# Global Dynamics and Implications of an HBV Model with Proliferating Infected Hepatocytes
## 2 Model Development and Dynamics
### Infected Hepatocytes Proliferating at a Different Rate
In drawing the connections between (7)-(9) and (14)-(16), we start by discussing the dynamics if the proliferation rate of infected hepatocytes is not severely impaired. This implies that \(\rho>a\) and \(\rho>\frac{r}{R_{0}}\). Therefore, \(E_{i}\) exists and \(E^{*}\) does not. Proposition: If \(\rho>a\) and \(\rho>\frac{r}{R_{0}}\), then \(E_{f}\) is unstable and \(E_{i}\) is locally asymptotically stable. \({}_{\blacksquare}\)  
Proof: The Jacobian matrix of the vector field corresponding to (14)-(16) at \(E_{f}\) is:  
\[J(x,y,v)|_{E_{f}}=\left( \begin{array}{ccc}-r&-r&-\beta\\ 0&-a&\beta\\ 0&\gamma&-\mu\end{array} \right).\]  
The eigenvalues of the matrix are given by:  
\[\lambda_{1} = -r, \tag{18}\] \[\lambda_{2,3} = -\frac{1}{2}(a+\mu)\pm\frac{1}{2}\sqrt{(a+\mu)^{2}-4a\mu(1-R_{0} )}. \tag{19}\]  
Since:  
\[R_{0}>\frac{r}{\rho}>1,\]  
\(\lambda_{2,3}\) are positive and \(E_{f}\) is unstable. The Jacobian matrix of the vector field corresponding to (14)-(16) at \(E_{i}\) is:  
\[J(x,y,v)|_{E_{i}}=\left( \begin{array}{ccc}a\Big{(}\frac{r}{\rho}-R_{0}\Big{)} &0&0\\ -\rho+a+aR_{0}&a-\rho&0\\ 0&\gamma&-\mu\end{array} \right).\]  
Figure 3: The fraction of infected hepatocytes at equilibrium, as a function of \(\rho\), the infected hepatocyte proliferation rate. As this rate increases, more and more hepatocytes are infected, until a threshold is passed and the liver experiences 100% infection. When this point is crossed, there is also a stability switch as \(E_{i}^{*}\) shifts from stable to unstable, while \(E_{i}\) becomes stable. Other parameter values are fixed at \(r=1\), \(K=2\times 10^{11}\), \(\beta=0.0014\), \(\gamma=200\), \(a=0.0693\), \(\mu=0.693\). The eigenvalues are the following,  
\[\lambda_{1} = -\mu,\] \[\lambda_{2} = -(\rho-a),\] \[\lambda_{3} = -a\bigg{(}R_{0}-\frac{r}{\rho}\bigg{)}.\]  
If \(\rho>a\) and \(R_{0}>\frac{r}{\rho}\), then \(\lambda_{1,2,3}<0\) and \(E_{i}\) is locally asymptotically stable. According to Proposition 3, this model presents the following biological hypothesis: even if healthy and infected hepatocytes do not proliferate at the same rate, if the infected hepatocytes proliferate at a sufficiently high rate (\(\rho>a\) and \(\rho>\frac{1}{R_{0}}r\)), then there will not be a significant population of healthy cells in the liver during chronic infection. Furthermore, with a sufficiently high infected hepatocyte proliferation rate, patients would be unlikely to experience oscillations in viral load and liver mass. Impairing the maximum infected hepatocyte proliferation slightly impacts the following conditions: \(\rho>a\) and \(\rho>\frac{1}{R_{0}}r\). Although the chronic state \(E^{*}\) and the infected state \(E_{i}\) are both present, \(E_{i}\) is always unstable and the dynamics for (14)-(16) are similar to the model analyzed in [PERSON] et al. [36]. **Proposition 4**. : _For (14)-(16), if \(\rho<\frac{1}{R_{0}}r\), then the following results hold. If \(R_{0}<1\), then \(E_{f}\) is locally asymptotically stable and \(E_{i}\) is a saddle; \(b\). If \(R_{0}>1\) then \(E_{f}\) is unstable and \(E_{i}\) is a saddle._  
Since \(E_{i}\) exists, the global stability of \(E_{f}\) cannot be determined. As in [PERSON] et al. [36], the severity of disease is controlled by \(R_{0}\) and \(R^{*}\). The bifurcation diagram (Figure 4) shows that for realistic \(R_{0}\) values, there will be stable oscillations before passing through the Hopf bifurcation to experience liver failure. We prove the existence of the Hopf bifurcation point below. Due to the extensive computations, we assumed that \(\rho=0\). Figure 4 suggests that this proof is valid for small \(\rho\) as well. As the reproductive number crosses the bifurcation point of \(R_{0}=1\), the stability of  
Figure 4: Bifurcation diagram showing equilibrium total hepatocyte and virion populations and oscillation bounds as a function of \(R_{0}\), for the model with \(r=\rho\), and \(R_{0}\) controlled by changes in \(\beta\).