madden [21] and Murray [22] and, despite widespread adoption of discrete time approximations to van der Plank’s model in early simulations of plant disease, see Teng [23], Zadoks [24] and a number of often subtle mathematical analyses that followed by Jeger [25, 26], Kushalappa and Ludwig [27], Waggoner [28], the model is rarely used in theoretical studies.
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