Data-driven modeling of imported malaria in Morocco and the impact of population migration

Hamza Toufga; Ghassane Benrhmach; Mustapha Lhous; Abdessamad Tridane; Hamza Toufga; Ghassane Benrhmach; Mustapha Lhous; Abdessamad Tridane

doi:10.3934/mbe.2026042

Mathematical Biosciences and Engineering

2026, Volume 23, Issue 4: 1121-1155. doi: 10.3934/mbe.2026042

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Data-driven modeling of imported malaria in Morocco and the impact of population migration

1.
Laboratory of Mathematical Analysis, Algebra and Applications (LAM2A), Department of Mathematics and Computer Science, Faculty of Sciences Ain Chock, Hassan Ⅱ University of Casablanca, Morocco
2.
Department of Mathematics & Statistics, College of Engineering, Abu Dhabi University, Abu Dhabi, United Arab Emirates
3.
Department of Mathematics, College of Science, United Arab Emirates University, Al Ain, United Arab Emirates

Received: 25 September 2025 Revised: 05 January 2026 Accepted: 08 January 2026 Published: 11 March 2026

Malaria remains a very critical health-threatening issue worldwide, and increasingly, migration is viewed as a cause of re-emergence in malaria-free zones. This study proposes a data-driven, deterministic compartmental model that explicitly accounts for human immigration when modeling malaria transmission from humans to mosquito populations. The human host is categorized into susceptible, asymptomatic, infected, and recovered classes, while the vector population is stratified into susceptible and infected compartments. The paper presents a rigorous mathematical analysis to prove the positivity and uniqueness of solutions, and further demonstrates that the model only admits a globally stable endemic equilibrium. Using Morocco as a case study, key parameters are estimated using three complementary methods: least-squares fit, Extended Kalman Filter, and Long Short-Term Memory (LSTM) neural networks, which where applied to imported malaria data from 1990 through 2023. To identify the best control measures to prevent the possible re-establishment of malaria in Morocco, we investigate an optimal control problem that includes awareness, prophylactic treatment, treatment of the infected population, and vector control with insecticide. The goal is to identify the optimal effectiveness of these controls. Numerical results show a significant reduction in human and vector infections through a combination of control strategies, thus underscoring the importance of surveillance and control policies that account for migration. Therfore, it serves as a practical, flexible framework to analyze imported malaria and other vector-borne diseases across similarly high-mobility areas.
- imported malaria,
- human migration,
- estimation of parameters,
- optimal control,
- Pontryagin's maximum principle,
- numerical simulation
Citation: Hamza Toufga, Ghassane Benrhmach, Mustapha Lhous, Abdessamad Tridane. Data-driven modeling of imported malaria in Morocco and the impact of population migration[J]. Mathematical Biosciences and Engineering, 2026, 23(4): 1121-1155. doi: 10.3934/mbe.2026042

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Abstract

Malaria remains a very critical health-threatening issue worldwide, and increasingly, migration is viewed as a cause of re-emergence in malaria-free zones. This study proposes a data-driven, deterministic compartmental model that explicitly accounts for human immigration when modeling malaria transmission from humans to mosquito populations. The human host is categorized into susceptible, asymptomatic, infected, and recovered classes, while the vector population is stratified into susceptible and infected compartments. The paper presents a rigorous mathematical analysis to prove the positivity and uniqueness of solutions, and further demonstrates that the model only admits a globally stable endemic equilibrium. Using Morocco as a case study, key parameters are estimated using three complementary methods: least-squares fit, Extended Kalman Filter, and Long Short-Term Memory (LSTM) neural networks, which where applied to imported malaria data from 1990 through 2023. To identify the best control measures to prevent the possible re-establishment of malaria in Morocco, we investigate an optimal control problem that includes awareness, prophylactic treatment, treatment of the infected population, and vector control with insecticide. The goal is to identify the optimal effectiveness of these controls. Numerical results show a significant reduction in human and vector infections through a combination of control strategies, thus underscoring the importance of surveillance and control policies that account for migration. Therfore, it serves as a practical, flexible framework to analyze imported malaria and other vector-borne diseases across similarly high-mobility areas.

