Modelling Tuberculosis and Hepatitis B Co-infections

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 Department of Physics Humboldt Universitat zu Berlin, 12489 Berlin, Germany
⁴ UMI 209 IRD/UPMC UMMISCO, Bondy, Projet MASAIE INRIA Grand Est, France Projet Grimcape, LIRIMA, Cameroun

^* Corresponding author: E-mail: sbowong@gmail.com

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.