### Abstract

Chronic hepatitis B virus (HBV) infection is a major cause of human suffering, and a number of mathematical models have examined within-host dynamics of the disease. Most previous HBV infection models have assumed that: (a) hepatocytes regenerate at a constant rate from a source outside the liver; and/or (b) the infection takes place via a mass action process. Assumption (a) contradicts experimental data showing that healthy hepatocytes proliferate at a rate that depends on current liver size relative to some equilibrium mass, while assumption (b) produces a problematic basic reproduction number. Here we replace the constant infusion of healthy hepatocytes with a logistic growth term and the mass action infection term by a standard incidence function; these modifications enrich the dynamics of a well-studied model of HBV pathogenesis. In particular, in addition to disease free and endemic steady states, the system also allows a stable periodic orbit and a steady state at the origin. Since the system is not differentiable at the origin, we use a ratio-dependent transformation to show that there is a region in parameter space where the origin is globally stable. When the basic reproduction number, R_{0}, is less than 1, the disease free steady state is stable. When R_{0} > 1 the system can either converge to the chronic steady state, experience sustained oscillations, or approach the origin. We characterize parameter regions for all three situations, identify a Hopf and a homoclinic bifurcation point, and show how they depend on the basic reproduction number and the intrinsic growth rate of hepatocytes.

Original language | English (US) |
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Pages (from-to) | 573-590 |

Number of pages | 18 |

Journal | Journal Of Mathematical Biology |

Volume | 60 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 2010 |

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### Keywords

- HBV
- Homoclinic bifurcation
- Hopf bifurcation
- Logistic hepatocyte growth
- Origin stability
- Ratio-dependent transformation

### ASJC Scopus subject areas

- Modeling and Simulation
- Agricultural and Biological Sciences (miscellaneous)
- Applied Mathematics

### Cite this

*Journal Of Mathematical Biology*,

*60*(4), 573-590. https://doi.org/10.1007/s00285-009-0278-3