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Global stability and optimal control for a tuberculosis model with vaccination and treatment
1. | College of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062, China |
2. | College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, 710062 |
3. | Science College, Air Force Engineering University, Xi'an 710051, China, China, China |
References:
[1] |
R. Baltussen, K. Floyd and C. Dye, Achieving the millennium development goals for health: Cost effectiveness analysis of strategies for tuberculosis control in developing countries,, British Medical Journal, 331 (2005), 1364. Google Scholar |
[2] |
M. Bannon and A. Finn, BCG and tuberculosis,, Archives of Disease in Childhood, 80 (1999), 80. Google Scholar |
[3] |
C. Bhunu and W. Garira, Tuberculosis transmission model with chemoprophylaxis and treatment,, Bulletin of Mathematical Biology, 70 (2008), 1163.
doi: 10.1007/s11538-008-9295-4. |
[4] |
S. M. Blower, A. R. McLean, T. C. Porco, P. M. Small, P. C. Hopewell, M. A. Sanchez and A. R. Moss, The intrinsic transmission dynamics of tuberculosis epidemic,, Nature Medicine, 1 (1995), 815.
doi: 10.1038/nm0895-815. |
[5] |
S. M. Blower, P. M. Small and P. C. Hopewell, Control stretagies for tuberculosis epidemic: New models for old problems,, Science, 273 (1996), 497. Google Scholar |
[6] |
C. Castillo-Chavez and Z. Feng, To treat or not to treat: The case of tuberculosis,, Journal of Mathemtical Biology, 35 (1997), 629.
doi: 10.1007/s002850050069. |
[7] |
C. Castillo-Chavez and Z. Feng, Global stability of an age-structure model for TB and its applications to optimal vaccination strategies,, Mathematical Biosciences, 151 (1998), 135.
doi: 10.1016/S0025-5564(98)10016-0. |
[8] |
C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications,, Mathematical Biosciences and Engineering, 1 (2004), 361.
doi: 10.3934/mbe.2004.1.361. |
[9] |
P. Clayden, S. Collins, C. Daniels, M. Frick, M. Harrington, T. Horn, R. Jefferys, K. Kaplan, E. Lessem, L. McKenna and T. Swan, 2014 Pipeline Report: HIV, Hepatitis C Virus (HCV) and Tuberculosis Drugs, Diagnostics, Vaccines, Preventive Technologies, Research Toward a Cure, and Immune-Based and Gene Therapies in Development,, New York, (2014). Google Scholar |
[10] |
W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control,, Springer Verlag, (1975).
|
[11] |
E. Jung, S. Lenhart and Z. Feng, Optimal control of treatments in a two-strain tuberculosis model,, Discrete and Continous Dynamicals Systems Series B, 2 (2002), 473.
doi: 10.3934/dcdsb.2002.2.473. |
[12] |
J. P. LaSalle, The Stability of Dynamical Systems,, Regional Conference Series in Applied Mathematics, (1976).
|
[13] |
K. X. Mao, C. X. Zhen, H. Y. Lu, Q. Dan, M. X. Ming, C. Y. Zhang, X. B. Hu and J. H. Dan, Protective effect of vaccination of Bacille Calmette-Gnerin on children,, Chinese Journal Of Contemporary Pediatrics, 5 (2003), 325. Google Scholar |
[14] |
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes,, Translated from the Russian by K. N. Trirogoff; edited by L. W. Neustadt Interscience Publishers John Wiley & Sons, (1962).
|
[15] |
P. Rodriguesa, M. G. M. Gomes and C. Rebelo, Drug resistance in tuberculosis-a reinfection model,, Theoretical Population Biology, 71 (2007), 196.
doi: 10.1016/j.tpb.2006.10.004. |
[16] |
M. A. Sanchez and S. M. Blower, Uncertainty and sensitivity analysis of the basic reproductive rate: Tuberculosis as an example,, American Journal of Epidemiology, 145 (1997), 1127.
doi: 10.1093/oxfordjournals.aje.a009076. |
[17] |
J. M. Tchuenche, S. A. Khamis, F. B. Agusto and S. C. Mpeshe, Optimal control and sensitivity analysis of an influenza model with treatment and vaccination,, Acta Biotheoretica, 59 (2011), 1.
doi: 10.1007/s10441-010-9095-8. |
[18] |
C. Ted and M. Megan, Modeling epidemics of multidrug-resisitant M.tuberculosis of heterogeneous fitness,, Nature Medicine, 10 (2004), 1117. Google Scholar |
[19] |
P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Mathematical Biosciences, 180 (2002), 29.
doi: 10.1016/S0025-5564(02)00108-6. |
[20] |
WHO, Global Tuberculosis Report 2014,, World Health Organization Press, (2014). Google Scholar |
[21] |
Y. L. Yang, J. Q. Li, Z. E. Ma and Y. C. Zhou, Global stability of two models with incomplete treatment for tuberculosis,, Chaos, 43 (2010), 79.