References

[1]	World Health Organization (WHO), World malaria report 2023. Available from: https://www.who.int/teams/global-malaria-programme/reports/world-malaria-report-2023.
[2]	World Health Organization (WHO), World malaria report 2024. Available from: https://www.who.int/teams/global-malaria-programme/reports/world-malaria-report-2024.
[3]	Medicines for Malaria Venture, Causes of malaria, 2023. Available from: https://www.mmv.org/malaria/about-malaria/causes-malaria.
[4]	World Health Organization, Malaria fact sheet, 2023. Available from: https://www.who.int/news-room/fact-sheets/detail/malaria.
[5]	World Health Organization, Malaria data and statistics, 2023. Available from: https://www.who.int/data/gho/data/themes/malaria.
[6]	M. Roser, H. Ritchie, Malaria, Our World in Data, 2015. Available from: https://ourworldindata.org/malaria.
[7]	World Health Organization, Malaria: Questions and answers, World Health Organization, 2023. Available from: https://www.who.int/news-room/questions-and-answers/item/malaria.
[8]	Medicines for Malaria Venture, Malaria facts and statistics, 2025. Available from: https://www.mmv.org/malaria/about-malaria/malaria-facts-statistics-2025.
[9]	K. C. Mezieobi, E. U. Alum, O. Paul-Chima Ugwu, D. E. Uti, B. N. Alum, S. I. Egba, et al., Economic burden of malaria on developing countries: A mini review, Parasite Epidemiol. Control, 30 (2025), e00435. https://doi.org/10.1016/j.parepi.2025.e00435 doi: 10.1016/j.parepi.2025.e00435
[10]	F. Ricci, Social implications of malaria and their relationships with poverty, Mediterr. J. Hematol. Infect. Dis., 4 (2012), e2012048. https://doi.org/10.4084/MJHID.2012.048 doi: 10.4084/MJHID.2012.048
[11]	G. K. Amina, Socio-economic determinants and malaria risk: Assessing the impact of poverty, housing conditions, and healthcare accessibility in high-incidence regions, Sciences, 5 (2024), 120–124. https://doi.org/10.59298/nijrms/2024/5.3.120124 doi: 10.59298/nijrms/2024/5.3.120124
[12]	World Health Organization, Global technical strategy for malaria 2016–2030, World Health Organization, Geneva, 2015. Available from: https://www.who.int/publications/i/item/9789240031357.
[13]	Centers for Disease Control and Prevention, Strategies for reducing malaria's global impact, April 1, 2024. National Center for Emerging and Zoonotic Infectious Diseases (NCEZID). Available from: https://www.cdc.gov/malaria/php/public-health-strategy/index.html.
[14]	International Organization for Migration, Malaria and mobility, Information Sheet, Migration Health Division, International Organization for Migration, 24 April 2018. Available from: https://www.iom.int/sites/default/files/our_work/DMM/Migration-Health/mhd_infosheet_malaria_and_mobility_24.04.2018.pdf.
[15]	European Centre for Disease Prevention and Control, Malaria: Annual epidemiological report for 2022. ECDC Surveillance Report. Available from: https://www.ecdc.europa.eu/en/publications-data/malaria-annual-epidemiological-report-2022.
[16]	Z. Herrador, B. Fernández-Martinez, V. Quesada-Cubo, O. Diaz-Garcia, R. Cano, A. Benito, et al., Imported cases of malaria in spain: Observational study using nationally reported statistics and surveillance data, 2002–2015, Malar. J., 18 (2019), 230. https://doi.org/10.1186/s12936-019-2863-2 doi: 10.1186/s12936-019-2863-2
[17]	K. Nabah, N. Mezzoug, A. Aarab, H. Oufdou, K. Rharrabe, Epidemiological profile of imported malaria in the northern region of Morocco, 2014–2018, E3S Web Conf., 319 (2021), 01057. https://doi.org/10.1051/e3sconf/202131901057 doi: 10.1051/e3sconf/202131901057
[18]	A. Hanafi, I. Zouaoui, H. Abjabja, Y. Abercha, S. Aoufi, Imported malaria at the ibn sina university hospital in rabat: A retrospective study of 81 cases, Cureus, 16 (2024), e60253. https://doi.org/10.7759/cureus.60253 doi: 10.7759/cureus.60253
