Modelling Tuberculosis and Hepatitis B Co-infections

S. Bowong; J. Kurths

Mathematical Modelling of Natural Phenomena (2010)

  • Volume: 5, Issue: 6, page 196-242
  • ISSN: 0973-5348

Abstract

top
Tuberculosis (TB) is the leading cause of death among individuals infected with the hepatitis B virus (HBV). The study of the joint dynamics of HBV and TB present formidable mathematical challenges due to the fact that the models of transmission are quite distinct. We formulate and analyze a deterministic mathematical model which incorporates of the co-dynamics of hepatitis B and tuberculosis. Two sub-models, namely: HBV-only and TB-only sub-models are considered first of all. Unlike the HBV-only sub-model, which has a globally-asymptotically stable disease-free equilibrium whenever the associated reproduction number is less than unity, the TB-only sub-model undergoes the phenomenon of backward bifurcation, where a stable disease-free equilibrium co-exists with a stable endemic equilibrium, for a certain range of the associated reproduction number less than unity. Thus, for TB, the classical requirement of having the associated reproduction number to be less than unity, although necessary, is not sufficient for its elimination. It is also shown, that the full HBV-TB co-infection model undergoes a backward bifurcation phenomenon. Through simulations, we mainly find that i) the two diseases will co-exist whenever their partial reproductive numbers exceed unity; (ii) the increased progression rate due to exogenous reinfection from latent to active TB in co-infected individuals may play a significant role in the rising prevalence of TB; and (iii) the increased progression rates from acute stage to chronic stage of HBV infection have increased the prevalence levels of HBV and TB prevalences.

How to cite

top

Bowong, S., and Kurths, J.. "Modelling Tuberculosis and Hepatitis B Co-infections." Mathematical Modelling of Natural Phenomena 5.6 (2010): 196-242. <http://eudml.org/doc/197615>.

@article{Bowong2010,
abstract = {Tuberculosis (TB) is the leading cause of death among individuals infected with the hepatitis B virus (HBV). The study of the joint dynamics of HBV and TB present formidable mathematical challenges due to the fact that the models of transmission are quite distinct. We formulate and analyze a deterministic mathematical model which incorporates of the co-dynamics of hepatitis B and tuberculosis. Two sub-models, namely: HBV-only and TB-only sub-models are considered first of all. Unlike the HBV-only sub-model, which has a globally-asymptotically stable disease-free equilibrium whenever the associated reproduction number is less than unity, the TB-only sub-model undergoes the phenomenon of backward bifurcation, where a stable disease-free equilibrium co-exists with a stable endemic equilibrium, for a certain range of the associated reproduction number less than unity. Thus, for TB, the classical requirement of having the associated reproduction number to be less than unity, although necessary, is not sufficient for its elimination. It is also shown, that the full HBV-TB co-infection model undergoes a backward bifurcation phenomenon. Through simulations, we mainly find that i) the two diseases will co-exist whenever their partial reproductive numbers exceed unity; (ii) the increased progression rate due to exogenous reinfection from latent to active TB in co-infected individuals may play a significant role in the rising prevalence of TB; and (iii) the increased progression rates from acute stage to chronic stage of HBV infection have increased the prevalence levels of HBV and TB prevalences.},
author = {Bowong, S., Kurths, J.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {nonlinear dynamical systems; epidemiological models; tuberculosis; hepatitis B; stability},
language = {eng},
month = {9},
number = {6},
pages = {196-242},
publisher = {EDP Sciences},
title = {Modelling Tuberculosis and Hepatitis B Co-infections},
url = {http://eudml.org/doc/197615},
volume = {5},
year = {2010},
}

TY - JOUR
AU - Bowong, S.
AU - Kurths, J.
TI - Modelling Tuberculosis and Hepatitis B Co-infections
JO - Mathematical Modelling of Natural Phenomena
DA - 2010/9//
PB - EDP Sciences
VL - 5
IS - 6
SP - 196
EP - 242
AB - Tuberculosis (TB) is the leading cause of death among individuals infected with the hepatitis B virus (HBV). The study of the joint dynamics of HBV and TB present formidable mathematical challenges due to the fact that the models of transmission are quite distinct. We formulate and analyze a deterministic mathematical model which incorporates of the co-dynamics of hepatitis B and tuberculosis. Two sub-models, namely: HBV-only and TB-only sub-models are considered first of all. Unlike the HBV-only sub-model, which has a globally-asymptotically stable disease-free equilibrium whenever the associated reproduction number is less than unity, the TB-only sub-model undergoes the phenomenon of backward bifurcation, where a stable disease-free equilibrium co-exists with a stable endemic equilibrium, for a certain range of the associated reproduction number less than unity. Thus, for TB, the classical requirement of having the associated reproduction number to be less than unity, although necessary, is not sufficient for its elimination. It is also shown, that the full HBV-TB co-infection model undergoes a backward bifurcation phenomenon. Through simulations, we mainly find that i) the two diseases will co-exist whenever their partial reproductive numbers exceed unity; (ii) the increased progression rate due to exogenous reinfection from latent to active TB in co-infected individuals may play a significant role in the rising prevalence of TB; and (iii) the increased progression rates from acute stage to chronic stage of HBV infection have increased the prevalence levels of HBV and TB prevalences.
LA - eng
KW - nonlinear dynamical systems; epidemiological models; tuberculosis; hepatitis B; stability
UR - http://eudml.org/doc/197615
ER -

