Optimal control problem for a tuberculosis model with multiple infectious compartments and time delays





Optimal control, time delays, Tuberculosis, numerical simulations


In this paper, we formulate an optimal control problem based on a tuberculosis model with multiple infectious compartments and time delays. In order to have a more realistic model that allows highlighting the role of detection, loss to follow-up and treatment in TB transmission, we propose an extension of the classical SEIR model by dividing infectious patients in the compartment (I) into three categories: undiagnosed infected (I), diagnosed patients who are under treatment (T) and diagnosed patients who are lost to follow-up (L). We incorporate in our model delays representing the incubation period and the time needed for treatment. We also introduce three control variables in our delayed system which represent prevention, detection and the efforts that prevent the failure of treatment. The purpose of our control strategies is to minimize the number of infected individuals and the cost of intervention. The existence of the optimal controls is investigated, and a characterization of the three controls is given using the Pontryagin's maximum principle with delays. To solve numerically the optimality system with delays, we present an adapted iterative method based on the iterative Forward-Backward Sweep Method (FBSM). Numerical simulations performed using Matlab are also provided. They indicate that the prevention control is the most effective one. To the best of our knowledge, it is the first work to apply optimal control theory to a TB model which considers infectious patients diagnosis, loss to follow-up phenomenon and multiple time delays.


Download data is not yet available.

Author Biographies

Mohamed Elhia, Hassan II University

received the PhD degree in applied Mathematics from Hassan II University, Morocco. He is currently professor of Mathematics at the same university. His research interests include optimal control problem, population dynamics, epidemiological model and numerical solutions.

Omar Balatif, Chouaib Doukkali University

is currently working as a professor of mathematics at Chouaib Doukkali university. His research interests include optimal control problem, modelling and analysis of epidemiological and sociological phenomena and discrete dynamical systems.

Lahoucine Boujallal, Hassan II University

achieved his PhD degree applied mathematics at Hassan II unversity, Morocco. His research interests include optimal control problem, null-controllability and viability.

Mostafa Rachik, Hassan II University

is currently professor of Mathematics at Hassan II university. He has published several articles in various journals. His research interests include ordinary and partial differential equations, optimal control and system analysis.


Bernoulli, D. (1760). Essai d'une nouvelle analyse de la mortalite causee par la petite verole et des avantages de l'inoculation pour la prevenir. Histoire de l'Acad. Roy. Sci.(Paris) avec Mem. des Math. et Phys. and Mem, pages 1-45.

Anderson, R. M., May, R. M., and Anderson, B. (1992). Infectious diseases of humans: dynamics and control, volume 28. Wiley Online Library.

Bailey, N. T. et al. (1975). The mathematical theory of infectious diseases and its applications. Charles Grin & Company Ltd, 5a Crendon Street, High Wycombe, Bucks HP13 6LE

Hethcote, H. W. (2000). The mathematics of infectious diseases. SIAM review, 42(4):599-653.

Huppert, A. and Katriel, G. (2013). Mathematical modelling and prediction in infectious disease epidemiology. Clinical Microbiology and Infection, 19(11):999-1005.

Keeling, M. J. and Rohani, P. (2008). Modeling infectious diseases in humans and animals. Princeton University Press.

Kermark, M. and Mckendrick, A. (1927). Contributions to the mathematical theory of epidemics. part i. In Proc. R. Soc. A, volume 115, pages 700-721.

Brauer, F., Castillo-Chavez, C., and Castillo-Chavez, C. (2012). Mathematical models in population biology and epidemiology, volume 2. Springer.

Li, M. Y. (2018). An introduction to mathematical modeling of infectious diseases, volume 2. Springer.

World health organisation. https://www.who.int/news-room/fact-sheets/detail/ tuberculosis. Accessed: 2019-11-01.

Sharma, B. (2011). North America's #1 Homeopathic Guide to Natural Health: A Complete Handbook on Homeopathic Prescribing. Xlibris Corporation LLC. Optimal control problem for a tuberculosis model with multiple infectious compartments and time delays 95

Cruz-Knight, W. and Blake-Gumbs, L. (2013). Tuberculosis: an overview. Primary Care: Clinics in Office Practice, 40(3):743-756.

Holmes, K. K., Bertozzi, S., Bloom, B. R., and Jha, P. (2017). Tuberculosis Major Infectious Diseases. The International Bank for Reconstruction and Development/The World Bank.