[19]	H. El Amouri, Sub-saharan immigration to morocco: Some impacts of human mobility on a state's domestic policy and geopolitics, Int. Social Sci. Manage. J., 2022 (2022).
[20]	N. W. Ruktanonchai, P. DeLeenheer, A. J. Tatem, V. A. Alegana, T. T. Caughlin, E. zu Erbach-Schoenberg, et al., Identifying malaria transmission foci for elimination using human mobility data, Comput. Biol., 12 (2016), e1004846. https://doi.org/10.1371/journal.pcbi.1004846 doi: 10.1371/journal.pcbi.1004846
[21]	O. Abdelkrim, Z. Said, L. Souad, Anopheles mosquitoes in Morocco: Implication for public health and underlined challenges for malaria re-establishment prevention under current and future climate conditions, Pest. Manage. Sci., 80 (2024), 2085–2095. https://doi.org/10.1002/ps.7943 doi: 10.1002/ps.7943
[22]	J. C. Semenza, J. E. Suk, Vector-borne diseases and climate change: A European perspective, FEMS Microbiol. Lett., 365 (2018), fnx244. https://doi.org/10.1093/femsle/fnx244 doi: 10.1093/femsle/fnx244
[23]	R. Ross, Some a priori pathometric equations, Proc. R. Soc. London, Ser. A, 81 (1908), 465–486. https://doi.org/10.1136/bmj.1.2830.546. doi: 10.1136/bmj.1.2830.546
[24]	G. Macdonald, The Epidemiology and Control of Malaria, Oxford University Press, 1957.
[25]	K. Dietz, L. Molineaux, A. Thomas, A malaria model tested in the african savannah, Bull. World Health Organ., 50 (1974), 347–357.
[26]	A. I. Abioye, M. O. Ibrahim, O. J. Peter, H. A. Ogunseye, Optimal control on a mathematical model of malaria, Sci. Bull., Series A: Appl. Math. Phy., 83 (2020), 177–190.
[27]	G. Otieno, J. K. Koske, J. M. Mutiso, Transmission dynamics and optimal control of malaria in kenya, Discrete Dyn. Nat. Soc., 2016 (2016), 8013574. https://doi.org/10.1155/2016/8013574 doi: 10.1155/2016/8013574
[28]	S. Kim, A. Tridane, D. E. Chang, Human migrations and mosquito-borne diseases in Africa, Math. Popul. Stud., 23 (2016), 123–146. https://doi.org/10.1080/08898480.2015.1054221 doi: 10.1080/08898480.2015.1054221
[29]	S. Yacheur, A. Moussaoui, A. Tridane, Modeling the imported malaria to north africa and the absorption effect of the immigrants, Math. Biosci. Eng., 16 (2019), 967–989. https://doi.org/10.3934/mbe.2019045. doi: 10.3934/mbe.2019045
[30]	A. Rajnarayanan, M. Kumar, A. Tridane, Analysis of a mathematical model for malaria using data-driven approach, Sci. Rep., 15 (2025), 27272. https://doi.org/10.1038/s41598-025-12078-4 doi: 10.1038/s41598-025-12078-4
[31]	S. Olaniyi, K. O. Okosun, S. O. Adesanya, R. S. Lebelo, Modelling malaria dynamics with partial immunity and protected travellers: Optimal control and cost-effectiveness analysis, J. Biol. Dyn., 14 (2020), 90–115. https://doi.org/10.1080/17513758.2020.1722265 doi: 10.1080/17513758.2020.1722265
[32]	K. Y. Ng, M. M. Gui, COVID-19: Development of a robust mathematical model and simulation package with consideration for ageing population and time delay for control action and resusceptibility, Physica D: Nonlinear Phenom., 411 (2020), 132599. https://doi.org/10.1016/j.physd.2020.132599 doi: 10.1016/j.physd.2020.132599
[33]	United Nations Department of Economic and Social Affairs, Population Division, International migrant stock 2024 by sex and destination, 2024. Available from: https://www.un.org/development/desa/pd/content/international-migrant-stock.
[34]	Macrotrends LLC, Morocco immigration statistics — historical chart & data, 2025. Includes data points for 2000, 2005, 2010, and 2015. Available from: https://www.macrotrends.net/countries/MAR/morocco/immigration-statistics.
[35]	Ministry of Health of Morocco, National statistics on malaria, 1990–2001. Archived statistical data, 2002. Available through annual statistical reports of the Ministry of Health, Rabat. Available from: https://www.sante.gov.ma.
