In this section, we focus on the existence and stability of equilibria for system (1). The equilibrium points are obtained by equating the righthand side of each equation in (1) to zero and it is found that system (1) has two nonnegative equilibria.
Let \(( S^{*}, I^{*}, I_{D}^{*}, I_{DA}^{*}, S_{+}^{*}, M^{*} ) \) denote an equilibrium point. Clearly, from the third equation we have \(I_{D}^{*}=\frac{\rho \delta }{\mu_{2}} I^{*}\), then from the fourth equation \(I_{DA}^{*}= \frac{\delta }{\mu_{2}\mu_{3}} [\rho \xi '+(1\rho )\mu_{2} ]I^{*}\). Next, from the second equation we obtain two possibilities: either \(I^{*}=0\) or \(\beta S ^{*} [ 1 + \lambda_{1}\frac{\rho \delta }{\mu_{2}}+ \lambda_{2} \frac{ \delta }{\mu_{2}\mu_{3}} (\rho \xi '+(1\rho )\mu_{2} ) ] =\mu_{1} \). In the first case we subsequently obtain \(M^{*}=0\) from the sixth equation, \(S_{+}^{*}=0\) from the fifth equation, and \(S^{*}=\frac{ \varPi }{d}=:S_{0}\) from the first one, which gives us diseasefree equilibrium (DFE). In the latter case we easily see that \(S^{*}=\frac{ \mu_{1}}{\beta [ 1 + \lambda_{1}\frac{\rho \delta }{\mu_{2}}+ \lambda_{2} \frac{\delta }{\mu_{2}\mu_{3}} (\rho \xi '+(1\rho ) \mu_{2} ) ] }>0 \) and all other coordinates of this equilibrium point are dependent on the value of \(I^{*}\), namely \(S_{+}^{*}=\frac{c}{ \mu_{4}}S^{*}M^{*}\) and \(M^{*}=\frac{\mu \delta }{\mu_{0}\mu_{2}\mu _{3}} [ \rho (\mu_{3}+\xi ')+(1\rho )\mu_{2} ] I^{*}\), which allows to calculate \(I^{*}\) from the first equation. We obtain
$$\begin{aligned}& \varPi \mu_{1}I^{*}dS^{*}d \frac{c}{\mu_{4}}S^{*} \frac{\mu \delta }{\mu_{0}\mu_{2}\mu_{3}} \bigl[ \rho \bigl( \mu_{3} + \xi ' \bigr) +(1\rho )\mu _{2} \bigr] I^{*}=0, \\& \Longrightarrow \quad I^{*}=\frac{\varPi dS^{*}}{\mu_{1}+ \frac{ cd S^{*} \mu \delta }{\mu_{0}\mu_{2}\mu_{3} \mu_{4}} [ \rho (\mu_{3} + \xi ') +(1\rho )\mu_{2} ] }. \end{aligned}$$
It is obvious that \(I^{*}>0\) implies positivity of all other coordinates, and therefore \(S^{*}< S_{0}\) is the condition guaranteeing the existence of positive (endemic) equilibrium (EE).
Diseasefree equilibrium (DFE) and basic reproduction number \(\mathcal{R}_{0}\)
The model has always a diseasefree equilibrium \(E_{0} = (S_{0},0,0,0,0,0) = (\frac{\varPi }{d},0,0,0,0,0 )\). Local stability of \(E_{0}\) is governed by basic reproduction number \(\mathcal{R}_{0}\); cf. [15]. Biologically speaking, \(\mathcal{R}_{0}\) is an average number of new secondary infections generated by a single HIV infected individual, introduced into a susceptible population. The basic reproduction number \(\mathcal{R} _{0}\) could be determined using the next generation approach [15]. Using this approach we need to renumber the model variables in such a way that compartments reflecting infected individuals are at the beginning, so we have \(x=(I, I_{D}, I_{DA}, S, S_{+}, M)^{T}\), with the number of infected compartments equal to 3. Now, by \(\mathcal{X}_{S}\) we denote the set of all diseasefree states, \(\mathcal{X}_{S}=\{x \geq 0 : x_{i}=0 \text{ for } i=1, 2,3 \}\). System (1) shall be written in the form
$$ \dot{x}_{i} = f_{i}(x)= \mathcal{F}_{i}(x) \mathcal{V}_{i}(x), \quad i=1,2,\ldots, 6 , $$
where \(\mathcal{F}_{i}\) describes a rate of appearance of new infections in compartment i, while \(\mathcal{V}_{i}=\mathcal{V}_{i}^{}  \mathcal{V}_{i}^{+}\) and \(\mathcal{V}_{i}^{+}\) is a rate of transfer into the compartment i, \(\mathcal{V}_{i}^{}\) is a rate of transfer out of the compartment i. The following assumptions are to be posed:

(A1) \(\mathcal{F}_{i}(x) \geq 0\), \(\mathcal{V}_{i}^{+}(x) \geq 0\), \(\mathcal{V}_{i}^{}(x) \geq 0\) for any \(x \geq 0\);

(A2) if \(x_{i}=0\), then \(\mathcal{V}_{i}^{}=0\);

(A3) \(\mathcal{F}_{i}=0\) for \(i>3\);

(A4) if \(x \in \mathcal{X}_{S}\), then \(\mathcal{F}_{i}(x)=0\) and \(\mathcal{V}_{i}^{+}(x)=0\) for \(i=1, 2, 3\);

(A5) if \(x_{0}\) is DFE, then eigenvalues of the Jacobi matrix \(Df(x_{0})\) restricted to the subspace \(\mathcal{F}=0\) have all eigenvalues with negative real parts.