References

top
  1. Global Fund to Fight AIDS, Tuberculosis, and Malaria. Fighting Tuberculosis. Geneva, Switzerland: (2006). Retrieved September 9, 2006, , 2006.  URIhttp://www.theglobalfund.org/en/about/tuberculosis/default.asp
  2. World Health Organization. Global tuberculosis control: surveillance, planning, financing. Geneva, Switzerland: World Health Organization, 2009.  
  3. WHO. Hepatitis B. /S, revised August 2008, 2008.  URIhttp://www.who.int/mediacentre/factsheets/fs204/en/ index.html
  4. C. Dye B.G. Williams. Eliminating human tuberculosis in the twenty-first century. J. R. Soc. Interface, 5 (2008), 653-662. 
  5. C. Chintu A. Mwinga. An African perspective of tuberculosis and HIV/AIDS. Lancet, 353 (1999), 997-1005. 
  6. R. Williams. Global challenges in liver disease. Hepatol., 44 (2006), No. 3, 521-526. 
  7. T. Frieden R.C. Driver. Tuberculosis control: past 10 years and future progress. Tuberculosis, 83 (2003), 82-85. 
  8. K.M. De Cock R.E. Chaisson. Will DOTS do it? A reappraisal of tuberculosis control in countries with high rates of HIV infection. Int. J. Tuberc. Lung Dis., 3 (1999), 457-465. 
  9. Global Fund Against AIDS, TB and Malaria. The Global Tuberculosis Epidemic, Geneva, Switzerland, 2004.  
  10. D. Lavanchy. Hepatitis B virus epidemiology, disease burden, treatment and current and emerging prevention and control measures. J. Viral. Hepat., 11 (2004), 97-107. 
  11. W.J. Edmunds, G.F. Medley D.J. Nokes. The transmission dynamics and control of hepatitis B virus in the Gambia. Stat. Med., 15 (1996), 2215-2233. 
  12. W.J. Edmunds, G.F. Medley D.J. Nokes. Vaccination against hepatitis B virus in highly endemic area: waning vaccine-induced immunity and the need for booster doses. Trans. R. Soc. Trop. Med. Hyg., 90 (1996), 436-440. 
  13. W.J. Edmunds, G.F. Medley, D.J. Nokes, A.J. Hall H.C. Whittle. The influence of age on the development of the hepatitis B carrier state. Proc. R. Soc. Lond. B, 253 (1993), 197-201. 
  14. W.J. Edmunds, G.F. Medley, D.J. Nokes, A.J. Hall H.C. Whittle. Epidemiological patterns of hepatitis B virus (HBV) in highly endemic areas. Epidemiol. Infect., 117 (1996), 313-325. 
  15. S.T. Goldstein, F.J. Zhou, S.C. Hadler, B.P. Bell, E.E. Mast H.S. Margolis. A mathematical model to estimate global hepatitis B disease burden and vaccination impact. Int. J. Epidemiol., 34 (2005), 1329-1339. 
  16. S. Hahnea, M. Ramsaya, K. Balogun, W.J. Edmund P. Mortimer. Incidence and routes of transmission of hepatitis B virus in England and Wales, 1995-2000:implications for immunisation policy. J. Clin. Virol., 29 (2004), 211-220. 
  17. J. Hou, Z. Liu F. Gu. Epidemiology and prevention of hepatitis B virus infection. Int. J. Med. Sci., 2 (2005), No. 1, 50-57. 
  18. K.C. Hyams. Risks of chronicity following acute hepatitis B virus infection: a review. Clin. Infect. Dis., 20 (1995), 992-1000. 
  19. J.D. Jia H. Zhuang. The overview of the seminar on chronic hepatitis B. Chin. J. Hepatol., 12 (2004), 698-699. 
  20. D. Lavanchy. Hepatitis B virus epidemiology, disease burden, treatment and current and emerging prevention and control measures. J. Viral. Hepat., 11 (2004) 97-107.  
  21. C.A. Blal, S.R.L. Passos, C. Horn, I. Georg, M.G. Bonecini, V.C. Rolla L. D. Castro. High prevalence of hepatitis B virus among tuberculosis patients with and without HIV in Rio de Janeiro, Brazil. Eur. Soc. Clin. Micro., 24 (2005), 41-43. 