Ramirez-Lapausa, M., Menendez-Saldana, A., and Noguerado-Asensio, A. (2015). Extrapulmonary tuberculosis: an overview. Rev Esp Sanid Penit, 17(1):3-11.

Singh, P., Kant, S., Gaur, P., Tripathi, A., and Pandey, S. (2018). Extra pulmonary tuberculosis: An overview and review of literature. Int. J. Life. Sci. Scienti. Res, 4(1):1539-1541.

Organisation, W. H. (2018). Global tuberculosis report 2018. Technical report, World Health Organisation.

Waaler, H., Geser, A., and Andersen, S. (1962). The use of mathematical models in the study of the epidemiology of tuberculosis. American Journal of Public Health and the Nations Health, 52(6):1002-1013.

Bowong, S. and Tewa, J. J. (2009). Mathematical analysis of a tuberculosis model with di erential infectivity. Communications in Nonlinear Science and Numerical Simulation, 14(11):4010-4021.

Feng, Z., Huang, W., and Castillo-Chavez, C. (2001). On the role of variable latent periods in mathematical models for tuberculosis. Journal of dynamics and di erential equations, 13(2):425-452.

Liu, L., Zhao, X.-Q., and Zhou, Y. (2010). A tuberculosis model with seasonality. Bulletin of Mathematical Biology, 72(4):931-952.

McCluskey, C. C. and van den Driessche, P. (2004). Global analysis of two tuberculosis models. Journal of Dynamics and Di erential Equations, 16(1):139-166.

Silva, C. J. and Torres, D. F. (2013). Optimal control for a tuberculosis model with reinfection and post-exposure interventions. Mathematical Biosciences, 244(2):154-164.

Silva, C. J. and Torres, D. F. (2015). Optimal control of tuberculosis: a review. In Dynamics, games and science, pages 701-722. Springer.

Moualeu, D. P.,Weiser, M., Ehrig, R., and Deu hard, P. (2015). Optimal control for a tuberculosis model with undetected cases in cameroon. Communications in Nonlinear Science and Numerical Simulation, 20(3):986-1003.

Huo, H.-F. and Zou, M.-X. (2016). Modelling e ects of treatment at home on tuberculosis transmission dynamics. Applied Mathematical Modelling, 40(21):9474-9484.

Li, J. and Ma, M. (2016). Impact of prevention in a tuberculosis model with latent delay. Advances in Di erence Equations, 2016(1):220.

Silva, C. J., Maurer, H., and Torres, D. F. (2017). Optimal control of a tuberculosis model with state and control delays. Mathematical Biosciences & Engineering, 14(1551-0018 2017 1 321):321.

Mondal, P. K. and Kar, T. (2017). Optimal treatment control and bifurcation analysis of a tuberculosis model with effect of multiple re-infections. International Journal of Dynamics and Control, 5(2):367-380.

Yang, Y., Wu, J., Li, J., and Xu, X. (2017). Tuberculosis with relapse: a model. Mathematical Population Studies, 24(1):3-20.

Khan, M. A., Ullah, S., and Farooq, M. (2018). A new fractional model for tuberculosis with relapse via atangana{baleanu derivative. Chaos, Solitons & Fractals, 116:227-238.

Kim, S., Aurelio, A., and Jung, E. (2018). Mathematical model and intervention strategies for mitigating tuberculosis in the philippines. Journal of theoretical biology, 443:100-112.

Y?ld?z, T. A. (2019). A comparison of some control strategies for a non-integer order tuberculosis model. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 9(3):21-30.

Purwati, U. D., Riyudha, F., Tasman, H., et al. (2020). Optimal control of a discrete agestructured model for tuberculosis transmission. Heliyon, 6(1):e03030.

Baba, I. A., Abdulkadir, R. A., and Esmaili, P. (2020). Analysis of tuberculosis model with saturated incidence rate and optimal control. Physica A: Statistical Mechanics and its Applications, 540:123237.

Almeida, R., da Cruz, A. M. B., Martins, N., and Monteiro, M. T. T. (2019). An epidemiological mseir model described by the caputo fractional derivative. International Journal of Dynamics and Control, 7(2):776-784.

Cao, H. and Tan, H. (2015). The discrete tuberculosis transmission model with treatment of latently infected individuals. Advances in Di erence Equations, 2015(1):1-18.

Mettle, F. O., Osei Affi, P., and Twumasi, C. (2020). Modelling the transmission dynamics of tuberculosis in the ashanti region of ghana. Interdisciplinary Perspectives on Infectious Diseases, 2020.