According to Lemma 1 in [15]
\mathcal{F}({x}_{0})=\left(\begin{array}{cc}F& 0\\ 0& 0\end{array}\right),\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\mathcal{V}({x}_{0})=\left(\begin{array}{cc}V& 0\\ {J}_{3}& {J}_{4}\end{array}\right),
where F and V are squared matrices of dimension m and \(\mathcal{R}_{0}=\varrho ( F V^{1} ) \) (ϱ denotes a spectral radius). Eventually, according to Theorem 2 in [15], we know that \(x_{0}\) is locally asymptotically stable for \(\mathcal{R}_{0}<1\) and unstable for reverse inequality.
In the case of system (1),
\mathcal{F}=\left(\begin{array}{c}\beta (I+{\lambda}_{1}{I}_{D}+{\lambda}_{2}{I}_{DA})S\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\right)
and
{\mathcal{V}}^{+}=\left(\begin{array}{c}0\\ \rho \delta I\\ (1\rho )\delta I+{\xi}^{\prime}{I}_{D}\\ \Pi +\omega {S}_{+}\\ cSM\\ \mu ({I}_{D}+{I}_{DA})\end{array}\right),\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{\mathcal{V}}^{}=\left(\begin{array}{c}{\mu}_{1}I\\ {\mu}_{2}{I}_{D}\\ {\mu}_{3}{I}_{DA}\\ \beta (I+{\lambda}_{1}{I}_{D}+{\lambda}_{2}{I}_{DA})S+dS+cSM\\ {\mu}_{4}{S}_{+}\\ {\mu}_{0}M\end{array}\right).
It is obvious that \(\mathcal{F}\), \(\mathcal{V}^{+}\), \(\mathcal{V}^{}\) satisfy Assumptions (A1)–(A4). Moreover,
Df(x){}_{\mathcal{F}=0}=\left(\begin{array}{cccccc}{\mu}_{1}& 0& 0& 0& 0& 0\\ \rho \delta & {\mu}_{2}& 0& 0& 0& 0\\ (1\rho )\delta & {\xi}^{\prime}& {\mu}_{3}& 0& 0& 0\\ 0& 0& 0& dcM& \omega & cS\\ 0& 0& 0& cM& {\mu}_{4}& cS\\ 0& \mu & \mu & 0& 0& {\mu}_{0}\end{array}\right),
and all eigenvalues z of \(Df(E_{0}) _{\mathcal{F}=0}\) are real negative. Namely, \(z_{1}=\mu_{1}\), \(z_{2}=\mu_{2}\), \(z_{3}=\mu_{3}\), \(z_{4}=d\), \(z_{5}=\mu_{4}\), \(z_{6}=\mu_{0}\). Therefore, all assumptions posed in [15] are satisfied and \(E_{0}\) is locally stable if \(\mathcal{R}_{0}<1\).
