Optimal treatment control and bifurcation analysis of a tuberculosis model with effect of multiple re-infections



We derive and analyze a tuberculosis (TB) model including exogenous re-infection and endogenous reactivation, and the re-infection among the treated individuals. The disease-free equilibrium and the existence criterion of endemic equilibrium are investigated. The basic reproduction number \(R_0 \) is derived and it is found that the disease-free equilibrium is stable when \(R_0 <1\), unstable for \(R_0 >1\), and the system undergoes a transcritical bifurcation at the disease-free equilibrium when \(R_0 =1\). Furthermore, for \(R_0 <1\), there are two endemic equilibria, one of which is stable and other one is unstable, indicating the occurrence of backward bifurcation. The local stability analysis of the disease-free and the endemic equilibrium is shown. Also, we studied the sensitivity analysis of the system in refer to some crucial model parameters and the sensitivity indices of \(R_0\) to parameters for the TB model are obtained. Using Pontryagin’s maximum principle, we have discussed about the optimal control of the disease. Various simulation works are given throughout the paper to support our analytical results.


Tuberculosis Reproduction number Transcritical bifurcation Backward bifurcation  Optimal control 



Research of T. K. Kar is supported by the Council of Scientific and Industrial Research (CSIR) (Sanction No: 25(0224)/14/EMR-II, dated: 2/12/2014), Human Resource Development Group, India.


