mathematical modeling, stability analysis, optimal control, cochineal insect, cacti plants, Applied mathematics. Quantitative methods, T57-57.97

We propose a mathematical model, $SIM$ , that describes the dynamics of cochineal insect spread among cacti and examines the effects of various control strategies. The model is analyzed for the existence and unique-ness of solutions, and we investigate the equilibrium points and stability of the system using both local and global stability analyses. By perform-ing numerical simulations in $MATLAB$, we validate our theoretical find-ings. Furthermore, we propose an optimal control strategy to minimize the cochineal population in cacti fields. The optimal control problem is formulated using Pontryagin’s maximum principle, and the corresponding optimality system is solved iteratively. Our study compares three control strategies: cutting and burning infected cacti, insecticide spraying, and a combined approach. The results demonstrate that the combined strategy is the most effective in reducing the cochineal population. This research provides valuable insights into managing cochineal infestations and offers practical recommendations for farmers to control the spread of these pests.