# American Institute of Mathematical Sciences

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

 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.
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 and 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. [2] M. Bannon and A. Finn, BCG and tuberculosis, Archives of Disease in Childhood, 80 (1999), 80-83. [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. [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. [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. [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. [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. [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. [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. [10] W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer Verlag, New York, 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), 473-482. doi: 10.3934/dcdsb.2002.2.473. [12] J. P. LaSalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 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), 325-326. [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. [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. [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. [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. [18] C. Ted and M. Megan, Modeling epidemics of multidrug-resisitant M.tuberculosis of heterogeneous fitness, Nature Medicine, 10 (2004), 1117-1121. [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. [20] WHO, Global Tuberculosis Report 2014, World Health Organization Press, Switzerland, 2014. [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.

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. [2] M. Bannon and A. Finn, BCG and tuberculosis, Archives of Disease in Childhood, 80 (1999), 80-83. [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. [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. [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. [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. [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. [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. [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. [10] W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer Verlag, New York, 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), 473-482. doi: 10.3934/dcdsb.2002.2.473. [12] J. P. LaSalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 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), 325-326. [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. [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. [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. [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. [18] C. Ted and M. Megan, Modeling epidemics of multidrug-resisitant M.tuberculosis of heterogeneous fitness, Nature Medicine, 10 (2004), 1117-1121. [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. [20] WHO, Global Tuberculosis Report 2014, World Health Organization Press, Switzerland, 2014. [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.
 [1] Xiaomei Feng, Zhidong Teng, Kai Wang, Fengqin Zhang. Backward bifurcation and global stability in an epidemic model with treatment and vaccination. Discrete and Continuous Dynamical Systems - B, 2014, 19 (4) : 999-1025. doi: 10.3934/dcdsb.2014.19.999 [2] Hui Cao, Yicang Zhou. The basic reproduction number of discrete SIR and SEIS models with periodic parameters. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 37-56. doi: 10.3934/dcdsb.2013.18.37 [3] Xiao-Li Ding, Iván Area, Juan J. Nieto. Controlled singular evolution equations and Pontryagin type maximum principle with applications. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021059 [4] Shanjing Ren. Global stability in a tuberculosis model of imperfect treatment with age-dependent latency and relapse. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1337-1360. doi: 10.3934/mbe.2017069 [5] Nicolas Bacaër, Xamxinur Abdurahman, Jianli Ye, Pierre Auger. On the basic reproduction number $R_0$ in sexual activity models for HIV/AIDS epidemics: Example from Yunnan, China. Mathematical Biosciences & Engineering, 2007, 4 (4) : 595-607. doi: 10.3934/mbe.2007.4.595 [6] Nitu Kumari, Sumit Kumar, Sandeep Sharma, Fateh Singh, Rana Parshad. Basic reproduction number estimation and forecasting of COVID-19: A case study of India, Brazil and Peru. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2021170 [7] Tao Feng, Zhipeng Qiu. Global analysis of a stochastic TB model with vaccination and treatment. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2923-2939. doi: 10.3934/dcdsb.2018292 [8] Guy Barles, Ariela Briani, Emmanuel Trélat. Value function for regional control problems via dynamic programming and Pontryagin maximum principle. Mathematical Control and Related Fields, 2018, 8 (3&4) : 509-533. doi: 10.3934/mcrf.2018021 [9] Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations and Control Theory, 2022, 11 (2) : 347-371. doi: 10.3934/eect.2020110 [10] Torsten Lindström. Discrete models and Fisher's maximum principle in ecology. Conference Publications, 2003, 2003 (Special) : 571-579. doi: 10.3934/proc.2003.2003.571 [11] Mingshang Hu. Stochastic global maximum principle for optimization with recursive utilities. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 1-. doi: 10.1186/s41546-017-0014-7 [12] Gerardo Chowell, R. Fuentes, A. Olea, X. Aguilera, H. Nesse, J. M. Hyman. The basic reproduction number $R_0$ and effectiveness of reactive interventions during dengue epidemics: The 2002 dengue outbreak in Easter Island, Chile. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1455-1474. doi: 10.3934/mbe.2013.10.1455 [13] Huaiqiang Yu, Bin Liu. Pontryagin's principle for local solutions of optimal control governed by the 2D Navier-Stokes equations with mixed control-state constraints. Mathematical Control and Related Fields, 2012, 2 (1) : 61-80. doi: 10.3934/mcrf.2012.2.61 [14] H. O. Fattorini. The maximum principle in infinite dimension. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 557-574. doi: 10.3934/dcds.2000.6.557 [15] Tianhui Yang, Lei Zhang. Remarks on basic reproduction ratios for periodic abstract functional differential equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6771-6782. doi: 10.3934/dcdsb.2019166 [16] Hans Josef Pesch. Carathéodory's royal road of the calculus of variations: Missed exits to the maximum principle of optimal control theory. Numerical Algebra, Control and Optimization, 2013, 3 (1) : 161-173. doi: 10.3934/naco.2013.3.161 [17] Geni Gupur, Xue-Zhi Li. Global stability of an age-structured SIRS epidemic model with vaccination. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 643-652. doi: 10.3934/dcdsb.2004.4.643 [18] Carlo Orrieri. A stochastic maximum principle with dissipativity conditions. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5499-5519. doi: 10.3934/dcds.2015.35.5499 [19] Maria do Rosário de Pinho, Helmut Maurer, Hasnaa Zidani. Optimal control of normalized SIMR models with vaccination and treatment. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 79-99. doi: 10.3934/dcdsb.2018006 [20] Tom Burr, Gerardo Chowell. The reproduction number $R_t$ in structured and nonstructured populations. Mathematical Biosciences & Engineering, 2009, 6 (2) : 239-259. doi: 10.3934/mbe.2009.6.239

2020 Impact Factor: 1.327