# Global Dynamics and Implications of an HBV Model with Proliferating Infected Hepatocytes
## 2 Model Development and Dynamics
### Infected Hepatocytes Proliferating at a Different Rate
For \(R_{0}\) below about 8, the total hepatocyte population falls nearly linearly with \(R_{0}\), while sustained oscillations are observed for the approximate interval \(R_{0}\in(8,11)\). A nonzero equilibrium still exists for \(R_{0}\) up to the point that \(R_{0}=R^{*}\), where \(R^{*}\) is the cellular vitality index; beyond this, the system goes to the \(E_{0}\) extinction equilibrium. The parameter values are \(r=0.8\), \(\mu=0.693\), \(\gamma=300\), and \(K=2\times 10^{11}\). \(E_{f}\) is transferred to \(E^{*}\) as it crosses into the positive quadrant. Recall that \(E^{*}\) only exists in the positive quadrant when \(1\leq R_{0}\leq R^{*}\). For the condition \(R_{0}\leq R^{*}\) to hold, the proliferation rate has to be sufficiently large; specifically, \(r\geq\frac{\beta K-a_{H}}{\mu}\). **Theorem 5**. : _Let \(\phi=a^{2}\Big{(}\frac{R^{*}}{R_{0}}-1\Big{)}(R_{0}-1)+\frac{a^{2}}{R_{0}^{2}}( R_{0}-1)-\frac{ar}{R_{0}}(R_{0}-1)+a\mu\Big{(}\frac{R^{*}}{R_{0}}-1\Big{)}\), and \(\sigma=\frac{-(\mu a^{2}R_{0}+a^{3}R^{*})(R^{*}-R_{0})(R_{0}-1)}{R_{0}(\mu R_{0 }+aR^{*})}\). If \(\phi>\sigma\), then \(E^{*}\) is locally asymptotically stable._  
Proof. : \[J(x,y,v)|_{E_{f}}=\left( \begin{array}{ccc}r(1-\frac{2x^{*}+y^{*}}{K})-\frac {\beta R^{*}y^{*}}{(x^{*}+y^{*})^{2}}&-\frac{rx^{*}}{K}+\frac{\beta R^{*}x^{* }}{(x^{*}+y^{*})^{2}}&-\frac{\beta x^{*}}{x^{*}+y^{*}}\\ \frac{\beta R^{*}y^{*}}{(x^{*}+y^{*})^{2}}&-\frac{\beta v^{*}x^{*}}{(x^{*}+y^{* })^{2}}-a&\frac{\beta x^{*}}{x^{*}+y^{*}}\\ 0&\gamma&-\mu\end{array} \right).\]  
The eigenvalues of \(J\) satisfy:  
\[\lambda^{3}+a_{2}\lambda^{2}+a_{1}\lambda+a_{0}=0,\]  
where:  
\[a_{2} = \mu+a\frac{R^{*}}{R_{0}},\] \[a_{1} = 2a^{2}\Big{(}\frac{R^{*}}{R_{0}}-1\Big{)}(R_{0}-1)+\frac{a^{2}}{ R_{0}^{2}}(R_{0}-1)^{2}-\frac{ar}{R_{0}}(R_{0}-1)+a\mu\Big{(}\frac{R^{*}}{R_{0}}-1 \Big{)},\] \[a_{0} = a^{2}\mu(R_{0}-1)\Big{(}\frac{R^{*}}{R_{0}}-1\Big{)}.\]  
Clearly, \(a_{2}>0\) and \(a_{0}>0\) when \(E^{*}\) exists. Let:  
\[\phi=a^{2}\bigg{(}\frac{R^{*}}{R_{0}}-1\bigg{)}(R_{0}-1)+\frac{a^{2}}{R_{0}^{2 }}(R_{0}-1)-\frac{ar}{R_{0}}(R_{0}-1)+a\mu\bigg{(}\frac{R^{*}}{R_{0}}-1\bigg{)},\]  
where,  
\[a_{2}a_{1}-a_{0} = \mu a_{2}+a\frac{R^{*}}{R_{0}}a_{2}-a^{2}\mu(R_{0}-1)\bigg{(} \frac{R^{*}}{R_{0}}-1\bigg{)}, \tag{20}\] \[= \mu\phi+a\frac{R^{*}}{R_{0}}\phi+\bigg{(}\mu a^{2}+a^{3}\frac{R^{ *}}{R_{0}}\bigg{)}\bigg{(}\frac{R^{*}}{R_{0}}-1\bigg{)}(R_{0}-1)>0.\]  
Therefore, \(a_{2}a_{1}>a_{0}\) when:  
\[\phi>\frac{-(\mu a^{2}R_{0}+a^{3}R^{*})(R^{*}-R_{0})(R_{0}-1)}{R_{0}(\mu R_{0 }+aR^{*})}=\sigma.\]  
By the Routh-Hurwitz criteria, we determined a condition for \(E^{*}\) to be locally asymptotically stable. **Theorem 6**. : _If \(a<a(R_{0}-1)<r\), then there is a Hopf bifurcation at \(\phi=\sigma\)._  
Proof. : Let \(\Delta=a_{2}a_{1}-a_{0}\). Then:  
\[\Delta = \mu\phi+a\frac{R^{*}}{R_{0}}\phi+\bigg{(}\mu a^{2}+a^{3}\frac{R^{ *}}{R_{0}}\bigg{)}\bigg{(}\frac{R^{*}}{R_{0}}-1\bigg{)}(R_{0}-1).\]We showed above that \(a_{2}a_{1}=a_{0}\) when \(\phi=\sigma\). Therefore, there exists an \(r=r^{*}\) such that \(\Delta(r^{*})=0\). Taking the partial derivative of \(\Delta\) with respect to \(r\), we obtain:  
\[\frac{\partial\Delta}{\partial r} = \mu\frac{\partial\phi}{\partial r}+\frac{aR^{*}}{R_{0}}\frac{ \partial\phi}{\partial r}+\frac{\phi}{R_{0}}+\frac{\mu a}{R_{0}}(R_{0}-1)+ \frac{a^{2}R^{*}}{R_{0}^{2}}(R_{0}-1)+\frac{a^{2}}{R_{0}}(R_{0}-1)\bigg{(} \frac{R^{*}}{R_{0}}-1\bigg{)}, \tag{21}\] \[a_{2} = \mu+a\frac{R^{*}}{R_{0}},\]0 \[a_{2} = \mu+a\frac{R^{*}}{R_{0}},\]1  
When \(\Delta=0\), from (20), we obtain:  
\[a_{2} = \mu+a\frac{R^{*}}{R_{0}},\]2  
Plugging (22) into (21) yields,  
\[a_{2} = \mu+a\frac{R^{*}}{R_{0}},\]3 \[a_{2} = \mu+a\frac{R^{*}}{R_{0}},\]4 \[a_{2} = \mu+a\frac{R^{*}}{R_{0}},\]5  
Notice that,  
\[a_{2} = \mu+a\frac{R^{*}}{R_{0}},\]6 \[a_{2} = \mu+a\frac{R^{*}}{R_{0}},\]7  
Therefore,  
\[a_{2} = \mu+a\frac{R^{*}}{R_{0}},\]8 \[a_{2} = \mu+a\frac{R^{*}}{R_{0}},\]9  
The two criteria (CH.1) and (CH.2) from [PERSON] and [PERSON] [45] are satisfied.