  22. M.H. Kuniholm, J. Mark, M. Aladashvili, N. Shubladze, G. Khechinashvili, T. Tsertsvadze, C. del Rio, K.E. Nelson. Risk factors and algorithms to identify hepatitis C, hepatitis B, and HIV among Georgian tuberculosis patients. Int. Soc. Inf. Dis., (2007) doi:.  URI10.1016/j.ijid.2007.04.015
  23. R. Bellamy, C. Ruwende, T. Corrah, K.P.W.J. McAdam, M. Thursz, H.C. Whittle A.V.S. Hill. Tuberculosis and Chronic Hepatitis B Virus Infection in Africans and Variation in the Vitamin D Receptor Gene. J. Inf. Dis., 179 (1999), 721-724. 
  24. A.R. Lifson, D. Thai, A. O’Fallon, W.A. Mills K. Hang. Prevalence of tuberculosis, hepatitis B virus, and intestinal parasitic infections among refugees to Minnesota. Public Health Rep., 117 (2002), 69-77. 
  25. K.A. McGlynn, E.D. Lustbader W.T. London. Immune responses to hepatitis B virus and tuberculosis infections in Southeast Asian refugees. Amer. J. Epide., 122 (1985), 1032-1036. 
  26. P.A. Patel M.D. Voigt. Prevalence and interaction of hepatitis B and latent tuberculosis in Vietnamese immigrants to the United States. Amer. J. Gastr., 97 (2002), 1198-1203. 
  27. N.W.Y. Leung. Treatment Of Tuberculosis In Patients With Hepatitis. Hong Kong Practitioner, 19 (1997), 6-13. 
  28. W.O. Kermack A.G. McKendrick. A contribution to the mathematical theory of epidemics. Proc. Roy. Soc., A115 (1927), 700-721. 
  29. R.M. Anderson, R.M. May. Infectious Disease of Humans: Dynamics and Control. Oxford University Press, London/New York, 1992.  
  30. K.B. Blyuss Y.N. Kyrychko. On a basic model of a two-disease epidemic. Appl. Math. Comput., 160 (2005), 177-187. 
  31. R. Naresh, A. Tripathi. Modelling and analysis of HIV-TB co-infection in a variable size population. Math. Model. Anal., 10(3) (2005), 275-286.  
  32. E.F. Long, N.K. Vaidya, M.L. Brandeau. Controlling Co-epidemic: Analysis of HIV and tuberculosis infection analysis. Oper. Res., 56 (2008), No. 6, 1366-1381. doi:10.1287/opre.1080.0571.  
  33. N. Bacaer, R. Ouifki, C. Pretorious, R. Wood B. William. Modelling the joint epidemics of TB and HIV in a South African township. J. Math. Biol., 57 (2008), 557-593. 
  34. O. Sharomi, C.N. Podder, A.B. Gumel B. Song. Mathematical analysis of the transmission dynamics of HIV/TB co-infection in the presence of treatment. Math. Biosci. Eng., 5 (2008), 145-174. 
  35. Z. Mukandavire, A.B. Gumel, W. Garira J.M. Tchuenche. Mathematical analysis of a model for HIV-malaria co-infection. Math. Biosci. Engr., 6 (2009), 333-362. 
  36. E. Mtisi, H. Rwezaura, J.M. Tchuenche. A mathematical analysis of malaria and tuberculosis co-dynamics. Dis. Cont. Dyn. Syst. Series B, 12 (2009) 827-8642009 doi:10.3934/dcdsb.2009.12.827 
  37. L-I.W. Roeger, Z. Feng C.C. Chavez. Modelling TB and HIV co-infections. Math. Bios. Eng., 6 (2009), 815-837. 
  38. S. Bowong J.J. Tewa. Mathematical analysis of a tuberculosis model with differential infectivity. Com. Non. Sci. Num. Sim., 14 (2009), 4010-4021. 
  39. S. Hahnea, M. Ramsaya, K. Balogun, W.J. Edmund P. Mortimer. Incidence and routes of transmission of hepatitis B virus in England and Wales, 1995-2000: implications for immunization policy. J. Clin. Virol., 29 (2004), 211-220. 
  40. C. Dye, S. Schele. For the WHO global surveillance and monitoring project. Global burden of tuberculosis estimated incidence, prevalence and mortality by country. 282 (1999), 677-686.  
