Skip to main content

Advertisement

SpringerLink
Log in

Modeling and analysis of the transmission dynamics of tuberculosis without and with seasonality

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

A deterministic model of tuberculosis without and with seasonality is designed and analyzed into its transmission dynamics. We first present and analyze a tuberculosis model without seasonality, which incorporates the essential biological and epidemiological features of the disease. The model is shown to exhibit the phenomenon of backward bifurcation, where a stable disease-free equilibrium coexists with one or more stable endemic equilibria when the associated basic reproduction number is less than unity. The statistical data of tuberculosis (TB) cases show seasonal fluctuations in many countries. Then, the extension of our TB model by incorporating seasonality is developed and the basic reproduction ratio is defined. Parameter values of the model are estimated according to demographic and epidemiological data in Cameroon. The simulation results are in good accordance with the seasonal variation of the reported cases of active TB in Cameroon.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. World Health Organization: Global tuberculosis control: surveillance, planning, financing. World Health Organization, Geneva, Switzerland (2009)

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

    Article  Google Scholar 

  3. Raviglione, M.C., Dye, C., Schmizt, S., Kochi, A.: For the global surveillance and monitoring project: assessment of worldwide tuberculosis control. Lancet 350, 624–629 (1997)

    Article  Google Scholar 

  4. Raviglione, M.C.: Evolution of WHO, 1948–2001 policies for tuberculosis control. Lancet 359, 775–780 (2002)

    Article  Google Scholar 

  5. Frieden, T., Driver, R.C.: Tuberculosis control: past 10 years and future progress. Tuberculosis 83, 82–85 (2003)

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  7. Chiang, C.Y., Riley, L.W.: Exogenous reinfection in tuberculosis. Lancet Infect. Dis. 5, 629–636 (2005)

    Article  Google Scholar 

  8. Grassly, N.C., Fraser, C.: Seasonality infectious disease epidemiology. Proc. R. Soc. B 273, 2541–2550 (2006)

    Article  Google Scholar 

  9. Hethcote, H.W., Yorke, J.A.: Gonorrhea Transmission Dynamics and Control. Lecture Notes in Biomathematics, vol. 56, p. 105. Springer, Berlin (1984)

    MATH  Google Scholar 

  10. Schaaf, H.S., Nel, E.D., Beyers, N., Gie, R.P., Scott, F., Donald, P.R.: A decade of experience with Mycobacterium tuberculosis culture from children: a seasonal influence of children tuberculosis. Tuberc. Lung Dis. 77, 43–46 (1996)

    Article  Google Scholar 

  11. Douglas, A.S., Strachan, D.P., Maxwell, J.D.: Seasonality of tuberculosis: the reverse of other respiratory disease in the UK. Thorax 51, 944–946 (1996)

    Article  Google Scholar 

  12. Leung, C.C., Yew, W.W., Chan, T.Y.K., Tam, C.M., Chan, C.Y., Chan, C.K., Tang, N., Chang, K.C., Law, W.S.: Seasonal pattern of tuberculosis in Hong Kong. Int. J. Epidemiol. 34, 924–930 (2005)

    Article  Google Scholar 

  13. Rios, M., Garcia, J.M., Sanchez, J.A., Perez, D.: A statistical analysis of the seasonality in pulmonary tuberculosis. Eur. J. Epidemiol. 16, 483–488 (2000)

    Article  Google Scholar 

  14. Nagayama, N., Ohmori, M.: Seasonality in various forms of tuberculosis. Int. J. Tuberc. Lung Dis. 10, 1117–1122 (2006)

    Google Scholar 

  15. Thorpe, L.E., Frieden, T.R., Laserson, K.F., Wells, C., Khatri, G.R.: Seasonality of tuberculosis in India: is it real and what does it tell us? Lancet 364, 1613–1614 (2004)

    Article  Google Scholar 

  16. Akhtar, S., Mohammad, H.G.: Seasonality in pulmonary tuberculosis among migrant workers entering Kuwait BMC Infect. Dis. 8, 3–13 (2008). doi:10.1186/1471-2334-8-3

    Article  Google Scholar 

  17. Liu, L., Zhao, X.-Q., Zhou, Y.: A tuberculosis model with seasonality. Bull. Math. Biol. 72, 931–952 (2010). doi:10.1007/s11538-009-9477-8

