Jump to ContentJump to Main Navigation
Show Summary Details
More options …

International Journal of Nonlinear Sciences and Numerical Simulation

Editor-in-Chief: Birnir, Björn

Editorial Board: Angheluta-Bauer, Luiza / Chen, Xi / Chou, Tom / Grauer, Rainer / Marzocchella, Antonio / Rangarajan, Govindan / Trivisa, Konstantina / Weikard, Rudi / Yang, Xu

IMPACT FACTOR 2018: 1.033
5-year IMPACT FACTOR: 1.106

CiteScore 2018: 1.11

SCImago Journal Rank (SJR) 2018: 0.288
Source Normalized Impact per Paper (SNIP) 2018: 0.510

Mathematical Citation Quotient (MCQ) 2017: 0.12

See all formats and pricing
More options …
Volume 20, Issue 3-4


Modeling the Effects of Health Education and Early Therapy on Tuberculosis Transmission Dynamics

Hong Xiang
  • Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu, 730050, People’s Republic of China
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Ming-Xuan Zou
  • Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu, 730050, People’s Republic of China
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Hai-Feng Huo
  • Corresponding author
  • Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu, 730050, People’s Republic of China
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2019-03-13 | DOI: https://doi.org/10.1515/ijnsns-2016-0084


A new tuberculosis model with health education and early therapy is introduced. The early therapy is available for both detected latent and infective individuals. The basic reproduction number R0 is derived by the next generation matrix. Mathematical analyses show that the disease free equilibrium is globally asymptotically stable if R0<1, and the endemic equilibrium is globally asymptotically stable if R0>1. Numerical simulations are also carried out to illustrate our analytical results. Our results show that both health education and early therapy have the positive impact in reducing burden of tuberculosis.

Keywords: tuberculosis; early therapy; health education; stability

JEL Classification: 92D30


  • [1]

    C. Dye, S. Scheele, P. Dolin, V. Pathania and M. Raviglione, For the WHO global surveillance and monitoring project: global burden of tuberculosis estimated incidence, prevalence and mortality by country, JAMA, 282 (1999), 677–686.Google Scholar

  • [2]

    World Health Organization: Global tuberculosis report 2014, ISBN 978-92-4-156480-9,2 014, 91–105, 2008.Google Scholar

  • [3]

    J. M. Cramm, H. J. M. Finkenflügel, V. Møller and A. P. Nieboer, Tuberculosis treatment initiation and adherence in a South African community influenced more by perceptions than by knowledge of tuberculosis, BMC Public Health 10 (2010), 72.Web of ScienceGoogle Scholar

  • [4]

    World Health Organization: Global tuberculosis control, 2009.

  • [5]

    H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev. 42 (2000), 599–653.CrossrefGoogle Scholar

  • [6]

    J. P. Aparicio, A. F. Capurro and C. Castillo-Chavez, Markers of disease evolution: the case of tuberculosis, J. Theo. Biol. 215(2) (2002), 227–237.CrossrefGoogle Scholar

  • [7]

    C. Castillo-Chavez and Z. Feng, To treat or not to treat: the case of tuberculosis, J. Math. Biol. 35(6) (1997), 629–656.CrossrefGoogle Scholar

  • [8]

    Z. Feng, C. Castillo-Chavez and A. F. Capurro, A model for tuberculosis with exogenous reinfection, Theor. Pop. Biol. 57 (2000), 235–247.CrossrefGoogle Scholar

  • [9]

    Z. Feng, W. Huang and C. Castillo-Chavez, On the role of variable latent periods in mathematical models for tuberculosis, J. Dynam. Differ. Equ. 13(2) (2001), 425–452.CrossrefGoogle Scholar

  • [10]

    Z. Feng, M. Iannelli, and F. Milner. A two-strain TB model with age-structure, SIAM J. Appl. Math. 62(5) (2002), 1634–1656.Crossref

  • [11]

    E. Jung, S. Lenhart and Z. Feng, Optimal control of treatments in a two-strain tuberculosis model, Discrete Contin. Dyn. Syst. Ser. B 2(4) (2002), 473–482.CrossrefGoogle Scholar

  • [12]

    M. Martcheva and H. R. Thieme, Progression age enhanced back-ward bifurcation in an epidemic model with super-infection, J. Math. Biol. 46(5) (2003), 385–424.CrossrefGoogle Scholar

  • [13]

