Journal of Mathematical Biology

, Volume 60, Issue 4, pp 573–590 | Cite as

Rich dynamics of a hepatitis B viral infection model with logistic hepatocyte growth

  • Sarah Hews
  • Steffen Eikenberry
  • John D. Nagy
  • Yang Kuang
Article

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.

Keywords

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

Mathematics Subject Classification (2000)

34C23 34C25 92C50 

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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Sarah Hews
    • 1
  • Steffen Eikenberry
    • 1
  • John D. Nagy
    • 2
  • Yang Kuang
    • 1
  1. 1.Department of Mathematics and StatisticsArizona State UniversityTempeUSA
  2. 2.Department of BiologyScottsdale Community CollegeScottsdaleUSA

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