Résumé
The main objective of this paper is to present a mathematical model of pathogen-host interactions of Banana Black Sigatoka Disease that incorporates mating limitation and three control strategies to reduce disease incidence, namely the use of resistant plants, the use of systemic fungicides, and the use of cultural management. First, we study the system with constant variables associated with resistant plants, cropping practices, and fungal or bio-fungal treatments. Resistant varieties are used in a preventive way to prevent spore germination. Conversely, cropping practices and systemic fungicides are used as curative measures to reduce spore production and infected leaves. We perform a sensitivity analysis to assess the influence of all disease parameters. We observe that some parameters, like the recruitment rate of healthy leaves, the efficiency of resistance, the infection rate, the spore production rate, and the mortality of infected leaves highly influence the disease epidemic. We then rely on the optimal control theory to assess and optimize the effectiveness of fungicides or bio-fungicides and cropping practices combined with resistant plants. We show the existence of the optimal control for our model and rely on Pontryagin's maximum principle for its characterization. We show that the incidence of disease can be optimally reduced. Specifically, we demonstrate how combining different control modes reduces disease incidence more effectively. Finally, we provide numerical simulations to reinforce our analytical results.