3.1 Basic properties
All the mentioned parameters in model (1) are nonnegative because it deals with human population. It can be easily proved that the closed set \(D= \{ ( S_{L},S_{H},P,L,Q,I_{1},I_{2},H,T,R ) \in R_{+}^{10}:N\leq \frac{\pi }{\mu } \} \) is positively-invariant and attracting with respect to model (1).
3.2 Disease-free equilibrium and its stability
The disease-free equilibrium (DFE) of system (1) is given by
$$\begin{aligned} \varepsilon _{0} &= \bigl( S_{L}^{\ast },S_{H}^{\ast }, P^{\ast }, L ^{\ast }, Q^{\ast }, I_{1}^{\ast }, I_{2}^{\ast }, H^{\ast },T^{ \ast },R^{\ast } \bigr) \\ &= \biggl( S_{L}^{\ast },S_{H}^{\ast }, \frac{\sigma _{L}S_{L}^{\ast }+ \sigma _{H} S_{H}^{\ast }}{\mu },0,0,0,0,0,0,0 \biggr), \end{aligned}$$
(3)
with \(S_{L}^{\ast }=\frac{\pi ( 1-P ) }{\sigma _{L}+\mu }\) and \(S_{H}^{\ast }=\frac{\pi P}{\sigma _{H}+\mu }\).
Following the notation given in [27], the nonnegative matrix F consisting of the new infection terms and the matrix V of the progression terms involved in model (1) are given, respectively, by
$$\begin{aligned} &F= \begin{bmatrix} \beta \eta _{1}\varOmega & \beta \eta \varOmega & \beta \varOmega & \beta \eta _{2}\varOmega & \beta \eta _{3}\varOmega & \beta \eta _{4}\varOmega \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix} ,\\ & V= \begin{bmatrix} K_{1} & 0 & 0 & 0 & 0 & 0 \\ -\beta _{1} & K & 0 & 0 & 0 & 0 \\ -\alpha & 0 & K_{2} & 0 & 0 & 0 \\ 0 & 0 & -\gamma & K_{3} & 0 & 0 \\ 0 & -\eta & 0 & -\psi & K_{4} & 0 \\ 0 & 0 & -\tau _{1} & -\tau _{2} & 0 & K_{5} \end{bmatrix} , \end{aligned}$$
where \(K_{1}=\alpha +\beta _{1}+\mu \), \(K=\mu +\eta \), \(K_{2}=\tau _{1}+\gamma +\mu \), \(K_{3}=\tau _{2}+\psi +\phi _{I_{2}}+\mu +\delta \), \(K_{4}=\phi _{H}+\mu +\theta _{1}\delta \), \(K_{5}=\phi _{T}+\mu \), and \(\varOmega =\frac{S_{L}^{\ast }+\theta _{H}S_{H}^{\ast }+\theta _{P}P^{ \ast }}{N^{\ast }}\).
It pursues the control reproduction number, signified by \(R_{C}= \rho ( FV^{-1} )\), which is given as follows:
$$\begin{aligned} R_{C} ={}&\rho \bigl( FV^{-1} \bigr) \\ ={}&\frac{\varOmega \beta }{K_{1}}\biggl(\eta _{1}+\frac{\alpha }{K_{2}}+ \frac{ \beta _{1} \eta }{K}+\alpha \gamma \frac{\eta _{2}}{K_{2}K_{3}}+\frac{ \eta _{4}}{K_{2}K_{3}K_{5}} ( \alpha \gamma \tau _{2}+\alpha \tau _{1}K_{3} ) \\ &{}+\frac{1}{K}\frac{\eta _{3}}{K_{2}K_{3}K_{4}} ( K\alpha \gamma \psi +\eta \beta _{1}K_{2}K_{3} ) \biggr), \end{aligned}$$
where ρ denotes the overwhelming eigenvalue in the absolute value of \(FV^{-1}\). Utilizing Theorem 2 in [27], the accompanying outcome is set up as follows.
Lemma 3.1
The DFE of model (1), given by (3), is locally-asymptotically stable (LAS) if
\(R_{C} < 1\)and unstable if
\(R_{C} > 1\).
The control reproduction number \(R_{C}\) represents the average number of new cases generated by a primary infectious individual in a population where some susceptible individuals receive antiviral prophylaxis. Lemma (3.1) shows that, for \(R_{C}< \)1, the H1N1 pandemic can be removed from the population if the basin of attraction of DFE\((\varepsilon _{0} )\) contains the initial sub-populations. Global stability of the DFE is proved in the following theorem to ensure that illness can be destroyed totally if the control reproduction number is less than one.
