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Modelling Tuberculosis and Hepatitis B Co-infections

Published online by Cambridge University Press:  13 September 2010

S. Bowong  and
J. Kurths
S. Bowong*
Affiliation:
Laboratory of Applied Mathematics, Department of Mathematics and Computer Science, Faculty of Science, University of Douala, Douala, P.O. Box 24157, Cameroon Postdam Institute for Climate Impact Research (PIK), Telegraphenberg A 31, 14412 Potsdam, Germany UMI 209 IRD/UPMC UMMISCO, Bondy, Projet MASAIE INRIA Grand Est, France Projet Grimcape, LIRIMA, Cameroun
J. Kurths
Affiliation:
Postdam Institute for Climate Impact Research (PIK), Telegraphenberg A 31, 14412 Potsdam, Germany Department of Physics Humboldt Universitat zu Berlin, 12489 Berlin, Germany
*
*Corresponding author: E-mail: [email protected]
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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.

Keywords

nonlinear dynamical systemsepidemiological modelstuberculosishepatitis Bstability
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 5 , Issue 6: Ecology (Part 2) , 2010 , pp. 196 - 242
Copyright
© EDP Sciences, 2010

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References

Global Fund to Fight AIDS, Tuberculosis, and Malaria. Fighting Tuberculosis. Geneva, Switzerland: (2006). Retrieved September 9, 2006, http://www.theglobalfund.org/en/about/tuberculosis/default.asp, 2006.
World Health Organization. Global tuberculosis control: surveillance, planning, financing. Geneva, Switzerland: World Health Organization, 2009.
WHO. Hepatitis B. /http://www.who.int/mediacentre/factsheets/fs204/en/ index.htmlS, revised August 2008, 2008.
Dye, C. Williams, B.G.. Eliminating human tuberculosis in the twenty-first century . J. R. Soc. Interface, 5 (2008), 653-662.CrossRefGoogle ScholarPubMed
Chintu, C. Mwinga, A.. An African perspective of tuberculosis and HIV/AIDS . Lancet, 353 (1999), 997-1005.CrossRefGoogle ScholarPubMed
Williams, R.. Global challenges in liver disease . Hepatol., 44 (2006), No. 3, 521-526.CrossRefGoogle Scholar
Frieden, T. Driver, R.C.. Tuberculosis control: past 10 years and future progress . Tuberculosis, 83 (2003), 82-85.CrossRefGoogle Scholar
De Cock, K.M. Chaisson, R.E.. 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.Google ScholarPubMed
Global Fund Against AIDS, TB and Malaria. The Global Tuberculosis Epidemic, Geneva, Switzerland, 2004.
Lavanchy, D.. Hepatitis B virus epidemiology, disease burden, treatment and current and emerging prevention and control measures . J. Viral. Hepat., 11 (2004), 97-107.CrossRefGoogle Scholar
Edmunds, W.J., Medley, G.F. Nokes, D.J.. The transmission dynamics and control of hepatitis B virus in the Gambia . Stat. Med., 15 (1996), 2215-2233.3.0.CO;2-2>CrossRefGoogle ScholarPubMed
Edmunds, W.J., Medley, G.F. Nokes, D.J.. 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.CrossRefGoogle Scholar
Edmunds, W.J., Medley, G.F., Nokes, D.J., Hall, A.J. Whittle, H.C.. The influence of age on the development of the hepatitis B carrier state . Proc. R. Soc. Lond. B, 253 (1993), 197-201.CrossRefGoogle ScholarPubMed
Edmunds, W.J., Medley, G.F., Nokes, D.J., Hall, A.J. Whittle, H.C.. Epidemiological patterns of hepatitis B virus (HBV) in highly endemic areas . Epidemiol. Infect., 117 (1996), 313-325.CrossRefGoogle Scholar
Goldstein, S.T., Zhou, F.J., Hadler, S.C., Bell, B.P., Mast, E.E. Margolis, H.S.. A mathematical model to estimate global hepatitis B disease burden and vaccination impact . Int. J. Epidemiol., 34 (2005), 1329-1339.CrossRefGoogle ScholarPubMed
Hahnea, S., Ramsaya, M., Balogun, K., Edmund, W.J. Mortimer, P.. 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.CrossRefGoogle Scholar
