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**Rich dynamics of a hepatitis B viral infection model with logistic hepatocyte growth.**
*(English)*
Zbl 1311.92117

Summary: 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.

### MSC:

92C60 | Medical epidemiology |

34C23 | Bifurcation theory for ordinary differential equations |

34C25 | Periodic solutions to ordinary differential equations |

92C50 | Medical applications (general) |

### Keywords:

HBV; ratio-dependent transformation; logistic hepatocyte growth; origin stability; Hopf bifurcation; homoclinic bifurcation
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\textit{S. Hews} et al., J. Math. Biol. 60, No. 4, 573--590 (2010; Zbl 1311.92117)

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