Mathematical modeling, analysis, and optimal control of the cochineal insect impact on cacti plants

Iranian Journal of Numerical Analysis and Optimization

مقاله 15، دوره 15، Issue 2 - شماره پیاپی 33، شهریور 2025، صفحه 823-851   

نوع مقاله: Research Article

شناسه دیجیتال (DOI): 10.22067/ijnao.2025.91108.1558

نویسندگان

S. Khassal^{* 1}؛ E.M. Moumine²؛ O. Balatif¹

¹Department of Mathematics, Faculty of Sciences El Jadida, Chouaib Doukkali University, El Jadida, Morocco.

²Department of Mathematics and Computer Science, Faculty of science Ben M’sik, University Hassan II, Casablanca, Morocco.

چکیده

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.

کلیدواژه‌ها

Mathematical modeling؛ Stability analysis؛ Optimal control؛ Cochineal insect؛ Cacti plants

مراجع

[1] Agri-Mag. Cactus, Cochineal, and Biological Con-trol, https://www.agri-mag.com/2017/06/22/ cactus-cochenille-et-lutte-biologique/#:~:text=Le%20cactus% 20est%20pr%20sent%20dans,%C3%A9cosyst%C3%A8mes%20%C3%A0%
20travers%20le%20monde. 

[2] Bani-Yaghoub, M., Gautam, R., Shuai, Z., van den Driessche, P., and Ivanek, R. Reproduction numbers for infections with free-living pathogens growing in the environment, J. Biol. Dyn. 6 (2) (2012) 923–940.
[3] Birkhoff, G., and Rota, G.C. Ordinary differential equations, 4th ed., John Wiley and Sons, 1989.
[4] Chowell, G., Simonsen, L., Viboud, C., and Kuang, Y. Is west Africa approaching a catastrophic phase or is the 2014 Ebola epidemic slow-ing down? Different models yield different answers for Liberia, PLoS Currents, 6 (2014) ecurrents-outbreaks.

[5] El Baz, O., Ait Ichou, M., Laarabi, H., and Rachik, M. Stability analysis of a fractional model for the transmission of the cochineal, Math. Model. Comput. 10 (2) (2023) 379–386.
[6] Farkas, M., and Weinberger, H.F. Optimal control theory in mathemat-ical biology, Springer, 1998.
[7] Fleming, W.H., and Rishel, R.W. Deterministic and stochastic optimal control, Springer, 1975.
[8] Gibernau, M. Introduction to Cochineal and Its Role in Global Trade, J. Econ. Bot. 28 (3) (2019) 101–105.
[9] Aït Ichou, M., and Laarabi, H. Application of SIR model to study the spread of cochineal on cacti, Math. Biosci., 2021.
[10] Khaleque, A., and Sen, P. An empirical analysis of the Ebola outbreak in west Africa, Sci. Rep., 7 (2017) 42594.
[11] LaSalle, J.P. The stability of dynamical systems, Regional Conference Series in Applied Mathematics, 25, SIAM, 1976.
[12] Lotka, A.J. Elements of physical biology, Williams and Wilkins, 1925.
[13] De Jesus Mendez-Gallegos, S., Tiberi, R. and Panzavolta, T. Carmine Cochineal *Dactylopius Coccus* Costa (Rhynchota: Dactylopiidae): Sig-nificance, Production and Use, Adv. Hortic. Sci. 17 (3) (2003) 165–171.
[14] Pell, B., Kuang, Y., Viboud, C., and Chowell, G. Using phenomenological models for forecasting the 2015 Ebola challenge, Epidemics, 22 (2018) 62–70.
[15] Pontryagin, L. S., Boltyanskii, V.G., Gamkrelidze, R.V., and Mishchenko, E.F. The mathematical theory of optimal processes, Wi-ley, 1962.
[16] Rachik, M., and Laarabi, H. Mathematical modeling of pest dynamics in agricultural systems: A review, Int. J. Appl. Math. Comput. Sci. 2020.
[17] Verhulst, P.F. Notice on the law of population growth, Corresp Math. Phys. 10 (1838) 113–126.
[18] van den Driessche, P., and Watmough, J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180 (2002) 29–48.