# 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 & Continuous Dynamical Systems - A, 2015, 35 (9) : 4639-4663. doi: 10.3934/dcds.2015.35.4639
##### References:
 [1] AVERT, HIV & AIDS Information from AVERT.org,, , (). Google Scholar [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. doi: 10.1007/s00285-008-0177-z. Google Scholar [3] C. P. Bhunu, W. Garira and Z. Mukandavire, Modeling HIV/AIDS and tuberculosis coinfection,, Bul. Math. Biol., 71 (2009), 1745. doi: 10.1007/s11538-009-9423-9. Google Scholar [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. doi: 10.3934/mbe.2014.11.761. Google Scholar [5] S. Bowong, Optimal control of the transmission dynamics of tuberculosis,, Nonlinear Dynam., 61 (2010), 729. doi: 10.1007/s11071-010-9683-9. Google Scholar [6] J. Carr, Applications Centre Manifold Theory,, Springer-Verlag, (1981). Google Scholar [7] C. Castillo-Chavez and Z. Feng, To treat or not to treat: The case of tuberculosis,, J. Math. Biol., 35 (1997), 629. doi: 10.1007/s002850050069. Google Scholar [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. doi: 10.1007/978-1-4757-3667-0_13. Google Scholar [9] C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications,, Math. Biosc. Engrg., 1 (2004), 361. doi: 10.3934/mbe.2004.1.361. Google Scholar [10] L. Cesari, Optimization - Theory and Applications. Problems with Ordinary Differential Equations,, Applications of Mathematics 17, (1983). doi: 10.1007/978-1-4613-8165-5. Google Scholar [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. Google Scholar [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. doi: 10.1016/S0140-6736(13)61809-7. Google Scholar [13] O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases,, Wiley Series in Mathematical and Computational Biology, (2000). Google Scholar [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. doi: 10.1007/BF00178324. Google Scholar [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). doi: 10.1155/2011/398476. Google Scholar [16] W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control,, Springer Verlag, (1975). Google Scholar [17] R. Fourer, D. M. Gay and B. W. Kernighan, AMPL: A Modeling Language for Mathematical Programming,, Duxbury Press, (1993). Google Scholar [18] H. Getahun, C. Gunneberg, R. Granich and P. Nunn, HIV infection-associated tuberculosis: The epidemiology and the response,, Clin. Infect. Dis., 50 (2010). doi: 10.1086/651492. Google Scholar [19] K. Hattaf, M. Rachik, S. Saadi, Y. Tabit and N. Yousfi, Optimal control of tuberculosis with exogenous reinfection,, Appl. Math. Sci., 3 (2009), 231. Google Scholar [20] H. W. Hethcote, The mathematics of infectious diseases,, SIAM Rev., 42 (2000), 599. doi: 10.1137/S0036144500371907. Google Scholar [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. doi: 10.3934/dcdsb.2002.2.473. Google Scholar [22] D. Kirschner, Dynamics of co-infection with M. tuberculosis and HIV-1,, Theor. Pop. Biol., 55 (1999), 94. doi: 10.1006/tpbi.1998.1382. Google Scholar [23] D. Kirschner, S. Lenhart and S. Serbin, Optimal control of the chemotherapy of HIV,, J. Mathematical Biology, 35 (1996), 775. doi: 10.1007/s002850050076. Google Scholar [24] C. K. Kwan and J. D. Ernst, HIV and tuberculosis: A deadly human syndemic,, Clin. Microbiol. Rev., 24 (2011), 351. doi: 10.1128/CMR.00042-10. Google Scholar [25] V. Lakshmikantham, S. Leela and A. A. Martynyuk, Stability Analysis of Nonlinear Systems,, Marcel Dekker, (1989). Google Scholar [26] U. Ledzewicz and H. Schättler, On optimal singular controls for a general SIR-model with vaccination and treatment,, Discrete Contin. Dyn. Syst., II (2011), 981. doi: 10.3934/proc.2011.2011.981. Google Scholar [27] S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models,, Chapman & Hall/CRC, (2007). Google Scholar [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. doi: 10.1080/08898480903467241. Google Scholar [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. doi: 10.3846/1392-6292.2009.14.483-494. Google Scholar [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. doi: 10.1080/13926292.2005.9637287. Google Scholar [31] L. Pontryagin, V. Boltyanskii, R. Gramkrelidze