Mathematical modelling and optimal control of production losses caused by Miridae

Cocoa mirid, Sahlbergella singulars , is the major pest of cocoa ( Theobroma cacao ) responsible of several damage in plots in West Africa and particularly in Cameroon. Occasional damage accounts for 30 40% of pod losses. However, when miridae affect the foliage, gradual wilting occurs and eventually, tree death. A few studies have focused on describing the time evolution of Miridae in the plot in Cameroon, yet numerous questions remain. The aim of this paper is to estimate and control the losses of production caused by the bites of miridea. To do this, we will formulate and study a mathematical model for the dynamics of pods that takes into account the feeding and egg-laying of adults miridae on pods. We present the theoretical analysis of the model. More precisely, we compute equilibria and derive a threshold parameter that determines the presence or not of miridae in the plot. Throughout numerical simulations, we found that miridae can cause approximately 39.21% of production losses (which represents approximatively USD 1276.8 revenue losses) when initially, one has 1200 plants in the plot. After, we aim to increase cocoa production through optimal control. Optimal control consists in reducing density the number of nymphs and adults miridae in the plot. We studied the controlled model and we found that losses with control shrink to 20.58% which corresponds to USD670.32 income revenue.