# American Institute of Mathematical Sciences

September  2015, 35(9): 4639-4663. doi: 10.3934/dcds.2015.35.4639

## A TB-HIV/AIDS coinfection model and optimal control treatment

 1 Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal 2 CIDMA — Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal

Received  May 2014 Revised  November 2014 Published  April 2015

We propose a population model for TB-HIV/AIDS coinfection transmission dynamics, which considers antiretroviral therapy for HIV infection and treatments for latent and active tuberculosis. The HIV-only and TB-only sub-models are analyzed separately, as well as the TB-HIV/AIDS full model. The respective basic reproduction numbers are computed, equilibria and stability are studied. Optimal control theory is applied to the TB-HIV/AIDS model and optimal treatment strategies for co-infected individuals with HIV and TB are derived. Numerical simulations to the optimal control problem show that non intuitive measures can lead to the reduction of the number of individuals with active TB and AIDS.
Citation: Cristiana J. Silva, Delfim F. M. Torres. A TB-HIV/AIDS coinfection model and optimal control treatment. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4639-4663. doi: 10.3934/dcds.2015.35.4639
##### References:
 [1] AVERT, HIV & AIDS Information from AVERT.org,, , (). [2] N. Bacaër, R. Ouifki, C. Pretorius, R. Wood and B. Williams, Modeling the joint epidemics of TB and HIV in a South African township, J. Math. Biol., 57 (2008), 557-593. doi: 10.1007/s00285-008-0177-z. [3] C. P. Bhunu, W. Garira and Z. Mukandavire, Modeling HIV/AIDS and tuberculosis coinfection, Bul. Math. Biol., 71 (2009), 1745-1780. doi: 10.1007/s11538-009-9423-9. [4] M. H. A. Biswas, L. T. Paiva and MdR de Pinho, A SEIR model for control of infectious diseases with constraints, Mathematical Biosciences and Engineering, 11 (2014), 761-784. doi: 10.3934/mbe.2014.11.761. [5] S. Bowong, Optimal control of the transmission dynamics of tuberculosis, Nonlinear Dynam., 61 (2010), 729-748. doi: 10.1007/s11071-010-9683-9. [6] J. Carr, Applications Centre Manifold Theory, Springer-Verlag, New-York, 1981. [7] C. Castillo-Chavez and Z. Feng, To treat or not to treat: The case of tuberculosis, J. Math. Biol., 35 (1997), 629-656. doi: 10.1007/s002850050069. [8] C. Castillo-Chavez, Z. Feng and W. Huang, On the computation $R_0$ its role on global stability, Mathematical approaches for emerging and re-emerging infectious diseases. IMA, 125 (2002), 229-250. doi: 10.1007/978-1-4757-3667-0_13. [9] C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications, Math. Biosc. Engrg., 1 (2004), 361-404. doi: 10.3934/mbe.2004.1.361. [10] L. Cesari, Optimization - Theory and Applications. Problems with Ordinary Differential Equations, Applications of Mathematics 17, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4613-8165-5. [11] P. W. David, G. L. Matthew, E. G. Andrew, A. C. David and M. K. John, Relation between HIV viral load and infectiousness: A model-based analysis, The Lancet, 372 (2008), 314-320. [12] S. G. Deeks, S. R. Lewin and D. V. Havlir, The end of AIDS: HIV infection as a chronic disease, The Lancet, 382 (2013), 1525-1533. doi: 10.1016/S0140-6736(13)61809-7. [13] O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases, Wiley Series in Mathematical and Computational Biology, Wiley, Chichester, 