    Article  MATH  MathSciNet  Google Scholar 

  18. Janmeja, A.K., Mohapatra, P.R.: Seasonality of tuberculosis. Int. J. Tuberc. Lung Dis. 9, 704–705 (2005)

    Google Scholar 

  19. Aron, J.L., Schwartz, I.B.: Seasonality and period-doubling bifurcations in an epidemic model. J. Theor. Biol. 110, 665–679 (1984)

    Article  MathSciNet  Google Scholar 

  20. Greenman, J., Kamo, M., Boots, M.: External forcing of ecological and epidemiological systems: a resonance approach. Physica D 190, 136–151 (2004)

    Article  MATH  Google Scholar 

  21. Altizer, S., Dobson, A., Hosseini, P., Hudson, P., Pascual, M., Rohani, P.: Seasonality and the dynamics of infectious diseases. Ecol. Lett. 9, 467–484 (2006)

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  23. Anderson, R.M., May, R.M.: Infectious Disease of Humans, Dynamical and Control. Oxford University Press, Oxford (1992)

    Google Scholar 

  24. Anderson, R.M., May, R.M.: Population biology of infectious diseases: Part I. Nature 280, 361–367 (1879)

    Article  Google Scholar 

  25. Capasso, V.: Mathematical Structures of Epidemic Systems. Lecture Notes in Biomathematics, vol. 97. Springer, Berlin (1993)

    Book  MATH  Google Scholar 

  26. Thieme, H.R.: Mathematics in Population Biology, Princeton. Ser. Theor. Comput. Biol. Princeton University Press, Princeton (2003)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  28. Castillo-Chavez, C., Feng, Z.: To treat or not to treat: the case of tuberculosis. J. Math. Biol. 35, 629–635 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  29. Bhunu, C.P., Garira, W., Mukandawire, Z., Zimba, M.: Tuberculosis transmission model with chemoprophylaxis and treatment. Bull. Math. Biol. 70, 1163–1191 (2009). doi:10.1007/S11538-008-9295-4

    Article  Google Scholar 

  30. Blower, S.M., Small, P.M., Hopwell, P.C.: Control strategies for tuberculosis epidemics: new models for old problems. Science 273, 497–520 (1996)

    Article  Google Scholar 

  31. Murphy, B.M., Singer, B.H., Kirschner, D.: Comparing epidemic tuberculosis in demographically distinct populations. Math. Biosci. 180, 161–185 (2002)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  33. Bacaer, N., Ouifki, R., Pretorious, C., Wood, R., William, B.: Modelling the joint epidemics of TB and HIV in a South African township. J. Math. Biol. 57, 557–593 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  34. Blower, S.M., Small, P.M., Hopewell, P.C.: Control strategies for tuberculosis epidemics: new models for old problems. Science 273, 497–500 (1996)

    Article  Google Scholar 

  35. Blower, S.M., Porco, T.C., Lietman, T.M.: Tuberculosis: the evolution of antibiotic resistance and the design of epidemic control strategies. In: Horn, M.A., Simonett, G., Webb, G.F. (eds.) Mathematical Models in Medical and Health Science. Vanderbilt University Press, Nashville (1998)

    Google Scholar 

  36. Chintu, C., Mwinga, A.: An African perspective of tuberculosis and HIV/AIDS. Lancet 353, 997–1005 (1999)

    Article  Google Scholar 

  37. Bowong, S., Tewa, J.J.: Mathematical analysis of a tuberculosis model with differential infectivity. Commun. Nonlinear Sci. Numer. Simul. 14, 4010–4021 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  38. Bowong, S.: Optimal control of the transmission dynamics of tuberculosis. Nonlinear Dyn. 61, 729–748 (2010). doi:10.1007/s11071-010-9683-9

    Article  MATH  MathSciNet  Google Scholar 

  39. National Comittee of Fight Againts Tuberculosis: Guide de personnel de la santé (2001)

  40. Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences. SIAM, Philadelphia (1994)

    Book  MATH  Google Scholar 

  41. Jacquez, J.A., Simon, C.P.: Qualitative theory of compartmental systems. SIAM Rev. 35, 43–79 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  42. Smith, H.L., Waltmann, P.: The Theory of the Chemostat. Cambridge University Press, Cambridge (1995)

    Book  MATH  Google Scholar 

  43. Smith, H.L.: Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems. Mathematical Surveys and Monographs, vol. 41. Am. Math. Soc., Providence (1995)

    MATH  Google Scholar 

  44. National Institute of Statistics: Evolution des systèmes statistiques nationaux (2007)

  45. van den Driessche, P., Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–28 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  46. Feng, Z., Chavez, C., Capurro, A.: A model for tuberculosis with exogenous reinfection. Theor. Popul. Biol. 57, 235–247 (2000)

