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Modelling Tuberculosis and Hepatitis BCo-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 thehepatitis B virus (HBV). The study of the joint dynamics of HBV and TB present formidablemathematical challenges due to the fact that the models of transmission are quitedistinct. We formulate and analyze a deterministic mathematical model which incorporatesof the co-dynamics of hepatitis B and tuberculosis. Two sub-models, namely: HBV-only andTB-only sub-models are considered first of all. Unlike the HBV-only sub-model, which has aglobally-asymptotically stable disease-free equilibrium whenever the associatedreproduction number is less than unity, the TB-only sub-model undergoes the phenomenon ofbackward bifurcation, where a stable disease-free equilibrium co-exists with a stableendemic equilibrium, for a certain range of the associated reproduction number less thanunity. Thus, for TB, the classical requirement of having the associated reproductionnumber 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 bifurcationphenomenon. Through simulations, we mainly find that i) the two diseases will co-existwhenever their partial reproductive numbers exceed unity; (ii) the increased progressionrate due to exogenous reinfection from latent to active TB in co-infected individuals mayplay a significant role in the rising prevalence of TB; and (iii) the increasedprogression rates from acute stage to chronic stage of HBV infection have increased theprevalence 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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