The Burden of the Coinfection of HIV and TB in the Presence of Multi-drug Resistant Strains

  • Ana Carvalho
  • Carla M. A. Pinto
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 248)


We introduce a fractional-order model for the coinfection of the immunodeficiency virus and tuberculosis, in the presence of drug resistant tuberculosis strains and treatment for both diseases. We compute the reproduction number of the model. Numerical simulations show the different dynamics of the model for variation of relevant parameters. Moreover, the order of the fractional derivative plays an important role in the severity of the epidemics.


Tuberculosis HIV Resistant strains Coinfection 



The authors were partially funded by the European Regional Development Fund through the program COMPETE and by the Portuguese Government through the FCT - Fundação para a Ciência e a Tecnologia under the project PEst-C/MAT/UI0144/2013. The research of AC was partially supported by a FCT grant with reference SFRH/BD/96816/2013.


  1. 1.
    Madariaga, M.G., Lalloo, U.G., Swindells, S.: Extensively drug-resistant tuberculosis. Am. J. Med. 121(10), 835–844 (2008)CrossRefGoogle Scholar
  2. 2.
    The World Health Organization (WHO): Global Tuberculosis Report (2016)Google Scholar
  3. 3.
    Gandhi, N.R., Nunn, P., Dheda, K., Schaaf, H.S., Zignol, M., Van Soolingen, D., Jensen, P., Bayona, J.: Multidrug-resistant and extensively drug-resistant tuberculosis: a threat to global control of tuberculosis. Lancet 375, 1830–1843 (2010)CrossRefGoogle Scholar
  4. 4.
    Agusto, F.B., Adekunle, A.I.: Optimal control of a two-strain tuberculosis-HIV/AIDS co-infection model. BioSystems 119, 20–44 (2014)CrossRefGoogle Scholar
  5. 5.
    Bacaër, N.N., Ouifki, R., Pretorius, C., Wood, R., Williams, B.: Modeling the joint epidemics of TB and HIV in a South African township. J. Math. Biol. 57(4), 557–593 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bhunu, C.P., Garira, W., Mukandavire, Z.: Modeling HIV/AIDS and tuberculosis coinfection. Bull. Math. Biol. 71, 1745–1780 (2009)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Castillo-Chavez, C., Feng, Z.: To treat or not to treat: the case of tuberculosis. J. Math. Biol. 35, 629–656 (1997)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Denysiuk, R., Silva, C.J., Torres, D.F.M.: Multiobjective approach to optimal control for a tuberculosis model. Optim. Methods Softw. 30(5), (2015)Google Scholar
  9. 9.
    Sergeev, R., Colijn, C., Murray, M., Cohen, T.: Modeling the dynamic relationship between HIV and the risk of drug-resistant tuberculosis. Sci. Transl. Med. 4(135), 135–167 (2012)CrossRefGoogle Scholar
  10. 10.
    Silva, C.J., Torres, D.F.M.: A TB-HIV/AIDS coinfection model and optimal control treatment. Discrete Continuous Dyn. Syst. - A 35(9), 4639–4663 (2015)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Ezz-Eldien, S.S., Doha, E.H., Bhrawy, A.H., El-Kalaawy, A.A., Machado, J.T.A.: A new operational approach for solving fractional variational problems depending on indefinite integrals. Commun. Nonlinear Sci. Numer. Simul. 57, 246–263 (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Nigmatullin, R.R., Zhang, W., Gubaidullin, I.: Accurate relationships between fractals and fractional integrals: new approaches and evaluations. Fractional Calc. Appl. Anal. 20(5), 1263–1280 (2017)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Carvalho, A.R.M., Pinto, C.M.A., Baleanu, D.: HIV/HCV coinfection model: a fractional-order perspective for the effect of the HIV viral load. Adv. Differ. Equ. 2018(1), 1–22 (2018)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Copot, D., De Keyser, R., Derom, E., Ortigueira, M., Ionescu, C.M.: Reducing bias in fractional order impedance estimation for lung function evaluation. Biomed. Sig. Process. Control 39, 74–80 (2018)CrossRefGoogle Scholar
  15. 15.
    Oldham, K., Spanier, J.: The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order. Academic Press, New York, NY (1974)zbMATHGoogle Scholar
  16. 16.
    Samko, S., Kilbas, A., Marichev, O.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers, London (1993)zbMATHGoogle Scholar
  17. 17.
    Ventura, A., Tejado, I., Valério, D., Martins, J.: Fractional direct and inverse models of the dynamics of a human arm. J. Vibr. Control 22(9), 2240–2254 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Wu, G.C., Baleanu, D., Huang, L.L.: Novel Mittag-Leffler stability of linear fractional delay difference equations with impulse. Appl. Math. Lett. 82, 71–78 (2018)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Pinto, C.M.A., Carvalho, A.R.M.: New findings on the dynamics of HIV and TB coinfection models. Appl. Math. Comput. 242, 36–46 (2014)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Sardar, T., Rana, S., Bhattacharya, S., Al-Khaled, K., Chattopadhyay, J.: A generic model for a single strain mosquito-transmitted disease memory on the host and the vector. Math. Biosci. 263, 18–36 (2015)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Driessche, P., Watmough, P.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–48 (2002)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of SciencesUniversity of PortoPortoPortugal
  2. 2.School of EngineeringPolytechnic of Porto and Center for Mathematics of the University of PortoPortoPortugal

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