2000. [14] O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382. doi: 10.1007/BF00178324. [15] Y. Emvudu, R. Demasse and D. Djeudeu, Optimal control of the lost to follow up in a tuberculosis model, Comput. Math. Methods Med., 2011 (2011), Art. ID 398476, 12 pp. doi: 10.1155/2011/398476. [16] W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer Verlag, New York, 1975. [17] R. Fourer, D. M. Gay and B. W. Kernighan, AMPL: A Modeling Language for Mathematical Programming, Duxbury Press, Pacific Grove, CA, 1993. [18] H. Getahun, C. Gunneberg, R. Granich and P. Nunn, HIV infection-associated tuberculosis: The epidemiology and the response, Clin. Infect. Dis., 50 (2010), S201-S207. doi: 10.1086/651492. [19] K. Hattaf, M. Rachik, S. Saadi, Y. Tabit and N. Yousfi, Optimal control of tuberculosis with exogenous reinfection, Appl. Math. Sci., (Ruse) 3 (2009), 231-240. [20] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653. doi: 10.1137/S0036144500371907. [21] E. Jung, S. Lenhart and Z. Feng, Optimal control of treatments in a two-strain tuberculosis model, Discrete Contin. Dyn. Syst. Ser. B, 2 (2002), 473-482. doi: 10.3934/dcdsb.2002.2.473. [22] D. Kirschner, Dynamics of co-infection with M. tuberculosis and HIV-1, Theor. Pop. Biol., 55 (1999), 94-109. doi: 10.1006/tpbi.1998.1382. [23] D. Kirschner, S. Lenhart and S. Serbin, Optimal control of the chemotherapy of HIV, J. Mathematical Biology, 35 (1996), 775-792. doi: 10.1007/s002850050076. [24] C. K. Kwan and J. D. Ernst, HIV and tuberculosis: A deadly human syndemic, Clin. Microbiol. Rev., 24 (2011), 351-376. doi: 10.1128/CMR.00042-10. [25] V. Lakshmikantham, S. Leela and A. A. Martynyuk, Stability Analysis of Nonlinear Systems, Marcel Dekker, Inc., New York and Basel, 1989. [26] U. Ledzewicz and H. Schättler, On optimal singular controls for a general SIR-model with vaccination and treatment, Discrete Contin. Dyn. Syst., Dynamical systems, differential equations and applications. 8th AIMS Conference. Suppl., II (2011), 981-990. doi: 10.3934/proc.2011.2011.981. [27] S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models, Chapman & Hall/CRC, Boca Raton, FL, 2007. [28] G. Magombedze, W. Garira and E. Mwenje, Modeling the TB/HIV-1 Co-Infection and the Effects of Its Treatment, Math. Pop. Studies, 17 (2010), 12-64. doi: 10.1080/08898480903467241. [29] G. Magombedze, Z. Mukandavire, C. Chiyaka and G. Musuka, Optimal control of a sex structured HIV/AIDS model with condom use, Mathematical Modelling and Analysis, 14 (2009), 483-494. doi: 10.3846/1392-6292.2009.14.483-494. [30] R. Naresh and A. Tripathi, Modelling and analysis of HIV-TB co-infection in a variable size population, Math. Model. Anal., 10 (2005), 275-286. doi: 10.1080/13926292.2005.9637287. [31] L. Pontryagin, V. Boltyanskii, R. Gramkrelidze and E. Mischenko, The Mathematical Theory of Optimal Processes, Wiley Interscience, 1962. [32] PROPT, Matlab Optimal Control Software (DAE, ODE),, , (). [33] H. S. Rodrigues, M. T. T. Monteiro and D. F. M. Torres, Dynamics of dengue epidemics when using optimal control, Math. Comput. Modelling, 52 (2010), 1667-1673. doi: 10.1016/j.mcm.2010.06.034. [34] H. S. Rodrigues, M. T. T. Monteiro, D. F. M. Torres and A. Zinober, Dengue disease, basic reproduction number and control, Int. J. Comput. Math., 89 (2012), 334-346. doi: 10.1080/00207160.2011.554540. [35] P. Rodrigues, C. J. Silva and D. F. M. Torres, Cost-effectiveness