    Article  MATH  Google Scholar 

  47. Dushoff, J., Huang, W., Castillo-Chavez, C.: Backwards bifurcations and catastrophe in simple models of fatal diseases. J. Math. Biol. 36, 227–248 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  48. Brauer, F.: Backward bifurcation in simple vaccination models. J. Math. Anal. Appl. 298, 418–431 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  49. Chiang, C.Y., Riley, L.W.: Exogenous reinfection in tuberculosis. Lancet Infect. Dis. 5, 629–636 (2005)

    Article  Google Scholar 

  50. Arino, J., McCluskey, C.C., van den Driessche, P.: Global result for an epidemic model with vaccination that exhibits backward bifurcation. SIAM J. Appl. Math. 64, 260–276 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  51. Sharomi, O., Podder, C.N., Gumel, A.B., Elbasha, E.H., Watmough, J.: Role of incidence function in vaccine-induced backward bifurcation in some HIV models. Math. Biosci. 210, 436–463 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  52. Lakshmikantham, V., Leela, S., Martynyuk, A.A.: Stability Analysis of Nonlinear Systems. Marcel Dekker, New York (1989)

    MATH  Google Scholar 

  53. LaSalle, J.P.: The Stability of Dynamical Systems. Society for Industrial and Applied Mathematics, Philadelphia (1976). With an appendix: “Limiting equations and stability of nonautonomous ordinary differential equations” by Z. Artstein, Regional Conference Series in Applied Mathematics (1976)

    Book  MATH  Google Scholar 

  54. LaSalle, J.P.: Stability theory for ordinary differential equations. J. Differ. Equ. 41, 57–65 (1968)

    Article  MathSciNet  Google Scholar 

  55. Bhatia, N.P., Szegö, G.P.: Stability Theory of Dynamical Systems. Springer, Berlin (1970)

    MATH  Google Scholar 

  56. Wang, W., Zhao, X.-Q.: Threshold dynamics for compartmental epidemic models in periodic environments. J. Dyn. Differ. Equ. 20, 699–717 (2008)

    Article  MATH  Google Scholar 

  57. Saltelli, A., Chan, K., Scott, M. (eds.): Sensitivity Analysis. Probability and Statistics Series. Wiley, New York (2000)

    MATH  Google Scholar 

  58. Korobeinikov, A., Maini, P.K.: A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence. Math. Biosci. Eng. 1, 57–60 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  59. McCluskey, C.C.: Lyapunov functions for tuberculosis models with fast and slow progression. Math. Biosci. Eng. 3, 603–614 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  60. Bame, N., Bowong, S., Mbang, J., Sallet, G., Tewa, J.J.: Global stability for SEIS models with n latent classes. Math. Biosci. Eng. 5, 20–33 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  61. Iggidr, A., Kamgang, J.C., Sallet, G., Tewa, J.J.: Global analysis of new malaria intrahost models with a competitive exclusion principle. SIAM J. Appl. Math. 1, 260–278 (2007)

    MathSciNet  Google Scholar 

  62. Adda, P., Dimi, J.L., Iggidr, A., Kamgang, J.C., Sallet, G., Tewa, J.J.: General models of host-parasite systems, Global analysis. Discrete Contin. Dyn. Syst., Ser. B 8, 1–17 (2007)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratory of Applied Mathematics, Department of Mathematics and Computer Science, Faculty of Science, University of Douala, P.O. Box 24157, Douala, Cameroon

    Samuel Bowong

  2. Postdam Institute for Climate Impact Research (PIK), Telegraphenberg A 31, 14412, Potsdam, Germany

    Samuel Bowong & Jurgen Kurths

  3. Department of Physics, Humboldt Universitat zu Berlin, 12489, Berlin, Germany

    Jurgen Kurths

Authors
  1. Samuel Bowong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jurgen Kurths
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Samuel Bowong.

Additional information

Samuel Bowong is also with UMI 209 IRD/UPMC UMMISCO, Bondy, Projet MASAIE INRIA Grand Est, France and LIRIMA, Projet GRIMCAPE, Cameroon.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bowong, S., Kurths, J. Modeling and analysis of the transmission dynamics of tuberculosis without and with seasonality. Nonlinear Dyn 67, 2027–2051 (2012). https://doi.org/10.1007/s11071-011-0127-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-011-0127-y

Keywords

Search

Navigation