    Y. Zhou, K. Khan, Z. Feng and J. Wu, Projection of tuberculosis incidence with increasing immigration trends, J. Theoret. Biol. 254(2) (2008), 215–228.CrossrefGoogle Scholar

  • [14]

    H. Waaler, A. Geser and S. Andersen, The use of mathematical models in the study of the epidemiology of tuberculosis, Am. J. Publ. Health. 52 (1962), 1002–1013.CrossrefGoogle Scholar

  • [15]

    C. P. Bhunu, W. Garira, Z. Mukandavire and M. Zimba, Tuberculosis transmission model with chemoprophylaxis and treatment, Bull. Math. Biol. 70 (2008), 1163–1191.Web of ScienceCrossrefGoogle Scholar

  • [16]

    C. Castillo-Chavez, B. Song, Dynamical models of tuberculosis and their applications, Math. Biosci. Eng. 1 (2004), 361–404.CrossrefGoogle Scholar

  • [17]

    E. Ziv, C. L. Daley and S. Blower, Potential public health impact of new tuberculosis vaccines, Emerg. Infect. Dis. 10(9) (2004), 1529–1535.CrossrefGoogle Scholar

  • [18]

    S. Mushayabasa and C. P. Bhunu, Modeling the impact of early therapy for latent Tuberculosis patients and its optimal control analysis, J. Biol. Phys. 39 (2013), 723–747.Web of ScienceCrossrefGoogle Scholar

  • [19]

    X. Y. Zhou, X. Y. Shi and H. D. Cheng, Modelling and stability analysis for a tuberculosis model with healthy education and treatment, Comput. Appl. Math. 32 (2013), 245–260.CrossrefWeb of ScienceGoogle Scholar

  • [20]

    S. Del Valle, H. Hethcote, J. M. Hyman and C. Castillo-Chavez, Effects of behavioral changes in a smallpox attack model, Math. Biosci. 195 (2005), 228–251.CrossrefGoogle Scholar

  • [21]

    S. Del Valle, A. M. Evangelista, M. C. Velasco, C. M. Kribs-Zaleta and S. F. Hsu Schmitz, Effects of education, vaccination and treatment on HIV transmission in homosexuals with genetic heterogeneity, Math. Biosci. 187 (2004), 111–133.CrossrefGoogle Scholar

  • [22]

    Z. Mukandavire, W. Garira and J. M. Tchuenche, Modelling effects of public health educational campaigns on HIV/AIDS transmission dynamics, Appl. Math. Model. 33 (2009), 2084–2095.Web of ScienceCrossrefGoogle Scholar

  • [23]

    Y. Cai, J. Jiao, Z. Gui, Y. Liu and W. Wang, Environmental variability in a stochastic epidemic model, Appl. Math. Comput. 329 (2018), 210–226.Google Scholar

  • [24]

    H. Xiang, Y. Y. Wang and H. F. Huo, Analysis of the binge drinking models with demographics and nonlinear infectivity on networks, J. Appl. Anal. Comput. 8(5) (2018), 1535–1554.Web of ScienceGoogle Scholar

  • [25]

    B. M. Murphy, B. H. Singer and D. Kirschner. On treatment of tuberculosis in heterogeneous populations, J. Theor. Biol. 223 (2003), 391–404.CrossrefGoogle Scholar

  • [26]

    C. Castillo-Chavez, Z. Feng. Global stability of an age-structure model for TB and its applications to optimal vaccination strategies, Math. Biosci. 151 (1998), 135–154.CrossrefGoogle Scholar

  • [27]

    P. Rodrigues, M. G. Gomes and C. Rebelo. Drug resistance in tuberculosis–a reinfection model, Theor. Popul. Biol. 71 (2007), 196–212.Google Scholar

  • [28]

    C. Connell Mccluskey. Lyapunov functions for tuberculosis models with fast and slow progression, Math. Biosci. Eng. 3 (2006), 603–614.CrossrefGoogle Scholar

  • [29]

    J. Li, Y. Yang, and Y. Zhou, Global stability of an epidemic model with latent stage and vaccination, Nonlinear Anal.: Real World Appl. 12 (2011), 2163–2173.CrossrefGoogle Scholar

  • [30]

    H. F. Huo and L. X. Feng. Global stability for an HIV/AIDS epidemic model with different latent stages and treatment, Appl. Math. Model. 37 (2013), 1480–1489.CrossrefWeb of ScienceGoogle Scholar