[36]	World Health Organization Regional Office for the Eastern Mediterranean, Morocco: Malaria burden and elimination status, 2011. Available from: https://www.emro.who.int/mor/morocco-news/morocco-certified-malaria-free.html.
[37]	Ministry of Health of Morocco, Health in figures 2012–2013 edition (revised version), 2013. Available from: https://www.sante.gov.ma/Publications/Etudes_enquete.
[38]	Ministry of Health of Morocco, Health in figures 2013–2014 edition (revised version), 2014. Available from: https://www.sante.gov.ma/Publications/Etudes_enquete.
[39]	Ministry of Health of Morocco, Health in figures 2014–2015 edition, 2015. Available from: https://www.sante.gov.ma/Publications/Etudes_enquete.
[40]	Ministry of Health of Morocco, Health in figures 2015–2016 edition, 2016. Available from: https://www.sante.gov.ma/Publications/Etudes_enquete.
[41]	Ministry of Health of Morocco, Health in figures 2016, 2017. Available from: https://www.sante.gov.ma/Publications/Etudes_enquete.
[42]	Ministry of Health of Morocco, Health in figures 2017, 2018. Available from: https://www.sante.gov.ma/Publications/Etudes_enquete.
[43]	Ministry of Health of Morocco, Health in figures 2018, 2019. Available from: https://www.sante.gov.ma/Publications/Etudes_enquete.
[44]	Ministry of Health of Morocco, Health in figures 2019, 2020. Available from: https://www.sante.gov.ma/Publications/Etudes_enquete.
[45]	Ministry of Health and Social Protection of Morocco, Health in figures 2020, final version, 2021. Available from: https://www.sante.gov.ma/Publications/Etudes_enquete.
[46]	Ministry of Health and Social Protection of Morocco, Health in figures 2021, final version, 2024. Available from: https://www.sante.gov.ma/Publications/Etudes_enquete.
[47]	Ministry of Health and Social Protection of Morocco, Health in figures 2022, final version, 2024. Available from: https://www.sante.gov.ma/Publications/Etudes_enquete.
[48]	Ministry of Health and Social Protection of Morocco, Health in figures 2023, final version, 2025. Available from: https://www.sante.gov.ma/Publications/Etudes_enquete.
[49]	N. Chitnis, J. M. Hyman, J. M. Cushing, Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bull. Math. Biol., 70 (2008), 1272–1296. https://doi.org/10.1007/s11538-008-9299-0 doi: 10.1007/s11538-008-9299-0
[50]	C. Chiziba, S. Silal, Exploring measures to increase detection of malaria cases through reactive case detection in a southern province of zambia-like setup: A modelling study, medRxiv. https://doi.org/10.1101/2024.07.18.24310660
[51]	Hespress English, Health minister urges stronger malaria prevention among travelers as cases note increase, 2025.
[52]	E. W. Weisstein, Least squares fitting, MathWorld—A Wolfram Web Resource, Accessed 2022-01-15.
[53]	A. H. Jazwinski, Stochastic Processes and Filtering Theory, Academic Press, New York, 1970.
[54]	P. S. Maybeck, Stochastic Models, Estimation, and Control, Academic Press, New York, 1982.
[55]	S. Hochreiter, J. Schmidhuber, Long short-term memory, Neural Comput., 9 (1997), 1735–1780. https://doi.org/10.1162/neco.1997.9.8.1735.
[56]	K. Greff, R. K. Srivastava, J. Koutník, B. R. Steunebrink, J. Schmidhuber, Lstm: A search space odyssey, IEEE Trans. Neural Networks Learn. Syst., 28 (2017), 2222–2232. https://doi.org/10.1109/TNNLS.2016.2582924
[57]	W. H. Fleming, R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer-Verlag, Berlin, Heidelberg, 1975.
[58]	G. Birkhoff, G. C. Rota. Ordinary Differential Equations, Introductions to Higher Mathematics, Wiley, 1978.
[59]	H. Toufga, A. Sakkoum, L. Benahmadi, M. Lhous, Analysis of the dynamics and optimal control of cutaneous Leishmania during human immigration, Iran. J. Numer. Anal. Optim., 15 (2025), 311–345. https://doi.org/15(1):311–345, 2025.
[60]	J. P. LaSalle, Stability theory for ordinary differential equations, J. Differ. Equ., 4 (1968), 57–65.

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