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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 |
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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