In this section, we investigate the global stability of the disease-free equilibrium \(E_{f}\) and the chronic infection equilibrium \(E^{*}\). For the global stability of \(E_{f}\), we assume that \(a\geq d\). Biologically, this assumption is often satisfied because *a* represents the death rate of infected cells and includes the possibility of death by bursting of infected cells. Further, this assumption is considered by many authors; see, for example, [23–25]. Therefore, we have the following result.

### Theorem 4.1

*If*
\(R_{0}\leq1\), *then the disease*-*free equilibrium*
\(E_{f}\)
*is globally asymptotically stable*.

### Proof

Construct the Lyapunov functional

$$V(t)=y(t)+\frac{f(\frac{\lambda}{d},0,0)}{u}v(t). $$

Calculating the time derivative of \(V(t)\) along the positive solution of system (1), we have

$$ \dot{V}(t)|_{(\text{1})}= \biggl(f(x,y,v)-f\biggl(\frac {\lambda }{d},0,0 \biggr) \biggr)v+(a+\rho)y \biggl(\frac{kf(\frac{\lambda }{d},0,0)+ug(x,y)}{(a+\rho)u}-1 \biggr). $$

Note that \(\limsup_{t\rightarrow\infty} x(t)\leq\frac {\lambda}{d}\). This yields that all omega limit points satisfy \(x(t)\leq\frac{\lambda}{d}\). Hence, it suffices to consider solutions for which \(x(t)\leq\frac{\lambda}{d}\). Using the expression of \(R_{0}\) given in (3), we get

$$\begin{aligned} \dot{V}(t)|_{\text{(1)}} \leq& \biggl(f(x,0,0)-f\biggl(\frac{\lambda }{d},0,0 \biggr) \biggr)v+(a+\rho) (R_{0}-1 )y \\ \leq&(a+\rho) (R_{0}-1 )y. \end{aligned}$$

Consequently, \(\dot{V}|_{\text{(1)}}\leq0\) for \(R_{0}\leq1\). Further, it is easy to show that the largest compact invariant set in \(\{(x,y,v)| \dot{V}=0\}\) is the singleton \(\{E_{f}\}\). By the LaSalle invariance principle [26], the disease-free equilibrium \(E_{f}\) is globally asymptotically stable for \(R_{0}\leq 1\). □

Now, we will investigate the global dynamics of system (1) when \(R_{0}>1\). Firstly, we need the following lemma.

### Lemma 4.2

*If*
\(R_{0}>1\), *then system* (1) *is uniformly persistent*.

### Proof

This result follows from an application of Theorem 4.3 in [27] with \(X=\mathbb{R}^{3}\) and \(E=\Gamma\). The maximal invariant set *M* on the boundary *∂*Γ is the singleton \(\{E_{f}\}\), and it is isolated. From Theorem 4.3 in [27] we can see that the uniform persistence of system (1) is equivalent to the instability of the disease-free equilibrium \(E_{f}\). On the other hand, we have proved in Theorem 3.1 that \(E_{f}\) is unstable if \(R_{0}>1\). Thus, system (1) is uniformly persistent when \(R_{0}>1\). □

Next, we focus ourselves on the global stability of the chronic infection equilibrium \(E^{*}\) by assuming that \(R_{0}>1\) and the incidence function *f* satisfies the following hypothesis:

$$ (\mathrm{H}_{4})\quad f(x,y,v)+v\frac{\partial f}{\partial v}(x,y,v) \geq0\quad\textit{for all } x\geq0, y \geq0, \textit{and } v\geq0. $$

### Theorem 4.3

*Assume that*
\(R_{0}>1\)
*and* (H_{4}) *hold*. *Then the chronic infection equilibrium*
\(E^{*}\)
*is globally asymptotically stable*.

### Proof

To prove the global stability of \(E^{*}\), we will apply the geometrical approach developed by Li and Muldowney [28].

The second additive compound matrix of the Jacobian matrix *J*, given by (8), is defined by

$$ J^{[2]}=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{}} j_{11}+j_{22}& j_{23}& -j_{13}\\ j_{32}& j_{11}+j_{33} & j_{12}\\ -j_{31} & j_{21} & j_{22}+j_{33} \end{array}\displaystyle \right ), $$

(10)

where \(j_{kl}\) is the \((k,l)\)th entry of the matrix *J*.

