Fractional Model and Numerical Algorithms for Predicting Cactus Cochineal (Dactylopius opuntiae) Infestation on Prickly Pear with Quarantine and Treatment Strategies

Since 2014, the cactus cochineal Dactylopius opuntiae has been decimating prickly pear (Opuntia ficus-indica) plantations in Morocco and, more broadly, across the Maghreb, threatening nearly 160,000 hectares of cultivated land. The purpose of this paper is to investigate the mathematical modelling and dynamics of this plant pest using the Caputo–Fabrizio fractional derivative, in the presence of phytosanitary quarantine, treatment, and destruction strategies, as mandated by the 2017 Moroccan ministerial decree. The existence and uniqueness of solutions for the fractional model are proved using fixed-point iterations. The model exhibits a pest-free equilibrium and an endemic equilibrium, whose stability is governed by the effective reproduction number Re, derived here via a fully detailed next-generation-matrix method. We construct a fractional version of the four-step Adams–Bashforth method to numerically approximate the model, and use it to illustrate the evolution of the infestation for varying fractional orders and control parameters. A nonlinear least-squares estimation procedure is proposed to calibrate the model, and its practical identifiability is assessed on synthetic data by comparing recovered parameters against known ground-truth values. Finally, an optimal control problem combining a natural control (biological control by predatory ladybirds) and an artificial control (reinforced administrative quarantine) is formulated and solved via Pontryagin’s Maximum Principle, revealing a policy trade-off between effective containment and precautionary destruction of healthy plant material. Mathematics Subject Classification 34A08 · 92D30 · 65L05 · 49K15