and E. Mischenko, The Mathematical Theory of Optimal Processes,, Wiley Interscience, (1962). Google Scholar [32] PROPT, Matlab Optimal Control Software (DAE, ODE),, , (). Google Scholar [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. doi: 10.1016/j.mcm.2010.06.034. Google Scholar [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. doi: 10.1080/00207160.2011.554540. Google Scholar [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. doi: 10.1007/s11538-014-0028-6. Google Scholar [36] L. W. Roeger, Z. Feng and C. Castillo-Chavez, Modeling TB and HIV co-infections,, Math. Biosc. and Eng., 6 (2009), 815. doi: 10.3934/mbe.2009.6.815. Google Scholar [37] W. N. Rom and S. B. Markowitz, Environmental and Occupational Medicine,, Lippincott Williams & Wilkins, (2007). Google Scholar [38] H. Schättler and U. Ledzewicz, Geometric Optimal Control,, Springer, (2012). doi: 10.1007/978-1-4614-3834-2. Google Scholar [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. doi: 10.3934/dcdsb.2014.19.2657. Google Scholar [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. doi: 10.3934/mbe.2008.5.145. Google Scholar [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. doi: 10.3934/naco.2012.2.601. Google Scholar [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. doi: 10.1016/j.mbs.2013.05.005. Google Scholar [43] K. Styblo, State of art: Epidemiology of tuberculosis,, Bull. Int. Union Tuberc., 53 (1978), 141. Google Scholar [44] H. R. Thieme, Persistence under relaxed point-dissipaty (with applications to an epidemic model),, SIAM. J. Math. Anal. Appl., 24 (1993), 407. doi: 10.1137/0524026. Google Scholar [45] UNAIDS, Global Report: UNAIDS Report on the Global AIDS Epidemic 2013,, Geneva, (2013). Google Scholar [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. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar [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. doi: 10.1007/s10107-004-0559-y. Google Scholar [48] WHO, Global Tuberculosis Report 2013,, Geneva, (2013). Google Scholar [49] WHO, Tuberculosis,, Fact sheet no. 104, (2014). Google Scholar

show all references

##### References:
 [1] AVERT, HIV & AIDS Information from AVERT.org,, , (). Google Scholar [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. doi: 10.1007/s00285-008-0177-z. Google Scholar [3] C. P. Bhunu, W. Garira and Z. Mukandavire, Modeling HIV/AIDS and tuberculosis coinfection,, Bul. Math. Biol., 71 (2009), 1745. doi: 10.1007/s11538-009-9423-9. Google Scholar [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. doi: 10.3934/mbe.2014.11.761. Google Scholar [5] S. Bowong, Optimal control of the transmission dynamics of tuberculosis,, Nonlinear Dynam., 61 (2010), 729. doi: 10.1007/s11071-010-9683-9. Google Scholar [6] J. Carr, Applications Centre Manifold Theory,, Springer-Verlag, (1981). Google Scholar [7] C. Castillo-Chavez and Z. Feng, To treat or not to treat: The case of tuberculosis,, J. Math. Biol., 35 (1997), 629. doi: 10.1007/s002850050069. Google Scholar [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. doi: 10.1007/978-1-4757-3667-0_13. Google Scholar [9] C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications,, Math. Biosc. Engrg., 1 (2004), 361. doi: 10.3934/mbe.2004.1.361. Google Scholar [10] L. Cesari, Optimization - Theory and Applications. Problems with Ordinary Differential Equations,, Applications of Mathematics 17, (1983). doi: 10.1007/978-1-4613-8165-5. Google Scholar [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. Google Scholar [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. doi: 10.1016/S0140-6736(13)61809-7. Google Scholar [13] O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases,, Wiley Series in Mathematical and Computational Biology, (2000). Google Scholar [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. doi: 10.1007/BF00178324. Google Scholar [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). doi: 10.1155/2011/398476. Google Scholar [16] W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control,, Springer Verlag, (1975). Google Scholar [17] R. Fourer, D. M. Gay and B. W. Kernighan, AMPL: A Modeling