We consider the matrix \(P=\operatorname{diag} (1,\frac{y}{v},\frac {y}{v})\). It follows then that

$$P_{f}P^{-1}=\operatorname{diag} \biggl(0,\frac{\dot{y}}{y}- \frac{\dot{v}}{v},\frac {\dot{y}}{y}-\frac{\dot{v}}{v}\biggr), $$

where the matrix \(P_{f}\) is obtained by replacing each entry \(p_{ij}\) of *P* by its derivative in the direction of solution of (1). Furthermore, we have

$$B=P_{f}P^{-1}+PJ^{[2]}P^{-1}=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{}} B_{11} & B_{12} \\ B_{21} & B_{22} \end{array}\displaystyle \right ), $$

where

$$\begin{aligned}& B_{11} =-(a+d+\rho)-v\frac{\partial f}{\partial x}-y\frac{\partial g}{\partial x}+v \frac{\partial f}{\partial y}+y\frac{\partial g}{\partial y}+g(x,y), \\& B_{12} =\Bigl( \textstyle\begin{array}{@{}c@{\quad}c@{}} \frac{v}{y} (v\frac{\partial f}{\partial v}+f(x,y,v) )& \frac{v}{y} (v\frac{\partial f}{\partial v}+f(x,y,v) ) \end{array}\displaystyle \Bigr), \\& B_{21} =\left ( \textstyle\begin{array}{@{}c@{}} \frac{ky}{v} \\ 0 \end{array}\displaystyle \right ), \\& B_{22} =\left ( \textstyle\begin{array}{@{}c@{\quad}c@{}} \frac{\dot{y}}{y}-\frac{\dot{v}}{v}-u-d-v\frac{\partial f}{\partial x}-y\frac{\partial g}{\partial x} & \rho-v\frac{\partial f}{\partial y}-y\frac{\partial g}{\partial y}-g(x,y) \\ v\frac{\partial f}{\partial x}+y\frac{\partial g}{\partial x} & \frac{\dot{y}}{y}-\frac{\dot{v}}{v}-a-\rho-u +v\frac{\partial f}{\partial y}+y\frac{\partial g}{\partial y}+g(x,y) \end{array}\displaystyle \right ). \end{aligned}$$

Define the norm in \(\mathbb{R}^{3}\) as \(|(w_{1},w_{2},w_{3})|=\max\{ |w_{1}|,|w_{2}|+|w_{3}|\}\) for \((w_{1},w_{2},w_{3})\in\mathbb {R}^{3}\). Then the Lozinskii measure *μ* with respect to the norm \(|\cdot|\) can be estimated as follows (see [29]):

$$ \mu(B)\leq\sup\{g_{1},g_{2}\}, $$

(11)

where \(g_{1}=\mu_{1}(B_{11})+|B_{12}|\) and \(g_{2}=|B_{21}|+\mu_{1}(B_{22})\). Here \(\mu_{1}\) denotes the Lozinskii measure with respect to the \(l_{1}\) vector norm, and \(|B_{12}|\) and \(|B_{21}|\) are matrix norms with respect to the \(l_{1}\) norm. Moreover, we have

$$\begin{aligned}& \mu_{1}(B_{11}) = -(a+d+\rho)-v\frac{\partial f}{\partial x}-y \frac{\partial g}{\partial x}+v\frac{\partial f}{\partial y}+y\frac{\partial g}{\partial y}+g(x,y), \\& |B_{12}| = \biggl\vert \frac{v}{y}\biggl(v \frac{\partial f}{\partial v}+f(x,y,v)\biggr)\biggr\vert =\frac{\dot{y}}{y}+a+\rho+ \frac{v^{2}}{y}\frac{\partial f}{\partial v}-g(x,y), \\& |B_{21}| = \frac{ky}{v}=\frac{\dot{v}}{v}+u, \\& \mu_{1}(B_{22}) = \max\biggl\{ \frac{\dot{y}}{y}- \frac{\dot {v}}{v}-u-d,\frac{\dot{y}}{y}-\frac{\dot{v}}{v}-u-a\biggr\} \\& \hphantom{\mu_{1}(B_{22})} = \frac{\dot{y}}{y}-\frac{\dot {v}}{v}-u-\delta. \end{aligned}$$

Hence, we obtain

$$\begin{aligned} g_{1} =& \frac{\dot{y}}{y}-d+\frac{v^{2}}{y} \frac{\partial f}{\partial v}-v\frac{\partial f}{\partial x}-y\frac{\partial g}{\partial x} +v\frac{\partial f}{\partial y}+y \frac{\partial g}{\partial y} \\ \leq&\frac{\dot{y}}{y}-\delta \end{aligned}$$

(12)

and

$$ g_{2} = \frac{\dot{y}}{y}-\delta. $$

(13)

From (11)-(13) we get

$$\mu(B)\leq\frac{\dot{y}}{y}-\delta. $$

From Lemma 4.2 we know that system (1) is uniformly persistent when \(R_{0}>1\). Then there exists a compact absorbing set \(K\subset\Gamma\) [30]. Along each solution \((x(t),y(t),v(t) )\) of (1) with \(X_{0}= (x(0),y(0),v(0) )\in K\), we have

$$\frac{1}{t} \int_{0}^{t}\mu(B)\, ds\leq\frac{1}{t}\ln \biggl(\frac {y(t)}{y(0)} \biggr)-\delta, $$

which implies that

$$\bar{q}_{2}=\limsup_{t\rightarrow\infty}\sup_{X_{0}\in K} \frac {1}{t} \int_{0}^{t}\mu(B)\, ds< -\frac{\delta}{2}< 0. $$

Then, based on Theorem 3.5 of [28], we deduce that the chronic infection equilibrium \(E^{*}\) is globally asymptotically stable. This completes the proof. □