A Hepatitis B and C Virus Model with Age since Infection that Exhibits Backward Bifurcation

We propose a mathematical model describing the dynamics of hepatitis B or C virus infection. The method of characteristics reduces the age-structured model to a system of differential equations with distributed delay. The model is rigorously analyzed, and a detailed stability analysis is performed. The model, consisting of two mutually exclusive compartments representing the interaction of the virus in the liver and the blood, has a locally asymptotically stable disease-free equilibrium (DFE) whenever a certain epidemiological threshold, known as the basic reproduction number $R_0$, is less than unity. A sufficient condition for the global asymptotic stability of the trivial steady state is obtained. Further, under biologically realistic hypotheses, the model exhibits the phenomenon of backward bifurcation, where the stable DFE coexists with a stable endemic equilibrium. This means that reducing the basic reproduction number $R_0$ below 1 is not always sufficient to control HBV/HCV infection when the virus interacts with cells in both the liver and the blood. Numerical analysis confirms the results and stresses the role of some parameters involved in the elimination and persistence of the infection.