  • [31]

    J. M. Cushing and O. Diekmann. The many guises of R0. J. Theor. Biol. 404 (2016), 295–302.Web of ScienceGoogle Scholar

  • [32]

    H. Xiang, Y. L. Tang and H. F. Huo, A viral model with intracellular delay and humoral immunity, Bull. Malaysian Math. Sci. Soc. 40 (2017), 1011–1023.CrossrefGoogle Scholar

  • [33]

    C. Dye and B.G. William. Criteria for the control of drug-resistant tuberculosis, Proc. Natl. Acad. Sci. 97 (2000), 8180–8185.CrossrefGoogle Scholar

  • [34]

    M. W. Borgdorff. New measurable indicator for: tuberculosis case detection, Emerg. Infect Dis. 10(9) (2004), 1523–1528.CrossrefGoogle Scholar

  • [35]

    Z. L. Feng, C. Castillo-Chavez and A. F. Capurro, A model for tuberculosis with exogenous reinfection, Theor. Popul. Biol. 57 (2000), 235–247.CrossrefGoogle Scholar

  • [36]

    WHO (2001). Global Tuberculosis Control, WHO report 2001, Geneva.Google Scholar

  • [37]

    World Health Organization. Seventh Report. Expert Committee on Leprosy WHO Technical Report Series no. 874, World Health Organization Geneva, 1998.Google Scholar

  • [38]

    D. F. Wres, S. Singh, A. K. Acharya and R. Dangi. Non-adherence to tuberculosis treatment in the eastern Tarai of Nepal. Int. J. Tuberc. Lung. Dis. 7 (2003), 327–335.Google Scholar

  • [39]

    W. Fox. Problem of self-administration of drugs, with particular reference to pulmonary tuberculosis. Tubercle 39 (1958), 269-274.CrossrefGoogle Scholar

  • [40]

    T. L. Cheng, M. C. Ottolini and K. Baumhaft, et al, Strategies to increase adherence with tuberculosis test reading in a high-risk population. Pediatrics 100 (1997), 210–213.Google Scholar

  • [41]

    J. M. Cramm and H. J. M. Finkenflugel, et al, Tuberculosis treatment initiation and adherence in a South African community influenced more by perceptions than by knowledge of tuberculosis. BMC Public Health 10 (2010), 72.Google Scholar

  • [42]

    S. Munro, S. Lewin, H. Smith and M. Engel, et.al. Patient adherence to tuberculosis treatment: a systematic review of qualitative research. PLoS Med. 4(Suppl 7) (2007), e238.Google Scholar

  • [43]

    R. S. Wallis, Mathematical models of tuberculosis reactivation and relapse. Front Microbiol. 7 (2016), 669. doi: .CrossrefWeb of ScienceGoogle Scholar

  • [44]

    National TB control program Pakistan (NTP) http://www.ntp.gov.pk/webdatabase.php.

  • [45]

    P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180 (2002), 29–48.CrossrefGoogle Scholar

  • [46]

    H. L. Smith and H. R. Thieme. Dynamical systems and population persistence, AMS, Providence, 2011.Google Scholar

  • [47]

    H. R. Thieme, Global stability of the endemic equilibrium ininfinite dimension: Lyapunov functions and positive operators. J. Differ. Equ. 250(9) (2011), 3772–3801.CrossrefGoogle Scholar

  • [48]

    H. Guo, M. Li and Z. Shuai. Global dynamics of a general class of multistage models for infectious diseases. SIAM J. Appl. Math. 72(1) (2012), 261–279.Web of ScienceCrossrefGoogle Scholar

  • [49]

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

About the article

Received: 2016-06-07

Accepted: 2019-02-20

Published Online: 2019-03-13

Published in Print: 2019-05-26

Citation Information: International Journal of Nonlinear Sciences and Numerical Simulation, Volume 20, Issue 3-4, Pages 243–255, ISSN (Online) 2191-0294, ISSN (Print) 1565-1339, DOI: https://doi.org/10.1515/ijnsns-2016-0084.

Export Citation

© 2019 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Zhan‐Ping Ma, Hai‐Feng Huo, and Hong Xiang
Mathematical Methods in the Applied Sciences, 2020
Hong Xiang, Fang-Fang Cui, and Hai-Feng Huo
Journal of Biological Dynamics, 2019, Volume 13, Number 1, Page 621

Comments (0)

Please log in or register to comment.
Log in