Narula, P., Piratla, V., Bansal, A., Azad, S., and Lio, P. (2016). Parameter estimation of tuberculosis transmission model using ensemble kalman lter across indian states and union territories. Infection, Disease & Health, 21(4):184-191.

Syahrini, I., Hal ani, V., Yuni, S. M., Iskandar, T., Ramli, M., et al. (2017). The epidemic of tuberculosis on vaccinated population. In Journal of Physics: Conference Series, volume 890, page 012017. IOP Publishing.

Zhang, T., Kang, R., Wang, K., and Liu, J. (2015). Global dynamics of an seir epidemic model with discontinuous treatment. Advances in Di erence Equations, 2015(1):361.

Zhao, Y., Li, M., and Yuan, S. (2017). Analysis of transmission and control of tuberculosis in mainland china, 2005-2016, based on the age-structure mathematical model. International journal of environmental research and public health, 14(10):1192.

MacPherson, P., Houben, R. M., Glynn, J. R., Corbett, E. L., and Kranzer, K. (2013). Pretreatment loss to follow-up in tuberculosis patients in low-and lower-middle-income countries and high-burden countries: a systematic review and meta-analysis. Bulletin of the World Health Or- ganization, 92:126{138.

Elhia, M., Laaroussi, A., Rachik, M., Rachik, Z., and Labriji, E. (2014). Global stability of a susceptible-infected-recovered (sir) epidemic model with two infectious stages and treatment. Int J Sci Res, 3(5):114-121.

Flynn, J. L. and Chan, J. (2001). Tuberculosis: Latency and reactivation. Infection and Immunity, 69(7):4195-4201.

Haas, M. K. and Belknap, R. W. (2018). Updates in the treatment of active and latent tuberculosis. In Seminars in respiratory and critical care medicine, volume 39, pages 297{309. Thieme Medical Publishers.

Birkhoff, G. and Rota, G. (1989). Ordinary Differential Equations, volume 14. fourth ed. John Wiley & Sons, New York.

Van den Driessche, P. and Watmough, J. (2002). Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical biosciences, 180(1-2):29-48.

Diekmann, O., Heesterbeek, J. A. P., and Metz, J. A. (1990). On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations. Journal of mathematical biology, 28(4):365-382.

Luenberger, D. G. (1979). Introduction to dynamic systems; theory, models, and applications. Technical report.

van den Driessche, P. (2017). Reproduction numbers of infectious disease models. Infectious Disease Modelling, 2(3):288-303.

Gollmann, L., Kern, D., and Maurer, H. (2009). Optimal control problems with delays in state and control variables subject to mixed control-state constraints. Optimal Control Applications and Methods, 30(4):341-365.

Fleming, W. and Rishel, R. (1975). Deterministic and stochastic optimal control. Springer.

Lenhart, S. and Workman, J. T. (2007). Optimal control applied to biological models. CRC press.

McAsey, M., Mou, L., and Han, W. (2012). Convergence of the forward-backward sweep method in optimal control. Computational Optimization and Applications, 53(1):207-226.

Lienhardt, C. (2001). From exposure to disease: the role of environmental factors in susceptibility to and development of tuberculosis. Epidemiologic reviews, 23(2):288-301.

Trauer, J. M., Denholm, J. T., and McBryde, E. S. (2014). Construction of a mathematical model for tuberculosis transmission in highly endemic regions of the asia-paci c. Journal of theoretical biology, 358:74-84.

Emvudu, Y., Demasse, R., and Djeudeu, D. (2011). Optimal control of the lost to follow up in a tuberculosis model. Computational and mathematical methods in medicine, 2011.

Okuonghae, D. and Aihie, V. (2008). Case detection and direct observation therapy strategy (dots) in nigeria: its effect on TB dynamics. Journal of Biological Systems, 16(01):1-31.

Naidoo, P., Theron, G., Rangaka, M. X., Chihota, V. N., Vaughan, L., Brey, Z. O., and Pillay, Y. (2017). The south african tuberculosis care cascade: estimated losses and methodological challenges. The Journal of infectious diseases, 216(suppl 7):S702-S713.



DOI: 10.11121/ijocta.01.2021.00885
Published: 2021-01-03

How to Cite

Elhia, M., Balatif, O., Boujallal, L., & Rachik, M. (2021). Optimal control problem for a tuberculosis model with multiple infectious compartments and time delays. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 11(1), 75–91. https://doi.org/10.11121/ijocta.01.2021.00885



Research Articles