Modeling tuberculosis transmission dynamics in Algeria: Effects of vaccination, exogenous reinfection, and endogenous reactivation

Rayane Boucherma; Mohammed-Salah Abdelouahab; Messaoud Berkal; Taha Radwan; Yakup Yildirim; Karim K. Ahmed; Rayane Boucherma; Mohammed-Salah Abdelouahab; Messaoud Berkal; Taha Radwan; Yakup Yildirim; Karim K. Ahmed

doi:10.3934/math.2026550

AIMS Mathematics

2026, Volume 11, Issue 5: 13339-13370. doi: 10.3934/math.2026550

Previous Article Next Article

Research article Special Issues

Modeling tuberculosis transmission dynamics in Algeria: Effects of vaccination, exogenous reinfection, and endogenous reactivation

1.
Laboratoire des Mathématiques Appliquées et Didactique, Ecole Normale Supérieure de Constantine, Constantine, Algeria
2.
Department of Mathematics and Computer Sciences, University Abdelhafid Boussouf, Mila 43000, Algeria
3.
Laboratory of Mathematics and Their Interactions, Abdelhafid Boussouf University, Mila 43000, Algeria
4.
Department of Management Information Systems, College of Business and Economics, Qassim University, Buraydah 51452, Saudi Arabia
5.
Department of Computer Engineering, Biruni University, Istanbul 34010, Turkey
6.
Mathematics Research Center, Near East University, Nicosia 99138, Cyprus
7.
Department of Mathematics, Faculty of Engineering, German International University (GIU), New Administrative Capital, Cairo, Egypt

Received: 25 February 2026 Revised: 26 April 2026 Accepted: 08 May 2026 Published: 13 May 2026
MSC : 34A55, 92D30, 62F10

This paper developed a novel nonlinear Susceptible–Vaccinated–Exposed–Infectious–Treated–Recovered–Susceptible (SVEITRS) compartmental model to investigate the transmission dynamics of tuberculosis (TB) in Algeria over the period between 1990–2024, explicitly accounting for partial Bacillus Calmette-Guérin (BCG) vaccine efficacy, endogenous reactivation of latent infection, and exogenous reinfection. The basic reproduction number $ \mathcal{R}_0 $ was derived using the next-generation matrix approach, and rigorous analytical results were established for the local and global stability of both the disease-free and endemic equilibrium points. Model parameters were estimated by fitting the model to Algerian TB surveillance data, achieving a coefficient of determination of $ R^2 = 0.84 $, which indicates a strong agreement between the model predictions and reported epidemiological trends. A global sensitivity analysis based on partial rank correlation coefficients (PRCCs) reveals that demographic recruitment and transmission-related parameters exert the greatest influence on TB dynamics, whereas intervention-related parameters such as vaccination and treatment rates have a comparatively weaker impact. These findings indicate that vaccination alone is insufficient to eliminate TB and highlight the need for integrated control strategies that combine sustained vaccination coverage, early case detection, and effective management of latent infections. The proposed modeling framework provides a quantitative basis for evaluating TB control policies in high-burden settings such as Algeria.
- mathematical model,
- tuberculosis,
- endogenous reactivation,
- exogenous reinfection,
- parameter estimation,
- partial rank correlation coefficient (PRCC)
Citation: Rayane Boucherma, Mohammed-Salah Abdelouahab, Messaoud Berkal, Taha Radwan, Yakup Yildirim, Karim K. Ahmed. Modeling tuberculosis transmission dynamics in Algeria: Effects of vaccination, exogenous reinfection, and endogenous reactivation[J]. AIMS Mathematics, 2026, 11(5): 13339-13370. doi: 10.3934/math.2026550

Related Papers:

