# Global Dynamics and Implications of an HBV Model with Proliferating Infected Hepatocytes
###### Abstract  
Chronic hepatitis B (HBV) infection is a major cause of human suffering, and a number of mathematical models have examined the within-host dynamics of the disease. Most previous models assumed that infected hepatocytes do not proliferate; however, the effect of HBV infection on hepatocyte proliferation is controversial, with conflicting data showing both induction and inhibition of proliferation. With a family of ordinary differential equation (ODE) models, we explored the dynamical impact of proliferation among HBV-infected hepatocytes. Here, we show that infected hepatocyte proliferation in this class of models generates a threshold that divides the dynamics into two categories. Sufficiently compromised proliferation in infected cells produces complex dynamics characterized by oscillating viral loads, whereas higher proliferation generates straightforward dynamics that always results in chronic infection, sometimes with liver failure. A global stability result of the liver failure state was included as it is unique to this class of models. Finally, the model analysis motivated a testable biological hypothesis: Healthy hepatocytes are present in chronic HBV infection if and only if the proliferation of infected hepatocytes is severely impaired. H +
Footnote †: journal: Applied Sciences  
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H +
Footnote †: journal: Applied Sciences  
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Footnote 1: School of Natural Science, Hampshire College, Amherst, MA 01002, USA; [EMAIL_ADDRESS]  
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Footnote 1: School of Natural Science, Hampshire College, Amherst, MA 01002, USA; sarahews@gmail. by HBV are generated by immune attack on infected cells [4; 5; 6]. Considerable evidence supports this hypothesis [4; 5; 7; 8], However, the extent and nature of the virus's cytopaticity remain unresolved [9]. In particular, very little is known about how viral infection affects proliferation. In the experimental literature, the effect of HBV infection on hepatocyte proliferation is controversial, with conflicting data showing both induction and inhibition of proliferation [10]. In addition, infection has been correlated with both pro- and anti-apoptotic effects on hepatocytes [11]. HBV X protein has severely impaired liver regeneration in some mouse models [11; 12; 13], but had little effect in others [14]. It is possible that natural variation in the HBV virus itself may explain these conflicting results [10]. Here, we study the dynamical implications of the proliferation of infected hepatocytes. Mathematical models have been applied to a wide variety of viral illnesses, and HBV is no exception, with scores of modeling works published in recent years (see, e.g., recent reviews by [PERSON] et al. [15] and [PERSON] [16]). Previous models have mainly, though not exclusively, focused on chronic infections and were originally adapted from HIV models, including at least three state variables: healthy hepatocytes, \(x\), infected hepatocytes, \(y\), and free virions, \(v\). The basic virus infection model (BVIM), presented by [PERSON] et al. [17; 18], assumes that healthy hepatocytes are produced by some constant influx term, \(\lambda\), healthy hepatocytes die at per-capita rate \(d\), infected hepatocytes die at per-capita rate \(a\), and infection occurs according to _mass action kinetics_, with coefficient \(\beta\). Infectious virions are produced by infected hepatocytes at rate \(\gamma\) and die at rate \(\mu\), giving the model:  
\[\frac{dx}{dt} = \lambda-dx-\beta xv, \tag{1}\] \[\frac{dy}{dt} = \beta xv-ay,\] (2) \[\frac{dv}{dt} = \gamma y-\mu v. \tag{3}\]  
In this model, the immune response to infection is represented by an elevated death rate in infected hepatocytes, \(a>d\), and by the destruction of free virions at rate \(\mu\). Such simple linear dynamics are clearly inappropriate for modeling the adaptive immune system's response to acute infection, but may be a reasonable first approximation in the case of established, chronic infection.