Assessing the effects of treatment in HIV-TB co-infection model

  • Sachin Kumar
  • Shikha JainEmail author
Regular Article


We propose a population model for HIV-TB co-infection dynamics by considering treatments for HIV infection, active tuberculosis and co-infection. The HIV-only and TB-only models are analyzed separately, as well as full model. The basic reproduction numbers for TB ( \( {R}_0^T\) and HIV ( \( {R}_0^H\) and overall reproduction number for the system \( {R}_0= \max\{{R}_0^T, {R}_0^H\}\) are computed. The equilibria and their stability are studied. The main model undergoes supercritical transcritical bifurcation at \( {R}_0^T=1\) and \( {R}_0^H=1\) , whereas the parameters \( \beta^{\ast}=\beta e\) and \( \lambda^{\ast}=\lambda \sigma\) act as bifurcation parameters, respectively. Numerical simulation claims the existence of interior equilibrium when both the reproduction numbers are greater than unity. We explore the effect of early and late HIV treatment on disease-induced deaths during the TB treatment course. Mathematical analysis of our model shows that successful disease eradication requires treatment of single disease, that is, treatment for HIV-only- and TB-only-infected individuals with addition to co-infection treatment and in the absence of which disease eradication is extremely difficult even for R < 0. When both the diseases are epidemic, the treatment for TB-only-infected individuals is very effective in reducing the total infected population and disease-induced deaths in comparison with the treatment for HIV-infected individuals while these are minimum when both the single-disease treatments are given with co-infection treatment.


  1. 1.
  2. 2.
  3. 3.
  4. 4.
  5. 5.
    De Cock, M. Kevin, R.E. Chaisson, Int. J. Tuberculosis Lung Disease 3, 457 (1999)Google Scholar
  6. 6.
    G. Guzzetta et al., J. Theoret. Biol. 289, 197 (2011)MathSciNetCrossRefGoogle Scholar
  7. 7.
    J.M. Trauer, J.T. Denholm, E.S. McBryde, J. Theor. Biol. 358, 74 (2014)CrossRefGoogle Scholar
  8. 8.
    E.F. Long, N.K. Vaidya, M.L. Brandeau, Oper. Res. 56, 1366 (2008)MathSciNetCrossRefGoogle Scholar
  9. 9.
    L.-I.W. Roeger, Z. Feng, C. Castillo-Chavez, Math. Biosci. Eng. 6, 815 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    C.J. Silva, D.F.M. Torres, Dyn. Syst. 35, 4639 (2015)Google Scholar
  11. 11.
    C.P. Bhunu, W. Garira, Z. Mukandavire, Bull. Math. Biol. 71, 1745 (2009)MathSciNetCrossRefGoogle Scholar
  12. 12.
    R. Naresh, D. Sharma, A. Tripathi, Math. Comput. Modell. 50, 1154 (2009)CrossRefGoogle Scholar
  13. 13.
    S. Gakkhar, N. Chavda, Appl. Math. Comput. 218, 9261 (2012)MathSciNetGoogle Scholar
  14. 14.
    N. Kaur, M. Ghosh, S.S. Bhatia, J. Adv. Res. Dyn. Control Syst. 7, 66 (2015)MathSciNetGoogle Scholar
  15. 15.
    A. Mallela, S. Lenhart, N.K. Vaidya, J. Comput. Appl. Math. 307, 143 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    M.M. Bosma-den Boer, M.L. van Wetten, L. Pruimboom, Nutr. Metab. 9, 32 (2012)CrossRefGoogle Scholar
  17. 17.
    K.A. Sepkowitz, Clin. Infect. Diseases 23, 954 (1996)CrossRefGoogle Scholar
  18. 18.
    O. Diekmann, J.A.P. Heesterbeek, J.A.J. Metz, J. Math. Biol. 28, 365 (1990)MathSciNetCrossRefGoogle Scholar
  19. 19.
    O. Diekmann, J.A.P. Heesterbeek, M.G. Roberts, J. R. Soc. Interface 7, 873 (2010)CrossRefGoogle Scholar
  20. 20.
    J.H. Jones, Notes on $\mathcal{R}_0$,
  21. 21.
    L. Perko, Differential equations and dynamical systems, third edition, Texts in Applied Mathematics, Vol. 7 (Springer-Verlag, New York, 2001)Google Scholar
  22. 22.
    C. Castillo-Chavez, Z. Feng, W. Huang, On the computation of $\mathscr{R}_0$ and its role on global stability, in Mathematical approaches for emerging and reemerging infectious diseases: an introduction (Minneapolis, MN, 1999), IMA Vol. Math. Appl., Vol. 125 (Springer, New York, 2002) pp. 229--250Google Scholar
  23. 23.
    C. Castillo-Chavez, B. Song, Math. Biosci. Eng. 1, 361 (2004)MathSciNetCrossRefGoogle Scholar
  24. 24.
  25. 25.
  26. 26.
    C. Castillo-Chavez, Z. Feng, J. Math. Biol. 35, 629 (1997)MathSciNetCrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of DelhiNew DelhiIndia

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