The role of vaccination in curbing tuberculosis epidemic

  • Ayinla Ally Yeketi
  • Wan Ainun Mior OthmanEmail author
  • M. A. Omar Awang
Original Article


This work studies the impact of vaccine in controlling tuberculosis (TB) epidemic using susceptible, vaccinated, exposed, infectious and recovered compartmental model. This is necessitated due to the acclaimed ineffectiveness of BCG vaccine in combatting TB. The model is formulated using a non-linear system of ordinary differential equation which is normalised to eliminate the natural death factor \((\mu )\) so as to focus on other factors. The disease-free equilibrium and endemic equilibrium point (EEP) of the system are established alongside their local and global stabilities. Although the local stability of the EEP could not be established analytically due to the cumbersomeness of the EEP obtained, it is, however, established numerically. It is shown with the aid of numerical simulation carried out on the model that vaccination helps in reducing the tuberculosis epidemic and in fact, if the rates of contact and infectivity are reduced, further reduction in the rate of incidence \((\lambda )\) can be achieved. Further more, the reason why there is the need for a better vaccine to replace BCG vis-á-vis provision of better immunity coverage \((\theta \rightarrow 0\) and \(\sigma \rightarrow 0)\) and also, the need for the development of drugs that confer permanent or long lasting immunity \((\delta _2\rightarrow 0)\) is as well established. More vaccination proportion gives better outcome \((\tau \rightarrow 1)\) and the introduced controls show their relevance in reducing the infection. The novelty of this research is the provision of guiding frame work for the pharmacists on the intrinsic features expected of any proposed vaccine to replace BCG while the expected recommendations from the doctors are established using optimal control.


Tuberculosis model Vaccination Optimal control Basic reproduction number Global stability Reinfection rate 



The funding has been recevied from IIRG, Universiti Malaya with Grand No.  IIRG001C.


