# Global Dynamics and Implications of an HBV Model with Proliferating Infected Hepatocytes
###### Abstract
Extensions and variations of the BVIM model have been introduced to study the rich dynamics of HIV infection and treatment [19; 20]. This basic framework has been extended and modified to study different aspects of HBV infection. Multiple works have modeled the immune response in at least some detail [16; 21; 22; 23; 24] and considered the interplay between the immune response and imposed treatments [25; 26]. Drug treatment, with or without the immune response, can also be incorporated into this basic framework--e.g., [27; 28; 29]. Some drugs interfere with virion production, which can be simply modeled as a decreased virion production rate when treatment is on, while others decrease the magnitude of the infection term [28]. While different biological behaviors can be added in different ways, special care to the basic model construction and its implicit assumptions is warranted. Restricting our attention to chronic infection and ignoring any adaptive or evolving immune response, the BVIM still makes at least three basic assumptions incompatible with biology that have major effects on the model dynamics:  
1. Infection is a mass action process;
2. Healthy hepatocytes are produced by a constant influx;
3. Infected hepatocytes do not reproduce. As shown by [PERSON] et al. [30], the assumption of mass action kinetics for infection yields a basic reproductive number that is dependent on the homeostatic liver size, \(\lambda/d\), and thus results in the biologically implausible prediction that an individual's susceptibility to infection depends on liver mass. Replacing the mass action term with a standard incidence term, as in [30; 31], eliminates this dependence. The standard incidence term in generic viral infection models also received theoretical treatment in [32; 33]. Furthermore, the concept of mass action kinetics is borrowed from chemical kinetics, where reaction rates increase in proportion to the concentration of all involved reactants [34]. As a solid organ with a highly stereotyped cellular architecture, the actual density of hepatocytes is not expected to vary appreciably between individuals or disease states, further arguing against the mass action formulation and for a standard incidence (or other) term better suited to the physical system. The injured liver is clearly not replenished by a constant influx of cells. Rather, liver regeneration is driven by widespread hepatocyte proliferation in a process that is modeled very well, at least heuristically, by logistic growth (see [PERSON] et al. [35] for a detailed justification of the logistic growth term). As shown in [35; 36], replacing the constant influx of healthy hepatocytes with logistic growth that depends on the total liver mass (both healthy and infected hepatocytes) greatly affects the model dynamics. The analysis of [PERSON] et al. [36] also derived two new indices, the cellular vitality index, \(R^{*}\), and the liver failure index, \(R_{f}\), that, in addition to the well-known basic reproductive number, \(R_{0}\), partition the parameter space into distinct dynamical regions. This model admits three potential steady states, corresponding to complete obliteration of the liver (\(x=y=v=0\)), chronic liver infection (\(x,y,v>0\)), and absence of disease (\(x>0\); \(y,v=0\)). Interestingly, there exists a region of the parameter space where all of the states are unstable. In this region, hepatocyte populations (both healthy and infected) and viral loads exhibit sustained oscillation. Furthermore, for each of these three steady states, there exist parameter regimes in which the steady state is asymptotically stable. [PERSON] et al. [22; 37] also considered logistic growth for hepatocytes in the more complex setting of acute infection, and logistic growth has been considered in other works as well [38; 39]; however, the precise dynamical effect of this term in these models is unclear.