American Institute of Mathematical Sciences

Advanced
May  2016, 21(3): 1009-1022. doi: 10.3934/dcdsb.2016.21.1009

Global stability and optimal control for a tuberculosis model with vaccination and treatment

Yali Yang 1, , Sanyi Tang 2, , Xiaohong Ren 3, , Huiwen Zhao 3, and  Chenping Guo 3,

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

Received  February 2015 Revised  July 2015 Published  January 2016

We formulate a mathematical model to explore the impact of vaccination and treatment on the transmission dynamics of tuberculosis (TB). We develop a technique to prove that the basic reproduction number is the threshold of global stability of the disease-free and endemic equilibria. We then incorporate a control term and evaluate the cost of control strategies, and then perform an optimal control analysis by Pontryagin's maximum principle. Our numerical simulations suggest that the maximum vaccination strategy should be enforced regardless of its efficacy.
Keywords: global stability, treatment, basic reproduction number, Pontryagin's maximum principle., vaccination, Tuberculosis.
Mathematics Subject Classification: Primary: 34D20, 34H05; Secondary: 49J1.
Citation: Yali Yang, Sanyi Tang, Xiaohong Ren, Huiwen Zhao, Chenping Guo. Global stability and optimal control for a tuberculosis model with vaccination and treatment. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 1009-1022. doi: 10.3934/dcdsb.2016.21.1009
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), 1364-1368. Google Scholar

[2]

M. Bannon and A. Finn, BCG and tuberculosis, Archives of Disease in Childhood, 80 (1999), 80-83. Google Scholar

[3]

C. Bhunu and W. Garira, Tuberculosis transmission model with chemoprophylaxis and treatment, Bulletin of Mathematical Biology, 70 (2008), 1163-1191. doi: 10.1007/s11538-008-9295-4.  Google Scholar

[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), 815-821. doi: 10.1038/nm0895-815.  Google Scholar

[5]

S. M. Blower, P. M. Small and P. C. Hopewell, Control stretagies for tuberculosis epidemic: New models for old problems, Science, 273 (1996), 497-500. 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), 629-656. doi: 10.1007/s002850050069.  Google Scholar

[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), 135-154. doi: 10.1016/S0025-5564(98)10016-0.  Google Scholar

[8]

C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications, Mathematical Biosciences and Engineering, 1 (2004), 361-404. doi: 10.3934/mbe.2004.1.361.  Google Scholar

[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, New York, 1975.  Google Scholar

[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), 473-482. doi: 10.3934/dcdsb.2002.2.473.  Google Scholar

[12]

J. P. LaSalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1976.  Google Scholar

[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), 325-326. 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, Inc. New York-London 1962.  Google Scholar

[15]

P. Rodriguesa, M. G. M. Gomes and C. Rebelo, Drug resistance in tuberculosis-a reinfection model, Theoretical Population Biology, 71 (2007), 196-212. doi: 10.1016/j.tpb.2006.10.004.  Google Scholar

[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), 1127-1137. doi: 10.1093/oxfordjournals.aje.a009076.  Google Scholar

[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), 1-28. doi: 10.1007/s10441-010-9095-8.  Google Scholar

[18]

C. Ted and M. Megan, Modeling epidemics of multidrug-resisitant M.tuberculosis of heterogeneous fitness, Nature Medicine, 10 (2004), 1117-1121. 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), 29-48. doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[20]

WHO, Global Tuberculosis Report 2014, World Health Organization Press, Switzerland, 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, Solitions and Fractals, 43 (2010), 79-85. doi: 10.1016/j.chaos.2010.09.002.  Google Scholar

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), 1364-1368. Google Scholar

[2]

M. Bannon and A. Finn, BCG and tuberculosis, Archives of Disease in Childhood, 80 (1999), 80-83. Google Scholar

[3]

C. Bhunu and W. Garira, Tuberculosis transmission model with chemoprophylaxis and treatment, Bulletin of Mathematical Biology, 70 (2008), 1163-1191. doi: 10.1007/s11538-008-9295-4.  Google Scholar

[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), 815-821. doi: 10.1038/nm0895-815.  Google Scholar

[5]

S. M. Blower, P. M. Small and P. C. Hopewell, Control stretagies for tuberculosis epidemic: New models for old problems, Science, 273 (1996), 497-500. 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), 629-656. doi: 10.1007/s002850050069.  Google Scholar

[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), 135-154. doi: 10.1016/S0025-5564(98)10016-0.  Google Scholar

[8]

C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications, Mathematical Biosciences and Engineering, 1 (2004), 361-404. doi: 10.3934/mbe.2004.1.361.  Google Scholar

[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, New York, 1975.  Google Scholar

[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), 473-482. doi: 10.3934/dcdsb.2002.2.473.  Google Scholar

[12]

J. P. LaSalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1976.  Google Scholar

[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), 325-326. 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, Inc. New York-London 1962.  Google Scholar

[15]

P. Rodriguesa, M. G. M. Gomes and C. Rebelo, Drug resistance in tuberculosis-a reinfection model, Theoretical Population Biology, 71 (2007), 196-212. doi: 10.1016/j.tpb.2006.10.004.  Google Scholar

[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), 1127-1137. doi: 10.1093/oxfordjournals.aje.a009076.  Google Scholar

[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), 1-28. doi: 10.1007/s10441-010-9095-8.  Google Scholar

[18]

C. Ted and M. Megan, Modeling epidemics of multidrug-resisitant M.tuberculosis of heterogeneous fitness, Nature Medicine, 10 (2004), 1117-1121. 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), 29-48. doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[20]

WHO, Global Tuberculosis Report 2014, World Health Organization Press, Switzerland, 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, Solitions and Fractals, 43 (2010), 79-85. doi: 10.1016/j.chaos.2010.09.002.  Google Scholar

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