  1. Aparicio JP, Castillo-Chavez C (2009) Mathematical modelling of tuberculosis epidemics. Math Biosci Eng 6(2):209–237. CrossRefGoogle Scholar
  2. Bhandari T (2016) Study helps explain why tuberculosis vaccines are ineffective. From Accessed 21 Sept 2018
  3. Bhunu CP, Garira W, Mukandavire Z, Zimba M (2008) Tuberculosis transmission model with chemoprophylaxis and treatment. Bull Math Biol 70:1163–1191. CrossRefGoogle Scholar
  4. Bhunu CP, Mushayabasa S, Tchuenche JM (2011) A theoretical assessment of the effects of smoking on the transmission dynamics of tuberculosis. Bull Math Biol 73:1333–1357. CrossRefGoogle Scholar
  5. Carrasco A, Leiva H (2007) Variation of constants formula for functional parabolic partial differential equations. Electron J Differ Equ 130:1–20Google Scholar
  6. Castillo-Chavez C, Feng Z, Huang W (2002) On the computation of \(R_0\) and its role on global stability. Mathematical approaches for emerging and reemerging infectious Disease: an introduction 1:229–250. From Accessed 21 Sept 2018
  7. Daniel TM (1997) Captain of death: the story of tuberculosis. University of Rochester Press, RochesterGoogle Scholar
  8. Daniel TM (2006) The history of tuberculosis. Respir Med 100:1862–1870. CrossRefGoogle Scholar
  9. Gerberry DJ (2009) Trade-off between BCG vaccination and the ability to detect and treat latent tuberculosis. J Theor Biol 261:548–560. CrossRefGoogle Scholar
  10. Gomes MGM, Franco AO, Gomes MC, Medley GF (2004) The reinfection threshold promotes variability in tuberculosis epidemiology and vaccine efficacy. Proc R Soc Lond 271:617–623. CrossRefGoogle Scholar
  11. Goswami NK, Srivastav AK, Ghosh M, Shanmukha B (2018) Mathematical modeling of zika virus disease with non-linear incidence and optimal control. IOP Conf Ser J Phys Conf Seri 1000:012114. CrossRefGoogle Scholar
  12. Jalan M (2018) Why is the BCG vaccine not always effective? From Accessed 21 Sept 2018
  13. Jia Z-W, Tang G-Y, Jin Z, Dye C, Vlas SJ, Xiao-Wen Li X-W, Dan F, Fang L-Q, Zhao W-J, Cao W-C (2008) Modeling the impact of immigration on the epidemiology of tuberculosis. Theore Popul Biol 73:437–448. CrossRefGoogle Scholar
  14. Kernodle DS (2010) Decrease in the effectiveness of Bacille Calmette–Guérin vaccine against pulmonary tuberculosis: a consequence of increased immune suppression by microbial antioxidants, not overattenuation. Clin Infect Dis 51(2):177–184. CrossRefGoogle Scholar
  15. Lenhart S, Workman JT (2007) Optimal control applied to biological model. Chapman & Hall/CRC, LondonGoogle Scholar
  16. Luca S, Mihaescu T (2013) History of BCG vaccine. MAEDICA J Clin Med 8(1):53–58Google Scholar
  17. Major RH (1945) Classic descriptions of disease, 3rd edn. Spring-field, IL: Charles C. ThomasGoogle Scholar
  18. McCluskey CC, van den Driessche P (2004) Global analysis of two tuberculosis models. J Dyn Differ Equ 16(1):139–166. CrossRefGoogle Scholar
  19. Mohajan HK (2015) Tuberculosis is a fatal disease among some developing countries of the world. Am J Infect Dis Microbiol 3(1):18–31. Google Scholar
  20. Moliva JI, Turner J, Torrelles JB (2015) Why does BCG fail to protect against tuberculosis? From Accessed 21 Sept 2018
  21. Mukandavire Z, Gumel AB, Garira W, Tchuenche JM (2009) Mathematical analysis of a model for HIV-malaria co-infection. Math Bio Eng 6(2):333–362CrossRefGoogle Scholar
  22. Murphy BM, Singer BH, Kirschner D (2003) On treatment of tuberculosis in heterogeneous populations. J Theo Biol 223:391–404. CrossRefGoogle Scholar
  23. Omondi EO, Mbogoa RW, Luboobia LS (2018) Mathematical modelling of the impact of testing, treatment and control of HIV transmission in Kenya. Cogent Math Stat. Google Scholar
  24. Perko L (2001) Differential equations and dynamical systems. Text in Applied Mathematics, 7, 3rd Eds, Springer, BerlinGoogle Scholar
  25. Pontryagin LS, Boltyanskii VG, Gamkrelidze RV, Mischenko EF (1962) The mathematical theory of optimal processes. Wiley, New YorkGoogle Scholar
  26. Pontryagin LS, Boltyanskii VG (1980) The mathematical theory of optimal processes. Golden and Breach Science Publishers, LondonGoogle Scholar
  27. Ragonnet R, Trauer JM, Denholm JT, Marais BJ, McBryde ES (2017) A user-friendly mathematical modelling web interface to assist local decision making in the fight against drug-resistant tuberculosis. BMC Infect Dis 17:374. CrossRefGoogle Scholar
  28. Raimundo SM, Yang HM (2006) Transmission of tuberculosis with exogenous re-infection and endogenous reactivation. Math Popul Stud 13:1–23. CrossRefGoogle Scholar
  29. Srivastav AK, Ghosh M (2016) Modeling and analysis of the symptomatic and asymptomatic infections of swine flu with optimal control. Model Earth Syst Environ. Google Scholar
  30. Trauer JM, Denholm JT, McBryde ES (2014) Construction of a mathematical model for tuberculosis transmission in highly endemic regions of the Asia-Pacific. J Theor Biol 358:74–84. CrossRefGoogle Scholar
  31. van den Driessche P, Watmough J (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math Biosci 180:29–48. CrossRefGoogle Scholar
  32. Varughese M, Long R, Li MY (2017) Efficacy of screening and treatment of latent tuberculosis infection. Math Popul Stud 24(1):21–36. CrossRefGoogle Scholar
  33. Vynnycky E, Sumner T, Fielding KL, Lewis JJ, Cox AP, Hayes RJ, Corbett EL, Churchyard GJ, Grant AD, White RG (2015) Tuberculosis control in South African gold mines: mathematical modeling of a trial of community-wide isoniazid preventive therapy. Am J Epidemiol 181(8):619–632. CrossRefGoogle Scholar
  34. WHO (2011) Global tuberculosis control 2011 WHOGoogle Scholar
  35. WHO (2012) Global tuberculosis report 2012 WHOGoogle Scholar
  36. WHO (2013) Global tuberculosis report 2013 WHOGoogle Scholar
  37. WHO (2014) Global tuberculosis report 2014 WHOGoogle Scholar
  38. WHO (2015) Global tuberculosis report 2015 WHOGoogle Scholar
  39. WHO (2016) Global tuberculosis report 2016 WHOGoogle Scholar
  40. WHO (2017) Global tuberculosis report 2017. Geneva. Licence: CC BY-NC-SA 3.0 IGO WHOGoogle Scholar
  41. WHO (2018) Global tuberculosis report 2018. Geneva. Licence: CC BY-NC-SA 3.0 IGO WHOGoogle Scholar
  42. WHO (2018) Tuberculosis vaccine development. From Accessed 20 Sept 2018
  43. WHO (2018) BCG vaccine: WHO position paper, February 2018. WHO 93(8):73–96Google Scholar
  44. Wu H (2003) A variation-of-constants for a for a linear abstract evolution equation in hilbert space. ANZIAM J 44:583–590CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institute of Mathematical SciencesUniversity of MalayaKuala LumpurMalaysia

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