On the Role of Early Case Detection and Treatment Failure in Controlling Tuberculosis Transmission: A Mathematical Modeling Study

Dipo Aldila; Derio A. Ramadhan; Chidozie W. Chukwu; Bevina D. Handari; Muhammad Shahzad; Putri Zahra Kamalia

doi:10.5614/cbms.2024.7.1.4

Authors

Dipo Aldila Department of Mathematics, Universitas Indonesia, Depok 16424, Indonesia
Derio A. Ramadhan Department of Mathematics, Universitas Indonesia, Depok 16424, Indonesia
Chidozie W. Chukwu Department of Mathematics, Wake Forest University, Winston-Salem, NC 27109, USA
Bevina D. Handari Department of Mathematics, Universitas Indonesia, Depok 16424, Indonesia
Muhammad Shahzad Department of Mathematics and Statistics, The University of Haripur, KP, Haripur, 22620, Pakistan
Putri Zahra Kamalia Department of Mathematics, Universitas Indonesia, Depok 16424, Indonesia

DOI:

https://doi.org/10.5614/cbms.2024.7.1.4

Keywords:

Tuberculosis, case detection, Treatment failure, Control reproduction number, Bifurcation, Sensitivity analysis

Abstract

Tuberculosis (TB) remains a pressing global health concern, demanding urgent attention to mitigate its spread and impact. In this study, we present a rigorous mathematical model of TB transmission that incorporates early case detection and addresses the critical issue of treatment failure. Through the development of a system of nonlinear ordinary differential equations, we conduct comprehensive analyses to assess the dynamics of TB transmission and the efficacy of intervention strategies. Our findings underscore the urgent need for effective TB control measures. Mathematical analyses reveal that the model exhibits a TB-free equilibrium, which is globally asymptotically stable only if the control reproduction number falls below one. However, we identify a concerning phenomenon: the model demonstrates a forward bifurcation when the control reproduction number equals one, suggesting that the disease-free equilibrium loses its stability, while simultaneously, the stable unique endemic equilibrium begins to emerge. Moreover, sensitivity analysis highlights the complex interplay between case detection rates, treatment failure probabilities, and TB transmission dynamics. Contrary to expectations, increasing case detection rates and minimizing treatment failure probabilities may not consistently reduce the basic reproduction number or the size of the infected population. Instead, there exists a critical threshold for intervention effectiveness, beyond which TB transmission can be significantly curtailed. Biologically, this phenomenon may occur if there is no balance between case detection and treatment efforts. If treatment quality does not improve, then case detection will not have a significant impact, and in the worst case scenario, it can exacerbate the intervention?s negative effects. These findings underscore the urgency of implementing targeted intervention strategies to combat TB transmission effectively. Failure to meet the critical intervention threshold risks undermining TB elimination efforts and exacerbating the global TB burden. Through numerical simulations, we elucidate potential intervention scenarios necessary for achieving TB elimination goals in human populations. In conclusion, our study highlights the urgent imperative for coordinated action to control TB transmission effectively. By elucidating the dynamics of TB spread and intervention efficacy, we provide valuable insights to inform evidence-based policy decisions and accelerate progress towards TB elimination on a global scale.

References

National Health Services: Causes of Tuberculosis. https://www.nhs.uk/conditions/tuberculosis-tb/causes/, Accessed on January 10, 2023.

Cleveland Clinic: Tuberculosis. https://my.clevelandclinic.org/health/diseases/11301-tuberculosis, Accessed on January 10, 2023.

World Health Organizatioin: Recent facts Tuberculosis. https://www.who.int/news-room/fact-sheets/detail/tuberculosis#:?:text=Worldwide%2C%20TB%20is%20the%2013th,all%20countries%20and%20age%20groups, Accessed on January 10, 2023.

American Lung Association: Treating and Managing Tuberculosis. https://www.lung.org/lung-health-diseases/lung-disease-lookup/tuberculosis/treating-and-managing#:?:text=How%20Is%20Active%20TB%20Treated,%E2%80%94rifampin%2C%20pyrazinamide%20and%20ethambutol, Accessed on January 10, 2023.