  41. National Committee of Fight Against Tuberculosis. Guide de personnel de la santé, Cameroon, 2008.  
  42. National Institute of Statistics. Evolution des systèmes statistiques nationaux, Cameroon, 2007.  
  43. G. Birkhoff, G. C. Rota. Ordinary Differential Equations. 4th edition, John Wiley & Sons, Inc., New York, 1989.  
  44. V. Hutson K. Schmitt. Permanence and the dynamics of biological systems. Math. Biosci., 111 (1992), 1-71. 
  45. H.W. Hethcothe. The mathematics of infectious disease. SIAM Review, 42 (2000), 599-653. 
  46. C.W. Shepard, E.P. Simard, L. Finelli, A.E. Fiore B.P. Bell. Hepatitis B virus infection: epidemiology and vaccination. Epidemiol. Rev., 28 (2006), 112-125. 
  47. V. Lakshmikantham, S. Leela, A. Martynyuk. Stability Analysis of Nonlinear Systems. Marcel Dekker Inc., New York and Basel, pp. 31, 1989.  
  48. H.L. Smith, P. Waltman. The Theory of the Chemostat. Cambridge University Press, 1995.  
  49. S.N. Zhang. Comparison theorems on boundedness. Funkcial. Ekvac., 31 (1988), 179-196. 
  50. S.M. Moghadas. Modelling the effect of imperfect vaccines on disease epidemiology. Dis. Cont. Dynam. Syst. Series B, 4 (2004), 999-1012. 
  51. O. Diekmann, J.A.P. Heesterbeek J.A.P. Metz. On the definition and computation of the basic reproduction ratio R0 in the model of infectious disease in heterogeneous populations. J. Math. Biol., 2 (1990), 265-382. 
  52. P. van den Driessche J. Watmough. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Bios., 180 (2002), 29-28. 
  53. J.P. LaSalle. The stability of dynamical systems. Society for Industrial and Applied Mathematics, Philadelphia, Pa, 1976.  
  54. J.P. LaSalle. Stability theory for ordinary differential equations. J. Differ. Equ., 41 (1968), 57-65. 
  55. N.P. Bhatia, G.P. Szegö. Stability Theory of Dynamical Systems. Springer-Verlag, 1970.  
  56. J. Carr. Applications Centre Manifold Theory. Springer-Verlag, New York, 1981.  
  57. C. Castillo-Chavez B. Song. Dynamical models of tuberculosis and their applications. Math. Bios. Eng., 1 (2004), 361-404. 
  58. J. Dushoff, W. Huang C. Castillo-Chavez. Backwards bifurcations and catastrophe in simple models of fatal diseases. J. Math. Biol., 36 (1998), 227-248. 
  59. J. Arino, C.C. McCluskey P. van den DriesscheGlobal result for an epidemic model with vaccination that exihibits backward bifurcation. J. Appl. Math., 64 (2003), 260-276. 
  60. F. Brauer. Backward bifurcation in simple vaccination models. J. Math. Ana. Appl., 298 (2004), 418-431. 
  61. Z. Feng, C. Castillo-Chavez A.F. Capurro. A model for tuberculosis with exogenous reinfection. Theor. Pop. Biol., 57 (2000), 235-247. 
  62. C.Y. Chiang L.W. Riley. Exogenous reinfection in tuberculosis. Lancet Infect. Dis., 5 (2005), 629-636. 
  63. S.M. Garba, A.B. Gumel M.R. Abu Bakar. Backward bifurcation in dengue transmission dynamics. Math. Bios., 215 (2008), 11-25. 
  64. O. Sharomi, C.N. Podder, A.B. Gumel, E.H. Elbasha J. Watmough. Role of incidence function in vaccine-induced backward bifurcation in some HIV models. Math. Biosci., 210 (2007), 436-463. 
  65. F. Brauer, C. Castillo-Chavez. Mathematical Models in Population Biology and Epidemiology. Text in Applied Mathematics Series, 40, Springer-Verlag, New York, 2001.  
  66. B.M. Murphy, B.H. Singer D. Kirschner. Comparing epidemic tuberculosis in demographically distinct populations. Maths. Biosci., 180 (2002), 161-185. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.