  1. 1.
    Anderson RM, May RM (1982) Population biology of infectious diseases. Springer, BerlinCrossRefGoogle Scholar
  2. 2.
    Diekmann O, Heesterbeek JAP (2000) Mathematical epidemiology of infectious diseases. In: Model building, analysis and interpretation, Wiley Series in Mathematical and Computational Biology, John Wiley and Sons, ChichesterGoogle Scholar
  3. 3.
    Ruan S, Wong W (2003) Dynamical behavior of an epidemic model with a nonlinear incidence rate. J Differ Equ 188:135–163MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Takeuchi Y, Liu X, Cui J (2007) Global dynamics of SIS models with transport related infection. J Math Anal Appl 329:1460–1471MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Behr MA (2004) Tuberculosis due to multiple strains: a concern for the patient? A concern for tuberculosis control? Am J Respir Crit Care Med 169:554–555CrossRefGoogle Scholar
  6. 6.
    Richardson M et al (2002) Multiple Mycobacterium tuberculosis strains in early cultures from patients in a high incidence community setting. J Clin Microbiol 40:2750–2754CrossRefGoogle Scholar
  7. 7.
    Warren RM, Victor TC, Streicher EM, Richardson M, Beyers N, Gey van Pittius NC, van Helden PD (2004) Patients with active tuberculosis often have different strains in the same sputum specimen. Am J Respir Crit Care Med 169:610–614CrossRefGoogle Scholar
  8. 8.
    Yang HM, Raimundo SM (2010) Assessing the effects of multiple infections and long latency in the dynamics of tuberculosis. Theor Biol Med Model 7:41CrossRefGoogle Scholar
  9. 9.
    Chaves F, Dronda F, Alonso-Sanz M, Noriega AR (1999) Evidence of exogenous re-infection and mixed infection with more than one strain of Mycobacterium TB among Spanish HIV-infected inmates. AIDS 13:615–620CrossRefGoogle Scholar
  10. 10.
    Small PM, Shafer RW, Hopewell PC, Murphy MJ, Desmond E, Sierra MF, Schoolnik GK (1993) Exogenous re-infection with multidrug-resistant Mycobacterium tuberculosis in patients with advanced HIV infection. N Engl J Med 328:1137–1144CrossRefGoogle Scholar
  11. 11.
    Nardell E, Mc Innis B, Thomas B, Weidhaas S (1986) Exogenous re-infection with tuberculosis in a shelter for the homeless. N Engl J Med 315:1570–1575CrossRefGoogle Scholar
  12. 12.
    Martcheva M, Thieme HR (2003) Progression age enhanced backward bifurcation in an epidemic model with super-infection. J Math Biol 46:385–424MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Feng Z, Castillo-Chavez C, Capurro AF (2000) A model for tuberculosis with exogenous reinfection. Theor Popul Biol 57:235CrossRefMATHGoogle Scholar
  14. 14.
    Gomez-Acevedo H, Li MY (2005) Backward bifurcation in a model for HTLV-I infection of CD4+ T cells. Bull Math Biol 67:101–114MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Greenhalgh D, Griffiths M (2009) Backward bifurcation, equilibrium and stability phenomena in a three-stage extended BRSV epidemic model. Math Biol 59:1–36MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Singer BH, Kirschner DE (2004) Influence of backward bifurcation on interpretation of \(R_0\) in a model of epidemic tuberculosis with reinfection. Math Biosci Eng 1(1):81–93MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Zhang X, Liu X (2009) Backward bifurcation and global dynamics of an SIS epidemic model with general incidence rate and treatment. Nonlinear Anal Real World Appl 10:565–575MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Arino J, McCluskey CC, Van den Driessche P (2003) Global results for an epidemic model with vaccination that exhibits backward bifurcation. SIAM J Appl Math 64:260–276MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Barrett S, Hoel M (2007) Optimal disease eradication. Environ Dev Econ 12:627–652CrossRefGoogle Scholar
  20. 20.
    Zaman G, Kang YH, Jung IH (2008) Stability analysis and optimal vaccination of an SIR epidemic model. Biosystems 93:240–249CrossRefGoogle Scholar
  21. 21.
    Claytona T, Duke-Sylvesterb S, Grossc LJ, Lenhartd S, Realb LA (2010) Optimal control of a rabies epidemic model with a birth pulse. J Biol Dyn 4(1):43–58MathSciNetCrossRefGoogle Scholar
  22. 22.
    Ding W (2007) Optimal control on hybrid ODE systems with application to a tick disease model. Math Biosci Eng (SCI) 4:633–659MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Gaff H, Schaefer E (2009) Optimal control applied to vaccination and treatment strategies for various epidemiological models. Math Biosci Eng 6:469–492MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Nanda S, Moore H, Lenhart S (2007) Optimal control of treatment in a mathematical model of chronic myelogenous leukemia. Math Biosci 210:143–156MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Okosun KO, Ouifki R, Marcus N (2011) Optimal control analysis of a malaria disease transmission model that includes treatment and vaccination with waning immunity. Biosystems 106:136–145. doi: 10.1016/j.biosystems.2011.07.006
  26. 26.
    Joshi H, Lenhart S, Li MY, Wang L (2006) Optimal control methods applied to disease models. Contempor Math 410:187–207MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Blayneh KW, Gumel AB, Lenhart S, Clayton T (2010) Backward bifurcation and optimal control in transmission dynamics of the West Nile virus. Bull Math Biol 72:1006–1028MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Emvudu Y, Mewoli B, jean jules Tewa JJ, Kouenkam JP (2011) Epidemiological model for the spread of anti-tuberculosis resistance. Int J Inf Syst Sci 7(4):279–301MathSciNetMATHGoogle Scholar
  29. 29.
    Kar TK, Mondal PK (2012) Global dynamics of a tuberculosis epidemic model and the influence of backward bifurcation. J Math Model Algorithms 11:433–459MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Chitnis N, Hyman JM, Cushing JM (2008) Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model. Bull Math Biol 70:1272–1296MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Kar TK, Mondal PK (2011) Global dynamics and bifurcation in delayed SIR epidemic model. Nonlinear Anal Real World Appl 12:2058–2068MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Bartl M, Li P, Schuster S (2010) Modelling the optimal timing in metabolic pathway activetion—use of Pontryagin’s maximum principle and role of the golden section. Biosystems 101(1):67–77CrossRefGoogle Scholar
  33. 33.
    Kar TK, Batabyal A (2011) Stability analysis and optimal control of an SIR epidemic model with vaccination. Biosystems 104:127–135Google Scholar
  34. 34.
    Lenhart S, Workman JT (2007) Optimal control applied to biological models. Chapman and Hall/CRC, LondonMATHGoogle Scholar
  35. 35.
    Joshi HR (2002) Optimal control of an HIV immunology model. Optim Control Appl Methods 23:199–213MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Lukes DL (1982) Differential equations: classical to controlled. In: Mathematics in science and Engineering, vol. 162, Academic Press, New York, p. 182Google Scholar
  37. 37.
    Fleming WH, Rishel RW (1975) Deterministic and stochastic optimal control. Springer, BerlinCrossRefMATHGoogle Scholar
  38. 38.
    Lenhart S, Workman JT (2007) Optimal control applied to biological models. Chapman and Hall/CRC, LondonMATHGoogle Scholar
  39. 39.
    Pontryagin LS, Boltyanskii VG, Gamkrelidze RV, Mishchenko EF (1962) The mathematical theory of optimal processes. Wiley, New YorkGoogle Scholar
  40. 40.
    Zaman G, Kang YH, Jung IH (2007) Optimal vaccination and treatment in the SIR epidemic model. Proc KSIAM 3:31–33Google Scholar
  41. 41.
    Blower SM, Dowlatabadi H (1994) Sensitivity and uncertainty analysis of complex models of disease transmission: an HIV model, as an example. International Stat Rev 62(2):229–243CrossRefMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of MathematicsShibpur Sri Ramkrishna VidyalayaHowrahIndia
  2. 2.Department of MathematicsIndian Institute of Engineering Science and TechnologyShibpur, HowrahIndia

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