Putra, I.W.G.A.E., Kurniasari, N.M.D., Dewi, N.P.E.P., Suarjana, I.K., Duana, I.M.K., Mulyawan, I.K.H., Riono, P., Alisjahbana, B., Probandari, A., Notobroto, H.B. and Wahyuni, C.U., The implementation of early detection in tuberculosis contact investigation to improve case finding, Journal of Epidemiology and Global Health, 9(3), p. 191, 2019.

Oktamianti P., Bachtiar A., Sutoto S., Trihandini I., Prasetyo S., Achadi A. and Efendi F., Tuberculosis control within Indonesia's hospital accreditation, Journal of Public Health Research, 10(3), 2021. doi:10.4081/jphr.2021.1979.

World Health Organization, Early detection of tuberculosis: an overview of approaches, guidelines and tools, 2011.

Everyday Health: How Tuberculosis Is Diagnosed, Screenings and Tests. https://www.everydayhealth.com/tuberculosis/guide/testing/. Accessed on January 10, 2023.

Kammerer, J.S., Shang, N., Althomsons, S.P., Haddad, M.B., Grant, J. and Navin, T.R., Using statistical methods and genotyping to detect tuberculosis outbreaks, International Journal of Health Geographics, 12(1), pp. 1-8, 2013.

Harries A, Maher D, and Graham S., TB/HIV a clinical manual, Geneva: World Health Organisation, Management of Patients with Tuberculosis, pp. 111-115, 2004.

Becerra, M.C., Freeman, J., Bayona, J., Shin, S.S., Kim, J.Y., Furin, J.J., Werner, B., Sloutsky, A., Timperi, R., Wilson, M.E. and Pagano, M., Using treatment failure under effective directly observed short-course chemotherapy programs to identify patients with multidrug-resistant tuberculosis, The International Journal of Tuberculosis and Lung Disease, 4(2), pp. 108-114, 2000.

Athithan, S. and Ghosh, M., Mathematical modelling of TB with the effects of case detection and treatment, International Journal of Dynamics and Control, 1(3), pp. 223-230, 2013.

Aldila, D., Fardian, B.L., Chukwu, C.W., Noor Aziz, M.H. and Kamalia, P.Z., Improving Tuberculosis Control: Assessing the Value of Medical Masks and Case Detection-A Multi-Country Study with Cost-Effectiveness Analysis, will be published in the Royal Society Open Science, 2024.

Okuonghae, D. and Omosigho, S.E., Analysis of a mathematical model for tuberculosis: What could be done to increase case detection, Journal of Theoretical Biology, 269(1), pp. 31-45, 2011.

Liu, L. and Gao, X., Qualitative study for a multi-drug resistant TB model with exogenous reinfection and relapse, International Journal of Biomathematics, 5(4), p. 1250031, 2012.

Okuonghae, D., Analysis of a stochastic mathematical model for tuberculosis with case detection, International Journal of Dynamics and Control, 10(3), pp. 734-747, 2022.

Fati, S.M., Senan, E.M. and ElHakim, N., Deep and Hybrid Learning Technique for Early Detection of Tuberculosis Based on X-ray Images Using Feature Fusion, Applied Sciences, 12(14), p. 7092, 2022.

Chukwu, C.W., Bonyah, E. and Juga, M.L., On mathematical modeling of fractional-order stochastic for tuberculosis transmission dynamics. Results in Control and Optimization, 11, p. 100238, 2023.

Giri, N., Joseph, R., Chavan, S., Heda, R., Israni, R. and Sethiya, R., AI-based prediction for early detection of Tuberculosis in India based on environmental factors, In 2020 19th IEEE International Conference on Machine Learning and Applications (ICMLA), IEEE, pp. 278-285, 2020.

Simorangkir, G., Aldila, D., Rizka, A., Tasman, H. and Nugraha, E.S., Mathematical model of tuberculosis considering observed treatment and vaccination interventions, Journal of Interdisciplinary Mathematics, 24(6), pp. 1717-1737, 2021.

