Modelling and optimal strategy to control coffee berry borer

Coffee berry borer Hypothenemus hampei (Coleoptera: Scolytidae), denoted CBB, is the most important insect pest of coffee worldwide, with a high impact on the economy of coffee producing countries. The insect spends a great part of its life cycle inside the coffee berry and causes severe crop losses. Biological control based on the use of an entomopathogenic fungus is a major alternative to chemical pesticides in order to control CBB. The fungus is sprayed on the coffee berries to kill CBB when the insects drill holes to penetrate inside the berries. Our aim in this work is to optimise the fungus application, using a modelling approach. We first formulate a mathematical model describing the infestation dynamics of coffee berries by CBB. We analyse the model and show that the stability of the pest-free and coexistence equilibria depends on the basic reproduction number. To introduce a control variable corresponding to the application of an entomopathogenic fungus, we then extend the model and include the fungus dynamics. We formulate an optimal problem which consists in maximising the coffee yield, while minimising the control cost, as well as the CBB population for the next cropping season. The existence of the optimal control and the necessary optimality condition are established using Pontryagin's Maximum Principle. The optimal control problem is solved numerically using the BOCOP software and simulations are provided, showing that the use of entomopathogenic fungus effectively controls CBB.