Optimal control issues in plant disease with host demographic factor and botanical fungicides

In this paper, we discuss a mathematical model of plant disease with the effect of fungicide. We assume that the fungicide is given as a preventive treatment to infectious plants. The model is constructed based on the development of the disease in which the monomolecular is monocyclic. We show the value of the Basic Reproduction Number (BRN) R-0 of the plant disease transmission. The BRN is computed from the largest eigenvalue of the next generation matrix of the model. The result shows that in the region where R-0 greater than one there is a single stable endemic equilibrium. However, in the region where R-0 less than one this endemic equilibrium becomes unstable. The dynamics of the model is highly sensitive to changes in contact rate and infectious period. We also discuss the optimal control of the infected plant host by considering a preventive treatment aimed at reducing the infected host plant. The obtaining optimal control shows that it can reduce the number of infected hosts compared to that without control. Some numerical simulations are also given to illustrate our analytical results.