# Global Dynamics and Implications of an HBV Model with Proliferating Infected Hepatocytes
## 2 Model Development and Dynamics
### Infected Hepatocytes Proliferating at a Different Rate  
Since the data are inconclusive as to the proliferation rate of infected hepatocytes, we next explored the dynamical implication of infected hepatocytes proliferating at a smaller rate than healthy hepatocytes. The only change made to (7)-(9) is that the maximum proliferation rate for the infected hepatocytes is changed from \(r\) to \(\rho\),  
\[\frac{dx}{dt} = rx(t)\bigg{(}1-\frac{x(t)+y(t)}{K}\bigg{)}-\frac{\beta v(t)x(t)}{x (t)+y(t)}, \tag{14}\] \[\frac{dy}{dt} = \rho y(t)\bigg{(}1-\frac{(x(t)+y(t)}{K}\bigg{)}+\frac{\beta v(t)x(t )}{x(t)+y(t)}-ay(t),\] (15) \[\frac{dv}{dt} = \gamma y(t)-\mu v(t), \tag{16}\]  
where \(\rho<r\). The basic reproduction number, \(R_{0}\), is again,  
\[R_{0}=\frac{\beta\gamma}{a\mu}.\]  
The cellular vitality index, \(R^{*}\), is slightly modified to account for the proliferation of infected hepatocytes. In particular,  
\[R^{*}=\frac{a+r-\rho}{a}.\]  
In addition to the three equilibria admitted by the model (7)-(9), the system (14)-(16) allows an additional steady state representing chronic infection, seen in [PERSON] et al. (2016). Figure 1: Bifurcation diagram showing, from top to bottom, equilibrium healthy hepatocyte, infected hepatocyte, and free virion populations as a function of \(R_{0}\), where \(R_{0}\) is varied via \(a\). Other parameter values are fixed at: \(r=0.3\), \(\mu=0.693\), \(\beta=0.0014\), \(\gamma=450\), \(K=2\times 10^{11}\). The entire set of equilibria therefore can include the following: \(E_{0}=(0,0,0)\), \(E_{f}=(K,0,0)\), \(E_{i}=\left(0,\frac{K(\rho-a)}{\rho},\frac{\gamma K(\rho-a)}{\rho\mu}\right)\), and \(E^{*}=(x^{*},y^{*},v^{*})\), where:  
\[x^{*} = \frac{Kra}{(r-\rho)^{2}}\left(1-\frac{\rho}{r}R_{0}\right)\left( \frac{R^{*}}{R_{0}}-1\right),\] \[y^{*} = \frac{Kra}{(r-\rho)^{2}}(R_{0}-1)\left(\frac{R^{*}}{R_{0}}-1\right),\] \[v^{*} = \frac{K\gamma ra}{\mu(r-\rho)^{2}}(R_{0}-1)\left(\frac{R^{*}}{R_ {0}}-1\right).\]  
Notice that \(E_{i}\) only exists if \(\rho>a\) and \(E^{*}\) only exists if \(1<R_{0}<R^{*}\) and the maximum proliferation rate of infected hepatocytes is sufficiently small, namely:  
\[R_{0}<\frac{r}{\rho}.\]  
Figure 2 highlights the relationship between \(R_{0}\) and \(\rho\). As \(R_{0}\) increases into the biologically relevant range of 6-8, The maximum value of \(\rho\) that admits a chronic steady state is quite small, less than 0.1. In contrast to the model studies by [PERSON] et al. [36] in which infected hepatocytes could not divide, the fraction of infected hepatocytes at the chronic steady state in the model (14)-(16) now depends on the maximum proliferation rate of hepatocytes, even when that rate is small. In particular, that fraction is:  
\[\frac{dy}{dt} = \rho y(t)\bigg{(}1-\frac{(x(t)+y(t)}{K}\bigg{)}+\frac{\beta v(t)x(t )}{x(t)+y(t)}-ay(t),\]0  
Notice that:  
\[\frac{dy}{dt} = \rho y(t)\bigg{(}1-\frac{(x(t)+y(t)}{K}\bigg{)}+\frac{\beta v(t)x(t )}{x(t)+y(t)}-ay(t),\]1  
which is identical to the fraction of infected hepatocytes in the model without proliferating infected cells [36]. Figure 3 shows the percentage of infected hepatocytes for values of \(\rho\)  
\[\frac{dy}{dt} = \rho y(t)\bigg{(}1-\frac{(x(t)+y(t)}{K}\bigg{)}+\frac{\beta v(t)x(t )}{x(t)+y(t)}-ay(t),\]2 The maximum value of \(\rho\) allowing the existence of a biologically relevant chronic steady state in the system (14)–(16) as a function of \(R_{0}\) values. It is seen that as \(R_{0}\) becomes larger, infected hepatocyte proliferation must be increasingly impaired if a chronic infection state with a nonzero number of healthy hepatocytes is to exist. Note that in this example, the healthy hepatocyte proliferation rate is fixed at \(r=1\). that yield a chronic state. Increasing \(\rho\) increases the fraction of infected hepatocytes. When \(\rho=\frac{r}{R_{0}}\), \(E^{*}\) collides with \(E_{i}\). For values of \(\rho>\frac{r}{R_{0}}\), \(E^{*}\) is no longer biologically relevant.