Let us calculate \(\mathcal{R}_{0}\). The matrices F and V for new infection terms and remaining transfer terms are respectively given by
F=\left(\begin{array}{ccc}\frac{\beta \Pi}{d}& {\lambda}_{1}\frac{\beta \Pi}{d}& {\lambda}_{2}\frac{\beta \Pi}{d}\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}V=\left(\begin{array}{ccc}{\mu}_{1}& 0& 0\\ \rho \delta & {\mu}_{2}& 0\\ (1\rho )\delta & {\xi}^{\prime}& {\mu}_{3}\end{array}\right).
The basic reproduction number \(\mathcal{R}_{0}\) is thus given by \(\mathcal{R}_{0}=\varrho (FV^{1}) = \frac{\beta \varPi }{d\mu_{1}} (1+\lambda_{1}\frac{ \rho \delta }{\mu_{2}}+\lambda_{2}\frac{ \delta [\rho \xi '+(1\rho )\mu_{2}]}{\mu_{2}\mu_{3}} )\).
Corollary 3.1
If
\(\frac{\beta \varPi }{d\mu_{1}} (1+\lambda_{1}\frac{ \rho \delta }{\mu_{2}}+\lambda_{2}\frac{\delta [\rho \xi '+(1\rho )\mu_{2}]}{\mu _{2}\mu_{3}} ) <1\), then the diseasefree equilibrium
\(E_{0}\)
is locally asymptotically stable.
Notice that, the inequality \(\mathcal{R}_{0}<1\) is equivalent to \(S^{*}>S_{0}\) which means that the positive equilibrium EE does not exist. On the other hand, if EE exists, then DFE is unstable.
Endemic equilibrium (EE) and its stability
System (1) has an endemic equilibrium \(E^{*}=(S^{*},I^{*},I _{D}^{*},I_{DA}^{*},S_{+}^{*},M^{*})\) with positive coordinates provided that \(\mathcal{R}_{0}>1\). From the analysis presented at the beginning of this section we have
$$\begin{aligned}& S^{*} = \frac{\varPi }{d \mathcal{R}_{0}}, \\& I^{*} = \frac{\varPi (\mathcal{R}_{0}  1)\mu_{0}\mu_{2}\mu_{3}\mu_{4}}{ \mu_{0}\mu_{1}\mu_{2}\mu_{3}\mu_{4} \mathcal{R}_{0}+ \varPi c \mu \delta [ \rho (\mu_{3}+\xi ')+(1\rho )\mu_{2} ]}, \\& I_{D}^{*} = \frac{\rho \delta }{\mu_{2}} I^{*}, \\& I_{DA}^{*} = \frac{\delta }{\mu_{2}\mu_{3}} \bigl[\rho \xi '+(1\rho ) \mu_{2} \bigr]I^{*}, \\& S^{*}_{+} = \frac{\varPi c \mu }{d \mathcal{R}_{0} \mu_{0} \mu_{1} \mu _{4}} \biggl[\rho \delta + \frac{\delta [\xi '+(1\rho )(d+\alpha_{I})]}{ \mu_{3}} \biggr] I^{*}, \\& M^{*} = \frac{\mu }{\mu_{0}\mu_{1}} \biggl[ \rho \delta + \frac{ \delta [\xi '+(1\rho )(d+\alpha_{I})]}{\mu_{3}} \biggr]I^{*}. \end{aligned}$$
Remark
Observe from the expression of \(I^{*}\) that when \(\mathcal{R}_{0}>1\), the endemic equilibrium exists. We thus have the existence condition of the endemic equilibria. It is also important to note that \(\frac{\partial I^{*}}{\partial c}\) and \(\frac{\partial I ^{*}}{\partial \delta }\) are negative. This means, as long as infected individuals become aware and go under treatment, the equilibrium level of unaware and untreated infectives starts to decrease. This shows that the dissemination rate and the treatment rate really have significant input in HIV control.
In order to attain full characterization of the endemic equilibrium \(E^{*}\), we study the asymptotic stability behavior using Lyapunov’s stability theory. If this function has only a single minimum, i.e., an equilibrium point, and it is strictly decreasing along all nonequilibrium solutions, then all solutions tend to the equilibrium point where the scalar function (Lyapunov function) is minimum. The results obtained by performing local and global stability analysis of the obtained equilibria are stated in the following theorems.
