Modelling Tuberculosis and Hepatitis B Co-infections

S. Bowong^1, 2, 3 ; J. Kurths^2, 4

¹ Laboratory of Applied Mathematics, Department of Mathematics and Computer Science, Faculty of Science, University of Douala, Douala, P.O. Box 24157, Cameroon
² Postdam Institute for Climate Impact Research (PIK), Telegraphenberg A 31, 14412 Potsdam, Germany
³ UMI 209 IRD/UPMC UMMISCO, Bondy, Projet MASAIE INRIA Grand Est, France Projet Grimcape, LIRIMA, Cameroun
⁴ Department of Physics Humboldt Universitat zu Berlin, 12489 Berlin, Germany

Mathematical modelling of natural phenomena, Tome 5 (2010) no. 6, pp. 196-242.

Résumé

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.

Détail

DOI : 10.1051/mmnp/20105610

Bibliographie

[1] 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.

[2] World Health Organization. Global tuberculosis control: surveillance, planning, financing. Geneva, Switzerland: World Health Organization, 2009.

[3] WHO. Hepatitis B. /http://www.who.int/mediacentre/factsheets/fs204/en/ index.htmlS, revised August 2008, 2008.

[4] C. Dye, B.G. Williams Eliminating human tuberculosis in the twenty-first century J. R. Soc. Interface 2008 653 662

[5] C. Chintu, A. Mwinga An African perspective of tuberculosis and HIV/AIDS Lancet 1999 997 1005

[6] R. Williams Global challenges in liver disease Hepatol. 2006 521 526

[7] T. Frieden, R.C. Driver Tuberculosis control: past 10 years and future progress Tuberculosis 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. 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. 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. 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. 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 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. 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. 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. 2004 211 220

[17] J. Hou, Z. Liu, F. Gu Epidemiology and prevention of hepatitis B virus infection Int. J. Med. Sci. 2005 50 57

[18] K.C. Hyams Risks of chronicity following acute hepatitis B virus infection: a review Clin. Infect. Dis. 1995 992 1000

[19] J.D. Jia, H. Zhuang The overview of the seminar on chronic hepatitis B Chin. J. Hepatol. 2004 698 699

[20] D. Lavanchy Hepatitis B virus epidemiology, disease burden, treatment and current and emerging prevention and control measures J. Viral. Hepat. 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. 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: 10.1016/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. 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. 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. 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. 2002 1198 1203

[27] N.W.Y. Leung Treatment Of Tuberculosis In Patients With Hepatitis Hong Kong Practitioner 1997 6 13

[28] W.O. Kermack, A.G. Mckendrick A contribution to the mathematical theory of epidemics Proc. Roy. Soc. 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. 2005 177 187

[31] R. Naresh, A. Tripathi Modelling and analysis of HIV-TB co-infection in a variable size population Math. Model. Anal. 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. 2008 1366 1381

[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. 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. 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. 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 2009 827 864

[37] L-I.W. Roeger, Z. Feng, C.C. Chavez Modelling TB and HIV co-infections Math. Bios. Eng. 2009 815 837

[38] S. Bowong, J.J. Tewa Mathematical analysis of a tuberculosis model with differential infectivity Com. Non. Sci. Num. Sim. 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. 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. 1992 1 71

[45] H.W. Hethcothe The mathematics of infectious disease SIAM Review 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. 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. 1988 179 196

[50] S.M. Moghadas Modelling the effect of imperfect vaccines on disease epidemiology Dis. Cont. Dynam. Syst. Series B 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. 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. 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. 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. 2004 361 404

[58] J. Dushoff, W. Huang, C. Castillo-Chavez Backwards bifurcations and catastrophe in simple models of fatal diseases J. Math. Biol. 1998 227 248

[59] J. Arino, C.C. Mccluskey, P. Van Den Driessche Global result for an epidemic model with vaccination that exihibits backward bifurcation J. Appl. Math. 2003 260 276

[60] F. Brauer Backward bifurcation in simple vaccination models J. Math. Ana. Appl. 2004 418 431

[61] Z. Feng, C. Castillo-Chavez, A.F. Capurro A model for tuberculosis with exogenous reinfection Theor. Pop. Biol. 2000 235 247

[62] C.Y. Chiang, L.W. Riley Exogenous reinfection in tuberculosis Lancet Infect. Dis. 2005 629 636

[63] S.M. Garba, A.B. Gumel, M.R. Abu Bakar Backward bifurcation in dengue transmission dynamics Math. Bios. 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. 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. 2002 161 185