A TB-HIV/AIDS coinfection model and optimal control treatment. (English) Zbl 1365.92134
Summary: 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.
MSC:
| 92D30 | Epidemiology |
| 34D30 | Structural stability and analogous concepts of solutions to ordinary differential equations |
| 49N90 | Applications of optimal control and differential games |
Keywords:
tuberculosis; human immunodeficiency virus; coinfection; treatment; equilibrium; stability; optimal controlReferences:
| [1] | AVERT, HIV & AIDS Information from AVERT.org,, <a href= |
| [2] | N. Bacaër, Modeling the joint epidemics of TB and HIV in a South African township,, J. Math. Biol., 57, 557 (2008) · Zbl 1194.92052 · doi:10.1007/s00285-008-0177-z |
| [3] | C. P. Bhunu, Modeling HIV/AIDS and tuberculosis coinfection,, Bul. Math. Biol., 71, 1745 (2009) · Zbl 1173.92018 · doi:10.1007/s11538-009-9423-9 |
| [4] | M. H. A. Biswas, A SEIR model for control of infectious diseases with constraints,, Mathematical Biosciences and Engineering, 11, 761 (2014) · Zbl 1327.92055 · doi:10.3934/mbe.2014.11.761 |
| [5] | S. Bowong, Optimal control of the transmission dynamics of tuberculosis,, Nonlinear Dynam., 61, 729 (2010) · Zbl 1204.49044 · doi:10.1007/s11071-010-9683-9 |
| [6] | J. Carr, <em>Applications Centre Manifold Theory</em>,, Springer-Verlag (1981) · Zbl 0464.58001 |
| [7] | C. Castillo-Chavez, To treat or not to treat: The case of tuberculosis,, J. Math. Biol., 35, 629 (1997) · Zbl 0895.92024 · doi:10.1007/s002850050069 |
| [8] | C. Castillo-Chavez, On the computation \(R_0\) its role on global stability,, Mathematical approaches for emerging and re-emerging infectious diseases. IMA, 125, 229 (2002) · Zbl 1021.92032 · doi:10.1007/978-1-4757-3667-0_13 |
| [9] | C. Castillo-Chavez, Dynamical models of tuberculosis and their applications,, Math. Biosc. Engrg., 1, 361 (2004) · Zbl 1060.92041 · doi:10.3934/mbe.2004.1.361 |
| [10] | L. Cesari, <em>Optimization - Theory and Applications. Problems with Ordinary Differential Equations</em>,, Applications of Mathematics 17 (1983) · Zbl 0506.49001 · doi:10.1007/978-1-4613-8165-5 |
| [11] | P. W. David, Relation between HIV viral load and infectiousness: A model-based analysis,, The Lancet, 372, 314 (2008) |
| [12] | S. G. Deeks, The end of AIDS: HIV infection as a chronic disease,, The Lancet, 382, 1525 (2013) · doi:10.1016/S0140-6736(13)61809-7 |
| [13] | O. Diekmann, <em>Mathematical Epidemiology of Infectious Diseases</em>,, Wiley Series in Mathematical and Computational Biology (2000) · Zbl 0997.92505 |
| [14] | O. Diekmann, 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, 365 (1990) · Zbl 0726.92018 · doi:10.1007/BF00178324 |
| [15] | Y. Emvudu, Optimal control of the lost to follow up in a tuberculosis model,, Comput. Math. Methods Med., 2011 (2011) · Zbl 1227.92039 · doi:10.1155/2011/398476 |
| [16] | W. H. Fleming, <em>Deterministic and Stochastic Optimal Control</em>,, Springer Verlag (1975) · Zbl 0323.49001 |
| [17] | R. Fourer, <em>AMPL: A Modeling Language for Mathematical Programming</em>,, Duxbury Press (1993) |
| [18] | H. Getahun, HIV infection-associated tuberculosis: The epidemiology and the response,, Clin. Infect. Dis., 50 (2010) · doi:10.1086/651492 |
| [19] | K. Hattaf, Optimal control of tuberculosis with exogenous reinfection,, Appl. Math. Sci., 3, 231 (2009) · Zbl 1172.92023 |
| [20] | H. W. Hethcote, The mathematics of infectious diseases,, SIAM Rev., 42, 599 (2000) · Zbl 0993.92033 · doi:10.1137/S0036144500371907 |
| [21] | E. Jung, Optimal control of treatments in a two-strain tuberculosis model,, Discrete Contin. Dyn. Syst. Ser. B, 2, 473 (2002) · Zbl 1005.92018 · doi:10.3934/dcdsb.2002.2.473 |
| [22] | D. Kirschner, Dynamics of co-infection with M. tuberculosis and HIV-1,, Theor. Pop. Biol., 55, 94 (1999) · Zbl 0916.92018 · doi:10.1006/tpbi.1998.1382 |
| [23] | D. Kirschner, Optimal control of the chemotherapy of HIV,, J. Mathematical Biology, 35, 775 (1996) · Zbl 0876.92016 · doi:10.1007/s002850050076 |