analysis of optimal control measures for tuberculosis, Bull. Math. Biol., 76 (2014), 2627-2645. doi: 10.1007/s11538-014-0028-6. [36] L. W. Roeger, Z. Feng and C. Castillo-Chavez, Modeling TB and HIV co-infections, Math. Biosc. and Eng., 6 (2009), 815-837. doi: 10.3934/mbe.2009.6.815. [37] W. N. Rom and S. B. Markowitz, Environmental and Occupational Medicine, Lippincott Williams & Wilkins, 2007. [38] H. Schättler and U. Ledzewicz, Geometric Optimal Control, Springer, New York, 2012. doi: 10.1007/978-1-4614-3834-2. [39] H. Schättler, U. Ledzewicz and H. Maurer, Sufficient conditions for strong local optimality in optimal control problems with $L_2$-type objectives and control constraints, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2657-2679. doi: 10.3934/dcdsb.2014.19.2657. [40] O. Sharomi, C.N. Podder, A. B. Gumel and B. Song, Mathematical analysis of the transmission dynamics of HIV/TB coinfection in the presence of treatment, Math. Biosc. Eng., 5 (2008), 145-174. doi: 10.3934/mbe.2008.5.145. [41] C. J. Silva and D. F. M. Torres, Optimal control strategies for tuberculosis treatment: A case study in Angola, Numer. Algebra Control Optim., 2 (2012), 601-617. doi: 10.3934/naco.2012.2.601. [42] C. J. Silva and D. F. M. Torres, Optimal control for a tuberculosis model with reinfection and post-exposure interventions, Math. Biosci., 244 (2013), 154-164. doi: 10.1016/j.mbs.2013.05.005. [43] K. Styblo, State of art: Epidemiology of tuberculosis, Bull. Int. Union Tuberc., 53 (1978), 141-152. [44] H. R. Thieme, Persistence under relaxed point-dissipaty (with applications to an epidemic model), SIAM. J. Math. Anal. Appl., 24 (1993), 407-435. doi: 10.1137/0524026. [45] UNAIDS, Global Report: UNAIDS Report on the Global AIDS Epidemic 2013, Geneva, World Health Organization, 2013. [46] P. van den Driessche and J. Watmough, Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Math. Biosc., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [47] A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Program., 106 (2006), 25-57. doi: 10.1007/s10107-004-0559-y. [48] WHO, Global Tuberculosis Report 2013, Geneva, World Health Organization, 2013. [49] WHO, Tuberculosis, Fact sheet no. 104, http://www.who.int/mediacentre/factsheets/fs104/en, Updated October 2014.

show all references

##### References:
 [1] AVERT, HIV & AIDS Information from AVERT.org,, , (). [2] N. Bacaër, R. Ouifki, C. Pretorius, R. Wood and B. Williams, Modeling the joint epidemics of TB and HIV in a South African township, J. Math. Biol., 57 (2008), 557-593. doi: 10.1007/s00285-008-0177-z. [3] C. P. Bhunu, W. Garira and Z. Mukandavire, Modeling HIV/AIDS and tuberculosis coinfection, Bul. Math. Biol., 71 (2009), 1745-1780. doi: 10.1007/s11538-009-9423-9. [4] M. H. A. Biswas, L. T. Paiva and MdR de Pinho, A SEIR model for control of infectious diseases with constraints, Mathematical Biosciences and Engineering, 11 (2014), 761-784. doi: 10.3934/mbe.2014.11.761. [5] S. Bowong, Optimal control of the transmission dynamics of tuberculosis, Nonlinear Dynam., 61 (2010), 729-748. doi: 10.1007/s11071-010-9683-9. [6] J. Carr, Applications Centre Manifold Theory, Springer-Verlag, New-York, 1981. [7] C. Castillo-Chavez and Z. Feng, To treat or not to treat: The case of tuberculosis, J. Math. Biol., 35 (1997), 629-656. doi: 10.1007/s002850050069. [8] C. Castillo-Chavez, Z. Feng and