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

Document Type : Review Article

Authors

K. Sofiane

B. Omar

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

10.22067/ijnao.2025.90998.1557

Abstract

This paper introduces a mathematical model, denoted as SIRM C, aimed at understanding the dynamics of cochineal infestation in cacti plants. The model incorporates two control strategies: biological control through Hyperaspis trifurcata, a natural predator of cochineal, and chemical control via insecticide spraying. The objective is to reduce the number of infected cacti while also achieving a balance between minimizing infection, maximizing recovery over time, and minimizing the costs associated with the control measures. The proposed framework effectively integrates these strategies to manage cochineal dynamics. Optimal control strategies are derived using Pontryagin’s maximum principle, and numerical simulations conducted in MATLAB validate the theoretical results.

Keywords

Mathematical modeling

Optimal control theory

Pontryagin Maximum

Cochineal

Cacti plants

Subjects

• Modeling and Simulation

---

References

[1] Arba, M., Cherif Benismail, M., and Mokhtari, M. The cactus pear (Opuntia spp.) in Morocco: Main species and cultivar characterization, Acta Hortic. 581, (2002), 103–109.
[2] Birkhoff, G., and Rota, G.C. Ordinary differential equations, 4th ed. New York: John Wiley & Sons; 1989.
[3] Bouharroud, R., Amarraque, A., and Qessaoui, R. First report of the opuntia cochineal scale Dactylopius opuntiae (Hemiptera: Dactylopiidae) in Morocco, Bulletin OEPP/EPPO Bulletin, 46(2), (2016), 308–310.
[4] Chávez-Moreno, C. K., Tecante, A., and Casas, A. The opuntia (Cactaceae) and dactylopius (Hemiptera: Dactylopiidae) in Mexico: A historical perspective of use, interaction and distribution, Biodivers. Conserv. 18, (2009) 3337–3355.
[5] De Jesus Mendez-Gallegos, S., Tiberi, R. and Panzavolta, T. Carmine cochineal dactylopius coccus Costa (Rhynchota: Dactylopiidae): Significance, production and use, Adv. Hortic. Sci. 17(3) (2023) 165–171.
[6] Fleming, W.H., and Rishel, R.W. Deterministic and stochastic optimal control, New York, NY, USA, Springer, 1975.
[7] Gumel, A. B., Patidar, K. C., and Spiteri, R. J. Asymptotically consistent non-standard finite-difference methods for solving mathematical models arising in population biology, R.E. Mickens, World Scientific, Singapore, 2005.
[8] Gumel, A.B., Shivakumar, P.N., and Sahai, B.M. A mathematical model for the dynamics of HIV-1 during the typical course of infection, Third World Congress of Nonlinear Analysts, 47, (2001) 2073–2083.
[9] Hattaf, K., Rachik, M., Saadi, S., Tabit, Y., and Yousfi, N. Optimal control of tuberculosis with exogenous reinfection, Appl. Math. Sci. 3(5), (2009) 231–240.
[10] J. Karrakchou, M., Rachik, and Gourari, S. Optimal control and infectiology: Application to an HIV/AIDS model, Appl. Math. Comput. 177, (2006) 807–818.
[11] Lukes, D.L. Differential Equations: Classical to Controlled, Academic Press, New York, 1982.
[12] Mora, M., Cortés, M., Sanhueza, C., and Sáenz, C. Innovation requirements for the development of cactus pear for export: A new item to be incorporated to the chilean fruit export sector, Acta Hortic. 995, (2013) 229–236.
[13] Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., and Mishchenko, E.F. The mathematical theory of optimal processes, New York, NY, USA, Wiley, 1962.

[14] Ramadan, M.F., and Mörsel, J.T. Oil cactus pear (Opuntia ficus-indica L.), Food Chem. (2003) 339–345.
[15] Shetty, A.A., Rana, M.K., and Preetham, S.P. Cactus: A medicinal food, J. Food Sci. Technol. 49(5), (2012) 530–536.
[16] https://www.agriculture.gov.ma/ar/actualites/ albhth-alzray-tqdm-ham-fy-mkafht-alhshrt-alqrmzyt-almdmrt-llsbar.
[17] https://www.agriculture.gov.ma/ar/filieres-regions/ cactus-gon.