Abstract

This paper developed a novel nonlinear Susceptible–Vaccinated–Exposed–Infectious–Treated–Recovered–Susceptible (SVEITRS) compartmental model to investigate the transmission dynamics of tuberculosis (TB) in Algeria over the period between 1990–2024, explicitly accounting for partial Bacillus Calmette-Guérin (BCG) vaccine efficacy, endogenous reactivation of latent infection, and exogenous reinfection. The basic reproduction number $ \mathcal{R}_0 $ was derived using the next-generation matrix approach, and rigorous analytical results were established for the local and global stability of both the disease-free and endemic equilibrium points. Model parameters were estimated by fitting the model to Algerian TB surveillance data, achieving a coefficient of determination of $ R^2 = 0.84 $, which indicates a strong agreement between the model predictions and reported epidemiological trends. A global sensitivity analysis based on partial rank correlation coefficients (PRCCs) reveals that demographic recruitment and transmission-related parameters exert the greatest influence on TB dynamics, whereas intervention-related parameters such as vaccination and treatment rates have a comparatively weaker impact. These findings indicate that vaccination alone is insufficient to eliminate TB and highlight the need for integrated control strategies that combine sustained vaccination coverage, early case detection, and effective management of latent infections. The proposed modeling framework provides a quantitative basis for evaluating TB control policies in high-burden settings such as Algeria.

