# Global Dynamics and Implications of an HBV Model with Proliferating Infected Hepatocytes
## 4 Discussion  
The effectiveness of treatment may be measured by its effect on \(R_{0}\). Any reduction in \(R_{0}\) that fails to cross \(R_{0}=1\) from the right will not clear the infection and is a failure in this sense. However, it may still improve symptoms and reduce the likelihood of liver failure. If treatment reduces \(R_{0}<1\), then it is curative if this reduction can be maintained underthe assumptions of the model. In the case of HBV, treatment with nucleoside analogues usually eventually fails. Mathematically, \(R_{0}\) wanders under treatment, and while it may fall to less than one transiently, it eventually transits unity from the left. However, as a practical matter, \(R_{0}\) need not be calculated to identify treatment failure. The emergence of the cellular vitality index suggests a new way to classify the dynamics and plan the treatment of chronic HBV. Traditional therapies that target the virus affect only \(R_{0}\), but \(R^{*}\) also determines when the liver fails. Given the limitation of the current standard of care, new treatment options are needed. Our model suggests that increasing healthy hepatocyte proliferation, decreasing infected hepatocyte proliferation, and increasing healthy and infected hepatocytes by the same factor should delay liver failure and promote patient survival. Since patients become resistant to all nucleoside analogues over time, further treatment options are needed. Our model proposes that controlling the proliferation rate should prolong the life of the liver. Increasing the distance between the cell vitality index and the reproduction number moves the Hopf bifurcation and decreases the likelihood of the onset of dangerous oscillations in liver damage. Specifically, the closer \(R_{0}\) to \(R^{*}\), the more likely the system will experience an oscillation or a collapse; see Figure 4. Thus, increasing the distance, for example by controlling the proliferation rate, can prevent this dangerous onset. Treatment options that control the proliferation rate of healthy and infected hepatocytes are not currently available and, to the best of our knowledge, are not currently being explored. These model conclusions only hold if proliferation in infected cells leads to infected daughter cells. However, experimental evidence suggests that hepatocyte proliferation may destabilize and dilute viral DNA, thus aiding in clearing infection [49], and that hepatocyte proliferation may be inversely associated with viral loads in experimental settings. Furthermore, modeling work by [PERSON] et al. [42] suggests hepatocyte division may be an important mechanism in both clearing infection and protecting the liver from catastrophic cell loss. It is unclear, however, the degree to which these possible protective properties of proliferation are at work in acute vs. chronic infection. The model analysis suggests that the modeling decision to include proliferating infected hepatocytes in an HBV model should depend on whether or not the proliferation rate of infected hepatocytes is severely impaired. If it is not, then including a slightly different proliferation rate of infected hepatocytes compared to uninfected cells adds needless complexity; the key dynamics will be essentially unaltered, so it is safe to assume that infected and uninfected cells proliferate at the same rate. On the other hand, if infection significantly impairs the proliferative potential of hepatocytes, then one can safely assume that infected cells do not proliferate; again, the dynamical behavior is insensitive to variations in the rate at which infected cells divide. The analysis presented here also suggests a testable biological prediction, namely that one can determine if infected hepatocytes are proliferating to an appreciable degree, and yielding infected daughter cells, simply by searching for uninfected cells. If all cells in the liver are infected (beyond the acute phase of infection), then this model proposes that infected cells are proliferating at a high rate and generating infected progeny.