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

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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

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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 -

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