Xiang, H., Zou, M.X. and Huo, H.F., Modeling the effects of health education and early therapy on tuberculosis transmission dynamics, International Journal of Nonlinear Sciences and Numerical Simulation, 20(3-4), pp. 243-255, 2019.

Nyerere, N., Luboobi, L.S. and Nkansah-Gyekye, Y., Modeling the effect of screening and treatment on the transmission of tuberculosis infections, Mathematical Theory and Modeling, 4(7), pp. 51-62, 2014.

Fatmawati, M.A.K., Bonyah, E., Hammouch, Z. and Shaiful, E.M., A mathematical model of tuberculosis (TB) transmission with children and adults groups: A fractional model, Aims Math, 5(4), pp. 2813-2842, 2020.

Kuddus, M.A., McBryde, E.S., Adekunle, A.I., White, L.J. and Meehan, M.T., Mathematical analysis of a two-strain tuberculosis model in Bangladesh, Scientific Reports, 12(1), pp. 1-13, 2022.

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

Panickar, J. R. and Hoskyns, Treatment failure in tuberculosis, European Respiratory Journal 2007, 29(3), pp. 561-564, 2007.

Liu, L. and Wang, Y., A mathematical study of a TB model with treatment interruptions and two latent periods, Computational and Mathematical Methods in Medicine, 2014(1), p. 932186, 2014.

Lotfi, M., Jabbari, A. and Kheiri, H., A mathematical analysis of a tuberculosis epidemic model with two treatments and exogenous re-infection, International Journal of Biomathematics, 13(8), p. 2050082, 2020.

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

Badan Pusat Statistik, Hasil Sensus Penduduk 2020, 2020. https://www.bps.go.id/pressrelease/2021/01/21/1854/hasil-sensus-penduduk-2020.html, Accessed on February 15, 2022.

World Bank, Life expectancy at birth, total (years) - Indonesia, 2018. https://data.worldbank.org/indicator/SP.DYN.LE00.IN?locations=ID, Accessed on February 15, 2022.

Egonmwan, A.O. and Okuonghae, D., Analysis of a mathematical model for tuberculosis with diagnosis, Journal of Applied Mathematics and Computing, 59, pp. 129-162, 2019.

Maulana, D.C., Utoyo, M.I., Purwati, U.D. and Chukw, C.W., Parameter estimation and analysis on SIS-SEIS types model of tuberculosis transmission in East Java Indonesia, Communications in Mathematical Biology and Neuroscience, 2022(2022), pp. 1-15, 2022.

Zhao, Y., Li, M. and Yuan, S., 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), p. 1192, 2017.

Centers for Disease Control and Prevention (CDC), Tuberculosis (TB), 2016. https://www.cdc.gov/tb.html, Accessed on January 23, 2022.

Mase, S.R., and Chorba, T., Treatment of Drug-Resistant Tuberculosis, Clinics in Chest Medicine, 40(4), pp. 775-795, 2019.

Behr, M.A., Edelstein, P.H. and Ramakrishnan, L., Revisiting the timetable of tuberculosis, BMJ (Clinical Research ed.), 362(k2738), pp. 1-10, 2018.

Mathworks, lsqnonlin, 2023. https://www.mathworks.com/help/optim/ug/lsqnonlin.html, Acessed on March 17, 2024.

Diekmann, O. and Heesterbeek, J.A.P., Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation (Vol. 5), John Wiley & Sons, 2000.

Aldila, D., Ndii, M.Z., Anggriani, N., Tasman, H. and Handari, B.D., Impact of social awareness, case detection, and hospital capacity on dengue eradication in Jakarta: a mathematical model approach, Alexandria Engineering Journal, 64, pp. 691-707, 2023.

Aldila, D., Shahzad, M., Khoshnaw, S.H., Ali, M., Sultan, F., Islamilova, A., Anwar, Y.R. and Samiadji, B.M., Optimal control problem arising from COVID-19 transmission model with rapid-test, Results in Physics, 37, p. 105501, 2022.

Aldila, D., Dynamical analysis on a malaria model with relapse preventive treatment and saturated fumigation, Computational and Mathematical Methods in Medicine, 2022(1), p. 1135452, 2022.