doi: 10.1016/j.chaos.2010.09.002. |
show all references
References:
[1] |
R. Baltussen, K. Floyd and C. Dye, Achieving the millennium development goals for health: Cost effectiveness analysis of strategies for tuberculosis control in developing countries,, British Medical Journal, 331 (2005), 1364. Google Scholar |
[2] |
M. Bannon and A. Finn, BCG and tuberculosis,, Archives of Disease in Childhood, 80 (1999), 80. Google Scholar |
[3] |
C. Bhunu and W. Garira, Tuberculosis transmission model with chemoprophylaxis and treatment,, Bulletin of Mathematical Biology, 70 (2008), 1163.
doi: 10.1007/s11538-008-9295-4. |
[4] |
S. M. Blower, A. R. McLean, T. C. Porco, P. M. Small, P. C. Hopewell, M. A. Sanchez and A. R. Moss, The intrinsic transmission dynamics of tuberculosis epidemic,, Nature Medicine, 1 (1995), 815.
doi: 10.1038/nm0895-815. |
[5] |
S. M. Blower, P. M. Small and P. C. Hopewell, Control stretagies for tuberculosis epidemic: New models for old problems,, Science, 273 (1996), 497. Google Scholar |
[6] |
C. Castillo-Chavez and Z. Feng, To treat or not to treat: The case of tuberculosis,, Journal of Mathemtical Biology, 35 (1997), 629.
doi: 10.1007/s002850050069. |
[7] |
C. Castillo-Chavez and Z. Feng, Global stability of an age-structure model for TB and its applications to optimal vaccination strategies,, Mathematical Biosciences, 151 (1998), 135.
doi: 10.1016/S0025-5564(98)10016-0. |
[8] |
C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications,, Mathematical Biosciences and Engineering, 1 (2004), 361.
doi: 10.3934/mbe.2004.1.361. |
[9] |
P. Clayden, S. Collins, C. Daniels, M. Frick, M. Harrington, T. Horn, R. Jefferys, K. Kaplan, E. Lessem, L. McKenna and T. Swan, 2014 Pipeline Report: HIV, Hepatitis C Virus (HCV) and Tuberculosis Drugs, Diagnostics, Vaccines, Preventive Technologies, Research Toward a Cure, and Immune-Based and Gene Therapies in Development,, New York, (2014). Google Scholar |
[10] |
W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control,, Springer Verlag, (1975).
|
[11] |
E. Jung, S. Lenhart and Z. Feng, Optimal control of treatments in a two-strain tuberculosis model,, Discrete and Continous Dynamicals Systems Series B, 2 (2002), 473.
doi: 10.3934/dcdsb.2002.2.473. |
[12] |
J. P. LaSalle, The Stability of Dynamical Systems,, Regional Conference Series in Applied Mathematics, (1976).
|
[13] |
K. X. Mao, C. X. Zhen, H. Y. Lu, Q. Dan, M. X. Ming, C. Y. Zhang, X. B. Hu and J. H. Dan, Protective effect of vaccination of Bacille Calmette-Gnerin on children,, Chinese Journal Of Contemporary Pediatrics, 5 (2003), 325. Google Scholar |
[14] |
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes,, Translated from the Russian by K. N. Trirogoff; edited by L. W. Neustadt Interscience Publishers John Wiley & Sons, (1962).
|
[15] |
P. Rodriguesa, M. G. M. Gomes and C. Rebelo, Drug resistance in tuberculosis-a reinfection model,, Theoretical Population Biology, 71 (2007), 196.
doi: 10.1016/j.tpb.2006.10.004. |
[16] |
M. A. Sanchez and S. M. Blower, Uncertainty and sensitivity analysis of the basic reproductive rate: Tuberculosis as an example,, American Journal of Epidemiology, 145 (1997), 1127.
doi: 10.1093/oxfordjournals.aje.a009076. |
[17] |
J. M. Tchuenche, S. A. Khamis, F. B. Agusto and S. C. Mpeshe, Optimal control and sensitivity analysis of an influenza model with treatment and vaccination,, Acta Biotheoretica, 59 (2011), 1.
doi: 10.1007/s10441-010-9095-8. |
[18] |
C. Ted and M. Megan, Modeling epidemics of multidrug-resisitant M.tuberculosis of heterogeneous fitness,, Nature Medicine, 10 (2004), 1117. Google Scholar |
[19] |
P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Mathematical Biosciences, 180 (2002), 29.
doi: 10.1016/S0025-5564(02)00108-6. |
[20] |
WHO, Global Tuberculosis Report 2014,, World Health Organization Press, (2014). Google Scholar |
[21] |
Y. L. Yang, J. Q. Li, Z. E. Ma and Y. C. Zhou, Global stability of two models with incomplete treatment for tuberculosis,, Chaos, 43 (2010), 79.
doi: 10.1016/j.chaos.2010.09.002. |
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