Hou, J., Liu, Z. Gu, F.. Epidemiology and prevention of hepatitis B virus infection . Int. J. Med. Sci., 2 (2005), No. 1, 50-57.CrossRefGoogle ScholarPubMed
Hyams, K.C.. Risks of chronicity following acute hepatitis B virus infection: a review . Clin. Infect. Dis., 20 (1995), 992-1000.CrossRefGoogle Scholar
Jia, J.D. Zhuang, H.. The overview of the seminar on chronic hepatitis B . Chin. J. Hepatol., 12 (2004), 698-699.Google ScholarPubMed
Lavanchy, D.. Hepatitis B virus epidemiology, disease burden, treatment and current and emerging prevention and control measures . J. Viral. Hepat., 11 (2004) 97-107. CrossRefGoogle ScholarPubMed
Blal, C.A., Passos, S.R.L., Horn, C., Georg, I., Bonecini, M.G., Rolla, V.C. Castro, L. D.. 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.CrossRefGoogle Scholar
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: 10.1016/j.ijid.2007.04.015.
Bellamy, R., Ruwende, C., Corrah, T., McAdam, K.P.W.J., Thursz, M., Whittle, H.C. Hill, A.V.S.. Tuberculosis and Chronic Hepatitis B Virus Infection in Africans and Variation in the Vitamin D Receptor Gene . J. Inf. Dis., 179 (1999), 721-724.CrossRefGoogle ScholarPubMed
Lifson, A.R., Thai, D., O’Fallon, A., Mills, W.A. Hang, K.. Prevalence of tuberculosis, hepatitis B virus, and intestinal parasitic infections among refugees to Minnesota . Public Health Rep., 117 (2002), 69-77.CrossRefGoogle ScholarPubMed
McGlynn, K.A., Lustbader, E.D. London, W.T.. Immune responses to hepatitis B virus and tuberculosis infections in Southeast Asian refugees . Amer. J. Epide., 122 (1985), 1032-1036.CrossRefGoogle ScholarPubMed
Patel, P.A. Voigt, M.D.. Prevalence and interaction of hepatitis B and latent tuberculosis in Vietnamese immigrants to the United States . Amer. J. Gastr., 97 (2002), 1198-1203.CrossRefGoogle ScholarPubMed
Leung, N.W.Y.. Treatment Of Tuberculosis In Patients With Hepatitis . Hong Kong Practitioner, 19 (1997), 6-13.Google Scholar
Kermack, W.O. McKendrick, A.G.. A contribution to the mathematical theory of epidemics . Proc. Roy. Soc., A115 (1927), 700-721.CrossRefGoogle Scholar
R.M. Anderson, R.M. May. Infectious Disease of Humans: Dynamics and Control. Oxford University Press, London/New York, 1992.
Blyuss, K.B. Kyrychko, Y.N.. On a basic model of a two-disease epidemic . Appl. Math. Comput., 160 (2005), 177-187.Google Scholar
Naresh, R., Tripathi, A.. Modelling and analysis of HIV-TB co-infection in a variable size population . Math. Model. Anal., 10(3) (2005), 275-286. Google Scholar
Long, E.F., Vaidya, N.K., Brandeau, M.L.. Controlling Co-epidemic: Analysis of HIV and tuberculosis infection analysis. Oper. Res., 56 (2008), No. 6, 1366-1381. doi:10.1287/opre.1080.0571. CrossRefGoogle Scholar
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 (2008), 557-593.CrossRefGoogle Scholar
Sharomi, O., Podder, C.N., Gumel, A.B. Song, B.. Mathematical analysis of the transmission dynamics of HIV/TB co-infection in the presence of treatment . Math. Biosci. Eng., 5 (2008), 145-174.Google Scholar
Mukandavire, Z., Gumel, A.B., Garira, W. Tchuenche, J.M.. Mathematical analysis of a model for HIV-malaria co-infection . Math. Biosci. Engr., 6 (2009), 333-362.Google ScholarPubMed
Mtisi, E., Rwezaura, H., Tchuenche, J.M.. A mathematical analysis of malaria and tuberculosis co-dynamics . Dis. Cont. Dyn. Syst. Series B, 12 (2009) 827-864 2009 doi:10.3934/dcdsb.2009.12.827 CrossRefGoogle Scholar
Roeger, L-I.W., Feng, Z. Chavez, C.C.. Modelling TB and HIV co-infections . Math. Bios. Eng., 6 (2009), 815-837.Google Scholar
Bowong, S. Tewa, J.J.. Mathematical analysis of a tuberculosis model with differential infectivity . Com. Non. Sci. Num. Sim., 14 (2009), 4010-4021.CrossRefGoogle Scholar
Hahnea, S., Ramsaya, M., Balogun, K., Edmund, W.J. Mortimer, P.. 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.CrossRefGoogle Scholar
