# Global Dynamics and Implications of an HBV Model with Proliferating Infected Hepatocytes
## 2 Model Development and Dynamics
### Infected Hepatocytes Proliferating at the Same Rate  
We assumed that healthy and infected hepatocytes proliferate at the same rate and therefore set \(\rho=r\) and arrive at the first model:\[\frac{dx}{dt} = rx(t)\bigg{(}1-\frac{x(t)+y(t)}{K}\bigg{)}-\frac{\beta v(t)x(t)}{x(t) +y(t)}, \tag{7}\] \[\frac{dy}{dt} = ry(t)\bigg{(}1-\frac{(x(t)+y(t)}{K}\bigg{)}+\frac{\beta v(t)x(t)}{x (t)+y(t)}-ay(t),\] (8) \[\frac{dv}{dt} = \gamma y(t)-\mu v(t). \tag{9}\]  
As mentioned above, the basic reproduction number is:  
\[R_{0}=\frac{\beta\gamma}{a\mu},\]  
and the cellular vitality index is:  
\[R^{*}=\frac{r-r+a}{a}=1.\]  
If \(R_{0}\
eq 1\), there are three steady states of (7)-(9):  
\[E_{0}=(0,0,0),\qquad\quad E_{f}=(K,0,0),\qquad\quad E_{i}=\bigg{(}0,\frac{K(r- a)}{r},\frac{\gamma K(r-a)}{r\mu}\bigg{)}.\]  
If \(R_{0}=1\), there are infinitely many positive steady states. Hence, in the following, we assumed that \(R_{0}\
eq 1\). Let \(E_{0}\), \(E_{f}\), and \(E_{i}\) be the liver failure, disease-free, and infected states, respectively. The liver failure state exists since,  
\[\lim_{(x,y,e)\to(0,0,0)}\frac{\beta vx}{x+y}=0.\]  
Notice that the infected state \(E_{i}\) does not allow a population of healthy hepatocytes. Therefore, chronic infection in this model is always characterized by complete infection of the liver. It should be pointed out that models using a mass action term instead of the standard incidence for infection do admit chronic infections that maintain a healthy cell population, but we rejected these models for reasons already specified [30]. Notice that \(E_{i}\) only exists in the positive cone and is therefore biologically relevant only when \(r>a\). At \(E_{i}\), the total number of hepatocytes is \(\frac{K(r-a)}{r}\); therefore, increasing proliferation (\(r\)) or decreasing mortality (\(a\)) rates increases equilibrium liver mass (assumed to be proportional to the number of hepatocytes). If the proliferation rate is greater than the death rate of infected hepatocytes then whether the infection becomes chronic depends on the reproduction rate. If \(R_{0}<1\), the liver is free of infection, and if \(R_{0}>R^{*}=1\), the liver becomes completely infected and the hepatocyte population is reduced compared to the healthy liver. **Proposition 1**. : _Given the system (7)-(9), if \(r>a\), then \(E_{i}\) exists and the following results hold. a. If \(R_{0}<1\), then \(E_{f}\) is locally asymptotically stable and \(E_{i}\) is a saddle point; \(b\). If \(R_{0}>1\), then \(E_{f}\) is unstable and \(E_{i}\) is locally asymptotically stable._  
The proofs are omitted as they involve straightforward linearization techniques. Since \(E_{i}\) always exists when \(r>a\) and is a saddle when \(R_{0}<1\), global stability for \(E_{f}\) cannot be determined. Figure 1 shows a bifurcation diagram of \(a\) with \(r>a\). Healthy hepatocytes and infected hepatocytes are plotted separately so it is clear that the number of infected hepatocytes and virions is an increasing function of \(a\). If \(r<a\), then the only steady states are \(E_{0}\) and \(E_{f}\). As with traditional viral infection models, \(E_{f}\) is globally stable when \(R_{0}<1\). **Theorem 2**. : _If \(r<a\) and \(R_{0}<1\), \(E_{f}\) is globally stable._  
Proof. : Let \((y(t),v(t))\) be a solution of:  
\[\frac{dy(t)}{dt} \leq ry(t)+\beta v(t)-ay(t), \tag{10}\] \[\frac{dv(t)}{dt} \leq \gamma y(t)-\mu v(t). \tag{11}\]  
Let \((Y(t),V(t))\) be a solution of:  
\[\frac{dY(t)}{dt} = \beta V(t)-(a-r)Y(t), \tag{12}\] \[\frac{dy}{dt} = ry(t)\bigg{(}1-\frac{(x(t)+y(t)}{K}\bigg{)}+\frac{\beta v(t)x(t)}{x (t)+y(t)}-ay(t),\]0  
\((Y,V)=(0,0)\) is the only steady state of (12) and (13) and is locally asymptotically stable when \(r<a\) and \(R_{0}<1\). Since (12) and (13) is a cooperative system, and therefore monotone, all solutions of (12) and (13) will approach \((0,0)\). Let \(y_{0}=Y_{0}\), \(v_{0}=V_{0}\); then by the comparison theorem, \((y(t),v(t))\leq(Y(t),V(t))\) for \(t>0\).