References

[1]	S. Elaydi, R. Lozi, Global dynamics of discrete mathematical models of tuberculosis, J. Biol. Dynam., 18 (2024), 2323724. https://doi.org/10.1080/17513758.2024.2323724 doi: 10.1080/17513758.2024.2323724
[2]	S. Mandal, P. Biswas, W. Ansar, P. Mukherjee, J. J. Jawed, Tuberculosis of the central nervous system: Pathogenicity and molecular mechanism, In: A Review on Diverse Neurological Disorders, Elsevier, 2024, 93–102. https://doi.org/10.1016/B978-0-323-95735-9.00050-4
[3]	X. He, Threshold dynamics for a tuberculosis model with seasonality, Math. Biosci. Eng., 9 (2011), 111–122. https://doi.org/10.3934/mbe.2011.9.111 doi: 10.3934/mbe.2011.9.111
[4]	M. Shah, S. E. Dorman, Latent tuberculosis infection, New Engl. J. Med., 385 (2021), 2271–2280. https://doi.org/10.1056/NEJMra2039400 doi: 10.1056/NEJMra2039400
[5]	M. Adhikari, Tuberculosis and tuberculosis/HIV co-infection in pregnancy, Semin. Fetal Neonat. M., 14 (2009), 234–240. https://doi.org/10.1016/j.siny.2009.02.001 doi: 10.1016/j.siny.2009.02.001
[6]	P. E. M. Fine, BCG vaccination against tuberculosis and leprosy, Brit. Med. Bull., 44 (1988), 691–703.
[7]	A. Roy, M. Eisenhut, R. J. Harris, L. C. Rodrigues, S. Sridhar, S. Habermann, et al., Effect of BCG vaccination against Mycobacterium tuberculosis infection in children, BMJ, 349 (2014), g4643. https://doi.org/10.1136/bmj.g4643 doi: 10.1136/bmj.g4643
[8]	M. Igoe, R. Casagrandi, M. Gatto, C. M. Hoover, L. Mari, C. N. Ngonghala, et al., Reframing optimal control problems for infectious disease management in low-income countries, Bull. Math. Biol., 85 (2023), 31. https://doi.org/10.1007/s11538-023-01112-5 doi: 10.1007/s11538-023-01112-5
[9]	J. I. Irunde, J. Z. Ndendya, J. A. Mwasunda, P. K. Robert, Modeling the impact of screening and treatment on typhoid fever dynamics in unprotected population, Results Phys., 50 (2023), 107120. https://doi.org/10.1016/j.rinp.2023.107120 doi: 10.1016/j.rinp.2023.107120
[10]	M. De la Sen, S. Alonso-Quesada, A. Ibeas, On the stability of an SEIR epidemic model with distributed time-delay and a general class of feedback vaccination rules, Appl. Math. Comput., 268 (2015), 639–654. https://doi.org/10.1016/j.amc.2015.08.099 doi: 10.1016/j.amc.2015.08.099
[11]	C. Ozcaglar, A. Shabbeer, S. L. Vandenberg, B. Yener, K. P. Bennett, Epidemiological models of Mycobacterium tuberculosis complex infections, Math. Biosci., 236 (2012), 77–96. https://doi.org/10.1016/j.mbs.2012.02.003 doi: 10.1016/j.mbs.2012.02.003
[12]	M. S. Abdelouahab, A. Arama, R. Lozi, Bifurcation analysis of a model of tuberculosis epidemic with treatment of wider population suggesting a possible role in the seasonality of this disease, Chaos, 31 (2021), 123125. https://doi.org/10.1063/5.0057635 doi: 10.1063/5.0057635
[13]	F. O. Ochieng, SEIRS model for TB transmission dynamics incorporating the environment and optimal control, BMC Infect. Dis., 25 (2025), 490. https://doi.org/10.1186/s12879-025-10710-2 doi: 10.1186/s12879-025-10710-2
[14]	L. Liu, X. Q. Zhao, Y. Zhou, A tuberculosis model with seasonality, Bull. Math. Biol., 72 (2010), 931–952. https://doi.org/10.1007/s11538-009-9490-3 doi: 10.1007/s11538-009-9490-3
[15]	J. M. Trauer, J. T. Denholm, E. S. McBryde, Construction of a mathematical model for tuberculosis transmission, J. Theor. Biol., 358 (2014), 74–84.
[16]	R. S. Wallis, Mathematical models of tuberculosis reactivation and relapse, Front. Microbiol., 7 (2016), 669. https://doi.org/10.3389/fmicb.2016.00669 doi: 10.3389/fmicb.2016.00669
[17]	J. Zhang, Y. Li, X. Zhang, Mathematical modeling of tuberculosis data of China, J. Theor. Biol., 365 (2015), 159–163.
[18]	B. Chennaf, M. S. Abdelouahab, R. Lozi, Analysis of the dynamics of tuberculosis in Algeria, Computation, 11 (2023), 146. https://doi.org/10.3390/computation11030146 doi: 10.3390/computation11030146
[19]	B. Chennaf, M. S. Abdelouahab, R. Lozi, A novel compartmental VSLIT model for TB in Algeria and Ukraine, Ukr. Math. J., 75 (2023), 1709–1722.
[20]	R. Boucherma, M. S. Abdelouahab, R. Lozi, A fractional-order VSEITR model with Caputo derivative for analyzing tuberculosis dynamics and control strategies in Algeria, J. Appl. Nonlinear Dyn., 2026, in press.
[21]	B. Chennaf, M. S. Abdelouahab, Discrete-time epidemic modeling with chemoprophylaxis, J. Innov. Appl. Math. Comput. Sci., 4 (2024), 63–85.
[22]	H. Joshi, M. Yavuz, Fractional-order COVID-19 and TB coinfection model, Eur. Phys. J. Plus, 138 (2023), 468. https://doi.org/10.1140/epjp/s13360-023-03916-x doi: 10.1140/epjp/s13360-023-03916-x