W. Huang, On the computation $R_0$ its role on global stability, Mathematical approaches for emerging and re-emerging infectious diseases. IMA, 125 (2002), 229-250. doi: 10.1007/978-1-4757-3667-0_13. [9] C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications, Math. Biosc. Engrg., 1 (2004), 361-404. doi: 10.3934/mbe.2004.1.361. [10] L. Cesari, Optimization - Theory and Applications. Problems with Ordinary Differential Equations, Applications of Mathematics 17, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4613-8165-5. [11] P. W. David, G. L. Matthew, E. G. Andrew, A. C. David and M. K. John, Relation between HIV viral load and infectiousness: A model-based analysis, The Lancet, 372 (2008), 314-320. [12] S. G. Deeks, S. R. Lewin and D. V. Havlir, The end of AIDS: HIV infection as a chronic disease, The Lancet, 382 (2013), 1525-1533. doi: 10.1016/S0140-6736(13)61809-7. [13] O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases, Wiley Series in Mathematical and Computational Biology, Wiley, Chichester, 2000. [14] O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382. doi: 10.1007/BF00178324. [15] Y. Emvudu, R. Demasse and D. Djeudeu, Optimal control of the lost to follow up in a tuberculosis model, Comput. Math. Methods Med., 2011 (2011), Art. ID 398476, 12 pp. doi: 10.1155/2011/398476. [16] W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer Verlag, New York, 1975. [17] R. Fourer, D. M. Gay and B. W. Kernighan, AMPL: A Modeling Language for Mathematical Programming, Duxbury Press, Pacific Grove, CA, 1993. [18] H. Getahun, C. Gunneberg, R. Granich and P. Nunn, HIV infection-associated tuberculosis: The epidemiology and the response, Clin. Infect. Dis., 50 (2010), S201-S207. doi: 10.1086/651492. [19] K. Hattaf, M. Rachik, S. Saadi, Y. Tabit and N. Yousfi, Optimal control of tuberculosis with exogenous reinfection, Appl. Math. Sci., (Ruse) 3 (2009), 231-240. [20] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653. doi: 10.1137/S0036144500371907. [21] E. Jung, S. Lenhart and Z. Feng, Optimal control of treatments in a two-strain tuberculosis model, Discrete Contin. Dyn. Syst. Ser. B, 2 (2002), 473-482. doi: 10.3934/dcdsb.2002.2.473. [22] D. Kirschner, Dynamics of co-infection with M. tuberculosis and HIV-1, Theor. Pop. Biol., 55 (1999), 94-109. doi: 10.1006/tpbi.1998.1382. [23] D. Kirschner, S. Lenhart and S. Serbin, Optimal control of the chemotherapy of HIV, J. Mathematical Biology, 35 (1996), 775-792. doi: 10.1007/s002850050076. [24] C. K. Kwan and J. D. Ernst, HIV and tuberculosis: A deadly human syndemic, Clin. Microbiol. Rev., 24 (2011), 351-376. doi: 10.1128/CMR.00042-10. [25] V. Lakshmikantham, S. Leela and A. A. Martynyuk, Stability Analysis of Nonlinear Systems, Marcel Dekker, Inc., New York and Basel, 1989. [26] U. Ledzewicz and H. Schättler, On optimal singular controls for a general SIR-model with vaccination and treatment, Discrete Contin. Dyn. Syst., Dynamical systems, differential equations and applications. 