Language for Mathematical Programming,, Duxbury Press, (1993). Google Scholar [18] H. Getahun, C. Gunneberg, R. Granich and P. Nunn, HIV infection-associated tuberculosis: The epidemiology and the response,, Clin. Infect. Dis., 50 (2010). doi: 10.1086/651492. Google Scholar [19] K. Hattaf, M. Rachik, S. Saadi, Y. Tabit and N. Yousfi, Optimal control of tuberculosis with exogenous reinfection,, Appl. Math. Sci., 3 (2009), 231. Google Scholar [20] H. W. Hethcote, The mathematics of infectious diseases,, SIAM Rev., 42 (2000), 599. doi: 10.1137/S0036144500371907. Google Scholar [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. doi: 10.3934/dcdsb.2002.2.473. Google Scholar [22] D. Kirschner, Dynamics of co-infection with M. tuberculosis and HIV-1,, Theor. Pop. Biol., 55 (1999), 94. doi: 10.1006/tpbi.1998.1382. Google Scholar [23] D. Kirschner, S. Lenhart and S. Serbin, Optimal control of the chemotherapy of HIV,, J. Mathematical Biology, 35 (1996), 775. doi: 10.1007/s002850050076. Google Scholar [24] C. K. Kwan and J. D. Ernst, HIV and tuberculosis: A deadly human syndemic,, Clin. Microbiol. Rev., 24 (2011), 351. doi: 10.1128/CMR.00042-10. Google Scholar [25] V. Lakshmikantham, S. Leela and A. A. Martynyuk, Stability Analysis of Nonlinear Systems,, Marcel Dekker, (1989). Google Scholar [26] U. Ledzewicz and H. Schättler, On optimal singular controls for a general SIR-model with vaccination and treatment,, Discrete Contin. Dyn. Syst., II (2011), 981. doi: 10.3934/proc.2011.2011.981. Google Scholar [27] S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models,, Chapman & Hall/CRC, (2007). Google Scholar [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. doi: 10.1080/08898480903467241. Google Scholar [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. doi: 10.3846/1392-6292.2009.14.483-494. Google Scholar [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. doi: 10.1080/13926292.2005.9637287. Google Scholar [31] L. Pontryagin, V. Boltyanskii, R. Gramkrelidze and E. Mischenko, The Mathematical Theory of Optimal Processes,, Wiley Interscience, (1962). Google Scholar [32] PROPT, Matlab Optimal Control Software (DAE, ODE),, , (). Google Scholar [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. doi: 10.1016/j.mcm.2010.06.034. Google Scholar [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. doi: 10.1080/00207160.2011.554540. Google Scholar [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. doi: 10.1007/s11538-014-0028-6. Google Scholar [36] L. W. Roeger, Z. Feng and C. Castillo-Chavez, Modeling TB and HIV co-infections,, Math. Biosc. and Eng., 6 (2009), 815. doi: 10.3934/mbe.2009.6.815. Google Scholar [37] W. N. Rom and S. B. Markowitz, Environmental and Occupational Medicine,, Lippincott Williams & Wilkins, (2007). Google Scholar [38] H. Schättler and U. Ledzewicz, Geometric Optimal Control,, Springer, (2012). doi: 10.1007/978-1-4614-3834-2. Google Scholar [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. doi: 10.3934/dcdsb.2014.19.2657. Google Scholar [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. doi: 10.3934/mbe.2008.5.145. Google Scholar [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. doi: 10.3934/naco.2012.2.601. Google Scholar [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. doi: 10.1016/j.mbs.2013.05.005. Google Scholar [43] K. Styblo, State of art: Epidemiology of tuberculosis,, Bull. Int. Union Tuberc., 53 (1978), 141. Google Scholar [44] H. R. Thieme, Persistence under relaxed point-dissipaty (with applications to an epidemic model),, SIAM. J. Math. Anal. Appl., 24 (1993), 407. doi: 10.1137/0524026. Google Scholar [45] UNAIDS, Global Report: UNAIDS Report on the Global AIDS Epidemic 2013,, Geneva, (2013). Google Scholar [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. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar [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. doi: 10.1007/s10107-004-0559-y. Google Scholar [48] WHO, Global Tuberculosis Report 2013,, Geneva, (2013). Google Scholar [49] WHO, Tuberculosis,, Fact sheet no. 104, (2014). Google Scholar
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