Dye, C., Schele, S.. For the WHO global surveillance and monitoring project. Global burden of tuberculosis estimated incidence, prevalence and mortality by country . 282 (1999), 677-686. Google Scholar
National Committee of Fight Against Tuberculosis. Guide de personnel de la santé, Cameroon, 2008.
National Institute of Statistics. Evolution des systèmes statistiques nationaux, Cameroon, 2007.
G. Birkhoff, G. C. Rota. Ordinary Differential Equations. 4th edition, John Wiley & Sons, Inc., New York, 1989.
Hutson, V. Schmitt, K.. Permanence and the dynamics of biological systems . Math. Biosci., 111 (1992), 1-71.CrossRefGoogle ScholarPubMed
Hethcothe, H.W.. The mathematics of infectious disease . SIAM Review, 42 (2000), 599-653.CrossRefGoogle Scholar
Shepard, C.W., Simard, E.P., Finelli, L., Fiore, A.E. Bell, B.P.. Hepatitis B virus infection: epidemiology and vaccination . Epidemiol. Rev., 28 (2006), 112-125.CrossRefGoogle ScholarPubMed
V. Lakshmikantham, S. Leela, A. Martynyuk. Stability Analysis of Nonlinear Systems. Marcel Dekker Inc., New York and Basel, pp. 31, 1989.
H.L. Smith, P. Waltman. The Theory of the Chemostat. Cambridge University Press, 1995.
Zhang, S.N.. Comparison theorems on boundedness . Funkcial. Ekvac., 31 (1988), 179-196.Google Scholar
Moghadas, S.M.. Modelling the effect of imperfect vaccines on disease epidemiology . Dis. Cont. Dynam. Syst. Series B, 4 (2004), 999-1012.CrossRefGoogle Scholar
Diekmann, O., Heesterbeek, J.A.P. Metz, J.A.P.. On the definition and computation of the basic reproduction ratio R 0 in the model of infectious disease in heterogeneous populations . J. Math. Biol., 2 (1990), 265-382.Google Scholar
van den Driessche, P. Watmough, J.. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission . Math. Bios., 180 (2002), 29-28.CrossRefGoogle ScholarPubMed
J.P. LaSalle. The stability of dynamical systems. Society for Industrial and Applied Mathematics, Philadelphia, Pa, 1976.
LaSalle, J.P.. Stability theory for ordinary differential equations . J. Differ. Equ., 41 (1968), 57-65.CrossRefGoogle Scholar
N.P. Bhatia, G.P. Szegö. Stability Theory of Dynamical Systems. Springer-Verlag, 1970.
J. Carr. Applications Centre Manifold Theory. Springer-Verlag, New York, 1981.
Castillo-Chavez, C. Song, B.. Dynamical models of tuberculosis and their applications . Math. Bios. Eng., 1 (2004), 361-404.CrossRefGoogle ScholarPubMed
Dushoff, J., Huang, W. Castillo-Chavez, C.. Backwards bifurcations and catastrophe in simple models of fatal diseases . J. Math. Biol., 36 (1998), 227-248.CrossRefGoogle ScholarPubMed
Arino, J., McCluskey, C.C. van den Driessche, P. Global result for an epidemic model with vaccination that exihibits backward bifurcation . J. Appl. Math., 64 (2003), 260-276.Google Scholar
Brauer, F.. Backward bifurcation in simple vaccination models . J. Math. Ana. Appl., 298 (2004), 418-431.CrossRefGoogle Scholar
Feng, Z., Castillo-Chavez, C. Capurro, A.F.. A model for tuberculosis with exogenous reinfection . Theor. Pop. Biol., 57 (2000), 235-247.CrossRefGoogle ScholarPubMed
Chiang, C.Y. Riley, L.W.. Exogenous reinfection in tuberculosis . Lancet Infect. Dis., 5 (2005), 629-636.CrossRefGoogle ScholarPubMed
Garba, S.M., Gumel, A.B. Abu Bakar, M.R.. Backward bifurcation in dengue transmission dynamics . Math. Bios., 215 (2008), 11-25.CrossRefGoogle ScholarPubMed
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 (2007), 436-463.CrossRefGoogle ScholarPubMed
F. Brauer, C. Castillo-Chavez. Mathematical Models in Population Biology and Epidemiology. Text in Applied Mathematics Series, 40, Springer-Verlag, New York, 2001.
Murphy, B.M., Singer, B.H. Kirschner, D.. Comparing epidemic tuberculosis in demographically distinct populations . Maths. Biosci., 180 (2002), 161-185.CrossRefGoogle Scholar