Global stability analysis of a delay cell-population model of hepatitis B infection with humoral immune response

Abstract

In this work, we propose and investigate a delay cell population model of hepatitis B virus (HBV) infection. We suppose spatial diffusion of free HBV particles, and use a Beddington-DeAngelis incidence function to describe viral infection. The model takes into account the exposed hepatocytes and the usually neglected humoral immune response. Moreover, a time delay is introduced to account for the transformation processes necessary for actual HBV production. We naturally find two threshold parameters, namely the basic reproduction number ${\mathcal{R}}_0$ and the humoral immune response reproduction number ${\mathcal{R}}_1$ which completely determine the global stability of the spatially homogeneous equilibria of the model obtained. By constructing appropriate Lyapunov functionals and using LaSalle's invariance principle we show that, if ${\mathcal{R}}_0\le 1,$ the disease-free equilibrium is globally asymptotically stable. Furthermore, we prove that the endemic equilibrium without humoral immune response and the endemic equilibrium with humoral immune response are globally asymptotically stable if ${\mathcal{R}}_1\le 1<{\mathcal{R}}_0$ and ${\mathcal{R}}_1>1,$ respectively. Finally, in one dimensional space, we perform some numerical simulations to illustrate the theoretical results obtained.

Keywords:

• HBV infection
• diffusion
• humoral immune response
• time delay
• Lyapunov functional
• global stability

Acknowledgements

C. Tadmon acknowledges support from the Abdus Salam International Centre for Theoretical Physics, where part of this work was done, and that of the Institute of Mathematics at the University of Mainz where the paper was finalized.

Disclosure statement

No potential conflict of interest was reported by the authors.