Mathematical modeling for <i>Hypothenemus hampei</i> and <i>Colletotrichum kahawae</i> co-dynamics with optimal control strategies

Several pests and diseases are major factors challenging the coffee industry worldwide. Particularly, production of <i>Coffee Arabica</i> in many African countries has been affected by <i>Hypothenemus hampei</i> and <i>Colletotrichum kahawae</i> in a coffee farm. Pest(s) and disease(s) are commonly inter-related and can interact, because pests and pathogens have the same biophysical requirements in ecosystems. Assessment of coffee berries damage due to multiple pests and diseases is a necessary step in designing appropriate control strategies. In this paper, we developed a mathematical model describing the co-dynamics of <i>Hypothenemus hampei</i> (coffee berry borer, CBB) and <i>Colletotrichum kahawae</i> (coffee berry disease, CBD). The model used a system of nonlinear ordinary differential equations to capture the interactions among the CBB pest population, the CBD fungal pathogen, and the healthy and infected coffee berry populations. Optimal control strategies were also incorporated to assess effective management approaches. Optimal control strategies were obtained by minimizing the number of pests and fungal pathogen population by incorporating two control variables such as biological control and cultural practices. The existence of optimal controls was examined using Pontryagin's minimum principle. The Hamiltonian was constructed, and adjoint equations were solved to minimize the cost functional. Lastly, from different scenarios, the numerical simulations were performed to illustrate the model's co-dynamics with and without optimal control strategies.