8th AIMS Conference. Suppl., II (2011), 981-990. doi: 10.3934/proc.2011.2011.981. [27] S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models, Chapman & Hall/CRC, Boca Raton, FL, 2007. [28] G. Magombedze, W. Garira and E. Mwenje, Modeling the TB/HIV-1 Co-Infection and the Effects of Its Treatment, Math. Pop. Studies, 17 (2010), 12-64. doi: 10.1080/08898480903467241. [29] G. Magombedze, Z. Mukandavire, C. Chiyaka and G. Musuka, Optimal control of a sex structured HIV/AIDS model with condom use, Mathematical Modelling and Analysis, 14 (2009), 483-494. doi: 10.3846/1392-6292.2009.14.483-494. [30] R. Naresh and A. Tripathi, Modelling and analysis of HIV-TB co-infection in a variable size population, Math. Model. Anal., 10 (2005), 275-286. doi: 10.1080/13926292.2005.9637287. [31] L. Pontryagin, V. Boltyanskii, R. Gramkrelidze and E. Mischenko, The Mathematical Theory of Optimal Processes, Wiley Interscience, 1962. [32] PROPT, Matlab Optimal Control Software (DAE, ODE),, , (). [33] H. S. Rodrigues, M. T. T. Monteiro and D. F. M. Torres, Dynamics of dengue epidemics when using optimal control, Math. Comput. Modelling, 52 (2010), 1667-1673. doi: 10.1016/j.mcm.2010.06.034. [34] H. S. Rodrigues, M. T. T. Monteiro, D. F. M. Torres and A. Zinober, Dengue disease, basic reproduction number and control, Int. J. Comput. Math., 89 (2012), 334-346. doi: 10.1080/00207160.2011.554540. [35] P. Rodrigues, C. J. Silva and D. F. M. Torres, Cost-effectiveness analysis of optimal control measures for tuberculosis, Bull. Math. Biol., 76 (2014), 2627-2645. doi: 10.1007/s11538-014-0028-6. [36] L. W. Roeger, Z. Feng and C. Castillo-Chavez, Modeling TB and HIV co-infections, Math. Biosc. and Eng., 6 (2009), 815-837. doi: 10.3934/mbe.2009.6.815. [37] W. N. Rom and S. B. Markowitz, Environmental and Occupational Medicine, Lippincott Williams & Wilkins, 2007. [38] H. Schättler and U. Ledzewicz, Geometric Optimal Control, Springer, New York, 2012. doi: 10.1007/978-1-4614-3834-2. [39] H. Schättler, U. Ledzewicz and H. Maurer, Sufficient conditions for strong local optimality in optimal control problems with $L_2$-type objectives and control constraints, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2657-2679. doi: 10.3934/dcdsb.2014.19.2657. [40] O. Sharomi, C.N. Podder, A. B. Gumel and B. Song, Mathematical analysis of the transmission dynamics of HIV/TB coinfection in the presence of treatment, Math. Biosc. Eng., 5 (2008), 145-174. doi: 10.3934/mbe.2008.5.145. [41] C. J. Silva and D. F. M. Torres, Optimal control strategies for tuberculosis treatment: A case study in Angola, Numer. Algebra Control Optim., 2 (2012), 601-617. doi: 10.3934/naco.2012.2.601. [42] C. J. Silva and D. F. M. Torres, Optimal control for a tuberculosis model with reinfection and post-exposure interventions, Math. Biosci., 244 (2013), 154-164. doi: 10.1016/j.mbs.2013.05.005. [43] K. Styblo, State of art: Epidemiology of tuberculosis, Bull. Int. Union Tuberc., 53 (1978), 141-152. [44] H. R. Thieme, Persistence under relaxed point-dissipaty (with applications to an epidemic model), SIAM. J. Math. Anal. Appl., 24 (1993), 407-435. doi: 10.1137/0524026. [45] UNAIDS, Global Report: UNAIDS Report on the Global AIDS Epidemic 2013, Geneva, World Health Organization, 2013. [46] P. van den Driessche and J. Watmough, Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Math. Biosc., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [47] A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Program., 106 (2006), 25-57. doi: 10.1007/s10107-004-0559-y. [48] WHO, Global Tuberculosis Report 2013, Geneva, World Health Organization, 2013. [49] WHO, Tuberculosis, Fact sheet no. 104, http://www.who.int/mediacentre/factsheets/fs104/en, Updated October 2014.
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