Mathematical modeling of Echinococcosis in humans, dogs and livestock with optimal control strategies

Iranian Journal of Numerical Analysis and Optimization

مقالات آماده انتشار، اصلاح شده برای چاپ، انتشار آنلاین از تاریخ 26 فروردین 1404   

نوع مقاله: Research Article

شناسه دیجیتال (DOI): 10.22067/ijnao.2025.90038.1530

نویسندگان

I. Sannaky^* ؛ M. Riouali؛ N. Ouldkhouia؛ I. El berrai؛ K. Adnaoui

Laboratory of Analysis, Modeling and Simulation, HassanII university, Faculty of sciences Ben M’sik, Casablanca, Morocco.

چکیده

In this paper, we present a deterministic model of Cystic Echinococcosis disease of eleven differential equations, describing complex interactions between three types of hosts, specifically humans, livestock and dogs. The model is analyzed; disease-free and endemic equilibrium exist and are globally asymptotically stable if the basic reproduction number is less than or greater than one, respectively. The optimal control represents the efficiency of health education and EG95 sheep vaccination. The existence of optimal controls is proven, the characterization is formulated using Pontriyagin’s maximum principle, and the optimal system is derived. For the numerical simulation, the data used are from some studies done in Morocco.

کلیدواژه‌ها

Cystic Echinococcosis؛ deterministic modeling؛ basic reproduction number؛ global stability؛ optimal control

مراجع

[1] Amarir, F., Rhalem, A., Sadak, A., Raes, M., Oukessou, M., Saadi, A., Bouslikhane, M., Gauci, C.G., Lightowlers, M.W., Kirschvink, N. and Marcotty, T. Control of cystic echinococcosis in the Middle Atlas, Morocco: Field evaluation of the EG95 vaccine in sheep and cesticide treatment in dogs, PLoS Negl. Trop. Dis. 15(3) (2021) e0009253.
[2] Amarir, F.E., Saadi, A., Marcotty, T., Rhalem, A., Oukessou, M., Sahibi, H. Obtel, M., Bouslikhane, M., Sadak, A. and Kirschvink, N. Cystic echinococcosis in three locations in the Middle Atlas, Morocco: Estimation of the infection rate in the dog reservoir’, Vector-Borne and Zoonotic Diseases, 20 (6) (2020) 436–443.
[3] Azlaf, R. and Dakkak, A. Epidemiological study of the cystic echinococcosis in Morocco, Vet. Parasitol. 137 (2006) 83–93.
[4] Birhan, G.B., Munganga, J.M.W. and Hassan, A.S. Mathematical modeling of echinococcosis in humans, dogs, and sheep. J. Appl. Math.2020 (2020) Article ID 8482696, 18 pages.
[5] Chacha, C.S., Stephano, M.A., Irunde, J.I. and Mwasunda, J.A. Cystic echinococcosis dynamics in dogs, humans and cattle: Deterministic and stochastic modeling, Results Phys. 51 (2023) 106697.