| [24] | C. K. Kwan, HIV and tuberculosis: A deadly human syndemic,, Clin. Microbiol. Rev., 24, 351 (2011) · doi:10.1128/CMR.00042-10 |
| [25] | V. Lakshmikantham, <em>Stability Analysis of Nonlinear Systems</em>,, Marcel Dekker (1989) · Zbl 0676.34003 |
| [26] | U. Ledzewicz, On optimal singular controls for a general SIR-model with vaccination and treatment,, Discrete Contin. Dyn. Syst., II, 981 (2011) · Zbl 1306.49056 · doi:10.3934/proc.2011.2011.981 |
| [27] | S. Lenhart, <em>Optimal Control Applied to Biological Models</em>,, Chapman & Hall/CRC (2007) · Zbl 1291.92010 |
| [28] | G. Magombedze, Modeling the TB/HIV-1 Co-Infection and the Effects of Its Treatment,, Math. Pop. Studies, 17, 12 (2010) · Zbl 1183.92048 · doi:10.1080/08898480903467241 |
| [29] | G. Magombedze, Optimal control of a sex structured HIV/AIDS model with condom use,, Mathematical Modelling and Analysis, 14, 483 (2009) · Zbl 1183.92058 · doi:10.3846/1392-6292.2009.14.483-494 |
| [30] | R. Naresh, Modelling and analysis of HIV-TB co-infection in a variable size population,, Math. Model. Anal., 10, 275 (2005) · Zbl 1082.92030 · doi:10.1080/13926292.2005.9637287 |
| [31] | L. Pontryagin, <em>The Mathematical Theory of Optimal Processes</em>,, Wiley Interscience (1962) · Zbl 0112.05502 |
| [32] | PROPT, Matlab Optimal Control Software (DAE, ODE),, <a href= |
| [33] | H. S. Rodrigues, Dynamics of dengue epidemics when using optimal control,, Math. Comput. Modelling, 52, 1667 (2010) · Zbl 1205.49051 · doi:10.1016/j.mcm.2010.06.034 |
| [34] | H. S. Rodrigues, Dengue disease, basic reproduction number and control,, Int. J. Comput. Math., 89, 334 (2012) · Zbl 1237.92042 · doi:10.1080/00207160.2011.554540 |
| [35] | P. Rodrigues, Cost-effectiveness analysis of optimal control measures for tuberculosis,, Bull. Math. Biol., 76, 2627 (2014) · Zbl 1330.92133 · doi:10.1007/s11538-014-0028-6 |
| [36] | L. W. Roeger, Modeling TB and HIV co-infections,, Math. Biosc. and Eng., 6, 815 (2009) · Zbl 1194.92054 · doi:10.3934/mbe.2009.6.815 |
| [37] | W. N. Rom, <em>Environmental and Occupational Medicine</em>,, Lippincott Williams & Wilkins (2007) |
| [38] | H. Schättler, <em>Geometric Optimal Control</em>,, Springer (2012) · Zbl 1276.49002 · doi:10.1007/978-1-4614-3834-2 |
| [39] | H. Schättler, 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, 2657 (2014) · Zbl 1304.49043 · doi:10.3934/dcdsb.2014.19.2657 |
| [40] | O. Sharomi, Mathematical analysis of the transmission dynamics of HIV/TB coinfection in the presence of treatment,, Math. Biosc. Eng., 5, 145 (2008) · Zbl 1140.92016 · doi:10.3934/mbe.2008.5.145 |
| [41] | C. J. Silva, Optimal control strategies for tuberculosis treatment: A case study in Angola,, Numer. Algebra Control Optim., 2, 601 (2012) · Zbl 1253.92035 · doi:10.3934/naco.2012.2.601 |
| [42] | C. J. Silva, Optimal control for a tuberculosis model with reinfection and post-exposure interventions,, Math. Biosci., 244, 154 (2013) · Zbl 1280.92028 · doi:10.1016/j.mbs.2013.05.005 |
| [43] | K. Styblo, State of art: Epidemiology of tuberculosis,, Bull. Int. Union Tuberc., 53, 141 (1978) |
| [44] | H. R. Thieme, Persistence under relaxed point-dissipaty (with applications to an epidemic model),, SIAM. J. Math. Anal. Appl., 24, 407 (1993) · Zbl 0774.34030 · doi:10.1137/0524026 |
| [45] | UNAIDS, <em>Global Report: UNAIDS Report on the Global AIDS Epidemic 2013</em>,, Geneva (2013) |
| [46] | P. van den Driessche, Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission,, Math. Biosc., 180, 29 (2002) · Zbl 1015.92036 · doi:10.1016/S0025-5564(02)00108-6 |
| [47] | A. Wächter, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming,, Math. Program., 106, 25 (2006) · Zbl 1134.90542 · doi:10.1007/s10107-004-0559-y |
| [48] | WHO, <em>Global Tuberculosis Report 2013</em>,, Geneva (2013) |
| [49] | WHO, Tuberculosis,, Fact sheet no. 104 (2014) |
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