Theorem 3.2
The endemic equilibrium
\(E^{*}\)
is locally asymptotically stable (LAS) in
\(\mathcal{D}\), provided the inequalities hold:
$$\begin{aligned}& \psi_{3}\xi ^{\prime 2} < \frac{4}{25} \psi_{2}\mu_{2}\mu_{3}, \end{aligned}$$
(3)
$$\begin{aligned}& \psi_{5}\mu^{2} < \frac{1}{5} \psi_{2} \mu_{0}\mu_{2}, \end{aligned}$$
(4)
$$\begin{aligned}& \psi_{5}\mu^{2} < \frac{1}{5} \psi_{3} \mu_{0}\mu_{3}, \end{aligned}$$
(5)
$$\begin{aligned}& \psi_{4}c^{2} {S^{*}}^{2} < \frac{1}{3} \psi_{5}\mu_{0}\mu_{4}, \end{aligned}$$
(6)
where
\(\psi_{i}\) (\(i=2,3,4,5\)) are chosen so that conditions (35), (36), (37), and (38) are satisfied.
In the next theorem, we show that the endemic equilibrium point \(E^{*}(S^{*},I^{*},I_{D}^{*}, I_{DA}^{*}, S_{+}^{*},M^{*})\) is globally asymptotically stable.
Theorem 3.3
The endemic equilibrium
\(E^{*}\)
is globally asymptotically stable (GAS) in
\(\mathcal{D}\), provided the following inequalities hold:
$$\begin{aligned}& 1\quad (d+\mu_{3})^{2} < \frac{2}{3} \frac{d^{2}\mu_{4}}{cM^{*}}; \\& \begin{aligned} 2\quad& \max \biggl\{ \frac{1}{11} \biggl[ \biggl( \frac{(2d+\alpha_{I})^{2}}{d}+\frac{c \alpha_{I}^{2} M^{*}}{d\mu_{4}} \biggr)\frac{c\alpha_{A}^{2}M^{*}}{\mu _{4}}+ \biggl(( \alpha_{I}+\alpha_{A})^{2} + \frac{(d_{I}+\alpha_{A})^{2} \xi ^{\prime 2}}{(1\rho )^{2} \delta^{2}} \biggr) \biggr]W, \\ &\quad \quad \frac{1}{8}\frac{\lambda^{2}_{1}(2d+d_{I})^{2} \varPi^{2}}{d^{2}(I ^{*}+\lambda_{1} I_{D}^{*}+\lambda_{2} I_{DA}^{*})^{2}} \biggl[\alpha _{A}+ \frac{(d_{I}+\alpha_{A})\mu_{3}}{(1\rho )\delta } \biggr], \frac{1}{8}\frac{\lambda^{2}_{2}(2d+d_{I})^{2} \varPi^{2}}{d^{2}(I^{*}+ \lambda_{1} I_{D}^{*}+\lambda_{2} I_{DA}^{*})^{2}} \biggl[ \alpha_{I} \\ &\quad\quad {} +\frac{(d_{I}+\alpha_{I})\mu_{2}}{\rho \delta } \biggr], \frac{1}{12} \biggl[ \alpha_{I}+\frac{(d_{I}+\alpha_{I})\mu_{2}}{\rho \delta } \biggr] \biggl[\alpha_{A}+ \frac{(d_{I}+\alpha_{A})\mu_{3}}{(1 \rho )\delta } \biggr] \biggr\} \\ &\quad < \frac{1}{36} \biggl[\alpha_{I}+\frac{(d_{I}+ \alpha_{I})\mu_{2}}{\rho \delta } \biggr] \biggl[\alpha_{A}+\frac{(d_{I}+ \alpha_{A})\mu_{3}}{(1\rho )\delta } \biggr]W; \end{aligned} \\& 3\quad \frac{dc\varPi^{2}}{\mu_{0}d^{2}\mu_{4}M^{*}} < \frac{1}{27}\min \biggl\{ \frac{\mu_{0}}{\mu^{2}} \biggl[\alpha_{I}+\frac{(d_{I}+\alpha_{I})\mu _{2}}{\rho \delta } \biggr], \frac{\mu_{0}}{\mu^{2}} \biggl[\alpha_{A}+\frac{(d _{I}+\alpha_{A})\mu_{3}}{(1\rho )\delta } \biggr] \biggr\} , \end{aligned}$$
where
\(W= [d_{I}\beta (I^{*}+\lambda_{1} I_{D}^{*}+\lambda_{2} I _{DA}^{*})+(2d+d_{I}) (\mu_{1} \frac{\varPi \beta }{d} ) ]\).
For the proof of Theorems 3.2 and 3.3 see Appendices 1 and 2, respectively.