Aldila, D. and Angelina, M., Optimal control problem and backward bifurcation on malaria transmission with vector bias, Heliyon, 7(4), p. e06824, 2021.

Aldila, D., Handari, B.D., Widyah, A. and Hartanti, G., Strategies of optimal control for hiv spreads prevention with health campaign, Communications in Mathematical Biology and Neuroscience, 2020(7), 2020.

Diekmann, O., Heesterbeek, J.A.P. and Roberts, M.G., The construction of next-generation matrices for compartmental epidemic models. Journal of the Royal Society Interface, 7(47), pp. 873-885, 2010.

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

Chavez C.C., Feng Z. and Huang, W., On the computation of R0 and its role on global stability, Mathematical Approaches for Emerging and Re-emerging Infection Diseases: An Introduction, 125, pp 31-65, 2002.

Castillo-Chavez, C. and Song, B., Dynamical models of tuberculosis and their applications, Mathematical Biosciences & Engineering, 1(2), pp. 361-404, 2004.

Gerberry, D.J., Practical aspects of backward bifurcation in a mathematical model for tuberculosis, Journal of Theoretical Biology, 388, pp.15-36, 2016.

Aldila, D., Saslia, B.R., Gayarti, W. and Tasman, H., Backward bifurcation analysis on tuberculosis disease transmission with saturated treatment, In Journal of Physics: Conference Series, IOP Publishing, 1821(1), p. 012002, 2021.

Benjamini, Y. and Hochberg, Y., Controlling the false discovery rate: a practical and powerful approach to multiple testing, Journal of the Royal statistical society: series B (Methodological), 57(1), pp. 289-300, 1995.

Marino, S., Hogue, I.B., Ray, C.J. and Kirschner, D.E., A methodology for performing global uncertainty and sensitivity analysis in systems biology, Journal of Theoretical Biology, 254(1), pp. 178-196, 2008.

Chukwu, C.W., Juga, M.L., Chazuka, Z. and Mushanyu, J., Mathematical analysis and sensitivity assessment of HIV/AIDSlisteriosis co-infection dynamics, International Journal of Applied and Computational Mathematics, 8(5), p. 251, 2022.

Handari, B.D., Ramadhani, R.A., Chukwu, C.W., Khoshnaw, S.H. and Aldila, D., An optimal control model to understand the potential impact of the new vaccine and transmission-blocking drugs for malaria: A case study in papua and west papua, indonesia, Vaccines, 10(8), p. 1174, 2022.

Chukwu, C.W., Juga, M.L., Chazuka, Z. and Mushanyu, J., Mathematical analysis and sensitivity assessment of HIV/AIDSlisteriosis co-infection dynamics, International Journal of Applied and Computational Mathematics, 8(5), p. 251, 2022.

Tasman, H., Aldila, D., Dumbela, P.A., Ndii, M.Z., Fatmawati, Herdicho, F.F. and Chukwu, C.W., Assessing the impact of relapse, reinfection and recrudescence on malaria eradication policy: a bifurcation and optimal control analysis, Tropical Medicine and Infectious Disease, 7(10), p. 263, 2022.

Altink, H., Fight TB with BCG: Mass Vaccination Campaigns in the British Caribbean, 1951-6. Medical History, 58(4), pp. 475-497, 2014.

Scriba, T.J., Netea, M.G. and Ginsberg, A.M., Key recent advances in TB vaccine development and understanding of protective immune responses against Mycobacterium tuberculosis, In Seminars in Immunology, Academic Press, 50, p. 101431, 2020.

Avilov, K.K. and Romanyukha, A.A., Mathematical modeling of tuberculosis propagation and patient detection, Automation and Remote Control, 68(9), pp. 1604-1617, 2007.

Das, D.K., Khajanchi, S. and Kar, T.K., Transmission dynamics of tuberculosis with multiple re-infections, Chaos, Solitons and Fractals, 130(2020), p. 109450, 2019.

Khajanchi, S., Das, D.K. and Kar, T.K., Dynamics of tuberculosis transmission with exogenous reinfections and endogenous reactivation, Physica A: Statistical Mechanics and its Applications, 497, pp. 52-71. 2018.