[6] Chebli, H., Laamrani El Idrissi, A., Benazzouz, M., Lmimouni, B.E., Nhammi, H., Elabandouni, M., Youbi, M., Afifi, R., Tahiri, S., Essayd El Feydi, A. and Settaf, A. Human cystic echinococcosis in Morocco: Ultrasound screening in the Mid Atlas through an Italian-Moroccan part-nership, PLoS Negl. Trop. Dis. 11(3) (2017) e0005384.
[7] Diekmann, O., Heesterbeek, J.A.P. and Metz, J.A.J. On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations. J. Math. Biol. 28(4) (1990) 365–382.
[8] El-Berbri, I. Epidémiologie de l’échinococcose/ hydatidose kystique dansla province de Sidi Kacem et évaluation d’actions de lutte conséquentes dans le cadre d’une approche intégrée associant la leishmaniose viscérale et la rage, PhD diss., Ph. D Thesis. IAV Hassan II, Rabat, Morocco, 2015.
[9] El-Berbri, I., Mahir, W., Fihri, O.F., Petavy, A.F., Dakkak, A. and Bouslikhane, M. Cystic echinococcosis in Morocco: epidemiology, socio-economic impact and control, J. Vet. Parasitol. 34(2) (2020) 72–78.
[10] Fleming, W.H. and Rishel. R.W. Deterministic and stochastic optimal control, Springer-Verlag, New York, 1975.
[11] Gong, W. and Wang, Z. Sliding motion control of Echinococcosis trans-mission dynamics Model, Math. Comput. Simul. 205 (2023) 468–482. 
[12] He, Y., Cui, Q. and Hu, Z. Modeling and analysis of the transmission dynamics of cystic echinococcosis: Effects of increasing the number of sheep, Math. Biosci. Eng. 20 (2023) 14596–14615.
[13] Jazouli, M., Lightowlers, M.W., Bamouh, Z., Gauci, C.G., Tadlaoui, K., Ennaji, M.M. and Elharrak, M. Immunological responses and potency of the EG95NC- recombinant sheep vaccine against cystic echinococcosis, Parasitol. Inter. 78 (2020) 102149.
[14] Lahmar, S., Debbek, H., Zhang, L.H., McManus, D.P., Souissi, A., Chelly, S. and Torgerson, P.R. Transmission dynamics of the Echinococ-cus granulosus sheep–dog strain (G1 genotype) in camels in Tunisia, Vet. Parasitol. 121(1-2) (2004) 151–156.
[15] Macrotrends. Morocco death rate 1950-2024. https://www. macrotrends.net/global-metrics/countries/MAR/morocco/ death-rate [16] Ouldkhouia, N., Elberrai, I., Adnaoui, K. and Benhachem, A. A general stochastic model for tumor growth: Simulating cardiac tumor (Myxoma) development, Math. Model. Eng. Probl. 11(12) (2024) 3323–3332.
[17] Ouldkhouia, N., Elberrai, I., Benhachem, A., Sannaky, I., Riouali, M. and Hachoum, S. On stochastic modelling: The impact of advertise-ment on the consumption-application on ChatGPT-3, Math. Model. Eng. Probl. 11 (10) (2024) 2705–2714.
[18] Saadi, A., Amarir, F., Filali, H., Thys, S., Rhalem, A., Kirschvink, N., Raes, M., Marcotty, T., Oukessou, M., Duchateau, L. and Sahibi, H. The socio-economic burden of cystic echinococcosis in morocco: A combination of estimation method, PLoS Negl. Trop. Dis., 14 (2020) 1–20.
[19] Tahiri, S., Naoui, H., Iken, M., Azelmat, S., Bouchrik, M. and Lmimouni, B.E. RETRAIT: Seroprevalence of cystic echinococcosis in the provinces of Ifrane and El Hajeb in Morocco, Med. Mal. Infect. (2020) 31928912.
[20] Thys, S., Sahibi, H., Gabriël, S., Rahali, T., Lefèvre, P., Rhalem, A., Marcotty, T., Boelaert, M. and Dorny, P. Community perception and knowledge of cystic echinococcosis in the High Atlas Mountains, Morocco. BMC Public Health 19 (2019) 1-15.
[21] Van den Driessche, P. and Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180(1-2) (2002) 29–48.
[22] Velasco-Tirado, V., Romero-Alegría, Á., Belhassen-García, M., Alonso-Sardón, M., Esteban-Velasco, C., López-Bernús, A., Carpio-Perez, A., Jimenez López, M.F., Muñoz Bellido, J.L., Muro, A. and Cordero-Sanchez, M. Recurrence of cystic echinococcosis in an endemic area: a retrospective study, BMC Infectious Diseases 17 (2017) 1–8.
[23] Wang, K., Teng, Z. and Zhang, X. Dynamical behaviors of an echinococ-cosis epidemic model with distributed delays, Math. Biosci. Eng. 14 (5& 6) (2017) 1425–1445.
[24] Wang, K., Zhang, X., Jin, Z., Ma, H., Teng, Z. and Wang, L. Modeling and analysis of the transmission of echinococcosis with application to Xinjiang Uygur Autonomous Region of China.,J .Theor. Biol. 333 (2013) 78–90.
[25] Weekly epidemiological record, (48) 2019 29. https://iris.who.int/ bitstream/handle/10665/330007/WER9448-574-579-eng-fre.pdf? sequence=1&isAllowed=y.
[26] World Health Organization. (n.d). Cystic and alveolar echinococ-cosis. https://www.who.int/docs/default-source/ntds/
echinococcosis/cystic-and-alveolar-echinococcosis.pdf? sfvrsn=fa73b7ca_8.
[27] World Health Organization. Echinococcosis fact sheet. 2020. https:// www.who.int/news-room/fact-sheets/detail/echinococcosis.