[23]	O. J. Peter, A. Abidemi, F. Fatmawati, M. M. Ojo, F. A. Oguntolu, Optimizing tuberculosis control: A comprehensive simulation of integrated interventions using a mathematical model, Math. Model. Numer. Simul. Appl., 4 (2024), 238–255.
[24]	W. Ahmad, A. I. K. Butt, M. Rafiq, Z. Asif, T. Ismaeel, N. Ahmad, Modeling, analyzing and simulating the measles transmission dynamics through efficient computational optimal control technique, Eur. Phys. J. Plus, 139 (2024), 345. https://doi.org/10.1140/epjp/s13360-024-0345-0 doi: 10.1140/epjp/s13360-024-0345-0
[25]	M. Yavuz, M. Akman, F. Usta, N. Özdemir, Effect of awareness in a fractional TB model, AIP Conf. Proc., 2483 (2022), 010001. https://doi.org/10.1063/5.0094321 doi: 10.1063/5.0094321
[26]	B. Bolaji, T. Onoja, A. C. Benedict, B. I. Omede, U. B. Odionyenma, Dynamical analysis of HIV-TB co-infection transmission model in the presence of treatment for TB, Bull. Biomath., 2 (2024), 21–56.
[27]	M. Yavuz, F. Özköse, M. Akman, Z. T. Taştan, TB epidemic under consciousness effect, Math. Model. Control, 3 (2023), 88–103.
[28]	Attaullah, K. Mahdi, M. Yavuz, S. Boulaaras, M. Haiour, V. T. Pham, Computational approaches on integrating vaccination and treatment strategies in the SIR model using Galerkin time discretization scheme, Math. Comput. Model. Dyn. Syst., 30 (2024), 758–791. https://doi.org/10.1080/13873954.2024.2405504 doi: 10.1080/13873954.2024.2405504
[29]	L. Boulaasair, H. Bouzahir, M. Yavuz, Global analysis of a patchy epidemic model, Int. J. Optim. Control, 14 (2024), 365–377.
[30]	M. Yavuz, K. Akyüz, N. B. Bayraktar, F. N. Özdemir, Fractional hepatitis-B modeling with real data, AIMS Biophys., 11 (2024), 378–402.
[31]	T. R. Nandi, A. K. Saha, S. Roy, Fractional-order TB model with vaccination and reinfection, Sci. Rep., 14 (2024), 28290. https://doi.org/10.1038/s41598-024-56345-7 doi: 10.1038/s41598-024-56345-7
[32]	World Health Organization, Algeria factsheets of health statistics 2010, 2022. Available from: https://www.afro.who.int/fr/node/5820.
[33]	World Bank Group, Tuberculosis treatment success rate, Algeria, 2022. Available from: https://data.worldbank.org/indicator/SH.TBS.CURE.ZS?locations = DZ.
[34]	J. Ripoll, J. Font, A discrete model for the evolution of infection prior to symptom onset, Mathematics, 11 (2023), 1092. https://doi.org/10.3390/math11051092 doi: 10.3390/math11051092
[35]	E. W. Tiemersma, M. J. van der Werf, M. W. Borgdorff, B. G. Williams, C. Dye, Natural history of tuberculosis: Duration and fatality of untreated pulmonary tuberculosis in HIV negative patients, PLoS One, 6 (2011), e17601. https://doi.org/10.1371/journal.pone.0017601 doi: 10.1371/journal.pone.0017601
[36]	O. Diekmann, J. A. P. Heesterbeek, J. A. J. Metz, 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 (1990), 365–382.
[37]	P. van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29–48.
[38]	J. P. LaSalle, Some extensions of Liapunov's second method, IRE T. Circ. Theor., 7 (1960), 520–527. https://doi.org/10.1109/TCT.1960.1086720 doi: 10.1109/TCT.1960.1086720
[39]	World Health Organization, Global tuberculosis report, 2022. Available from: https://extranet.who.int/tme/generateCSV.asp?ds = notifications.
[40]	Population growth in Algeria, 2024. Available from: https://www.donneesmondiales.com/afriquealgerie/croissance-population.php.
[41]	I. Abubakar, L. Pimpin, C. A. Ariti, R. Beynon, P. Mangtani, J. A. C. Sterne, et al., Systematic review and meta-analysis of the current evidence on the duration of protection by bacillus Calmette–Guérin vaccination against tuberculosis, Health Technol. Assess., 17 (2013), 1–372. https://doi.org/10.3310/hta17370 doi: 10.3310/hta17370
[42]	J. A. C. Sterne, G. Rodrigues, C. A. Glynn, J. F. Watson, A. P. Sutherland, P. Mangtani, Decline of BCG efficacy with time, Int. J. Tuberc. Lung Dis., 2 (1998), 200–207.
[43]	P. Mangtani, I. Abubakar, C. Ariti, R. Beynon, L. Pimpin, P. E. Fine, et al., Protection by BCG vaccine against tuberculosis: A systematic review of randomized controlled trials, Clin. Infect. Dis., 58 (2014), 470–480. https://doi.org/10.1093/cid/cit790 doi: 10.1093/cid/cit790
[44]	Trading Economics, Immunization, BCG, Algeria, 2022. Available from: https://www.tradingeconomics.com/algeria/immunization-bcg-percent-of-one-year-old-old-children-wb-data.html.
[45]	S. Selmane, The impact of treatment of latent tuberculosis on the incidence: The case of Algeria, Int. J. Math. Comput. Sci., 8 (2014), 961–966.

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)