Theorem 3.2
The DFE, \(\varepsilon _{0}\), of model (1) is GAS in D if
\(R_{C}\leq R^{\ast }=\frac{\varOmega }{\theta _{H}}\).
Proof
Consider the Lyapunov function
$$\begin{aligned} G(t)=g_{1}L+gQ+g_{2}I_{1} +g_{3}I_{2}+g_{4}H+g_{5}T, \end{aligned}$$
where
$$\begin{aligned} &g_{1} =\eta _{1}K_{2}K_{3}K_{4}K_{5}+ \alpha K_{3}k_{4}K_{5}+\alpha \gamma \eta _{2}K_{4}K_{5}+\alpha \gamma \tau _{2}\eta _{4}K_{4}+\alpha \tau _{1}\eta _{4}K_{3}K_{4} \\ &\phantom{g_{1} =}{}+\alpha \gamma \psi \eta _{3}K_{5}+ \frac{\beta _{1} \eta K_{2}K_{3}K _{4}K_{5}}{K}+\eta _{3}\eta \beta _{1} \frac{K_{5}K_{2}K_{3}}{K}, \\ &g =\frac{\eta \eta _{3}K_{1}K_{2}K_{3}K_{5}}{K}+\eta \frac{K_{1}K_{2}K _{3}K_{4}K_{5}}{K}, \\ &g_{2} =\gamma \tau _{2}\eta _{4}K_{1}K_{4}+ \gamma \psi \eta _{3}K_{1}K _{5}+\gamma \eta _{2}K_{1}K_{4}K_{5}+\tau _{1}\eta _{4}K_{1}K_{3}K_{4}+K _{1}K_{3}K_{4}K_{5}, \\ &g_{3} =\eta _{2}K_{1}K_{2}K_{4}K_{5}+ \psi \eta _{3}K_{1}K_{2}K_{5}+ \tau _{2}\eta _{4}K_{1}K_{2}K_{4}, \\ &g_{4} =\eta _{3}K_{1}K_{2}K_{3}K_{5}, \\ &g_{5} =\eta _{4}K_{1}K_{2}K_{3}K_{4}. \end{aligned}$$
The time derivative of \(G(t)\) is given by
$$\begin{aligned} G^{\prime }(t) ={}&g_{1}L^{\prime }+gQ^{\prime }+g_{2}I_{1}^{\prime } +g _{3}I_{2}^{\prime }+g_{4}H^{\prime }+g_{5}T^{\prime } \\ ={}& g_{1}\bigl(\lambda ( S_{L}+\theta _{H}S_{H}+\theta _{P}P ) -K _{1}L\bigr)+g(\beta _{1}L-KQ)+ g_{2}( \alpha L-K_{2}I_{1})+ g_{3}(\gamma I _{1}-K_{3}I_{2}) \\ &{}+g_{4}(\psi I_{2}+\eta Q-K_{4}H)+g_{5}( \tau _{1}I_{1}+\tau _{2}I_{2}-K _{5}T) \\ ={}&\biggl(\eta _{1}K_{2}K_{3}K_{4}K_{5}+ \alpha K_{3}K_{4}K_{5}+\alpha \gamma \eta _{2}K_{4}K_{5}+\alpha \gamma \tau _{2}K_{4}\eta _{4}+\alpha \tau _{1}K_{3}K_{4}\eta _{4}+ \alpha \gamma \psi \eta _{3}K_{5} \\ &{}+\frac{\beta _{1}\eta K_{2}K_{3}K_{4}k_{5}}{K}+\eta _{3}\eta \beta _{1} \frac{K_{5}}{K}K_{2}K_{3}\biggr)\lambda ( S_{L}+\theta _{H}S_{H}+ \theta _{P}P ) \\ &{}-K_{1}K_{2}K_{3}K_{4}K_{5} ( \eta _{1}L+\eta Q+I_{1}+\eta _{2}I _{2}+\eta _{3}H+\eta _{4}T ) \\ \leq{} & g_{1}\lambda \theta _{H}N-K_{1}K_{2}K_{3}K_{4}K_{5} \frac{ \lambda N}{\beta } \\ \leq{} & K_{1}K_{2}K_{3}K_{4}K_{5} \frac{ \lambda N}{\beta } \biggl( \frac{R _{C}}{R^{\ast }}-1 \biggr). \end{aligned}$$
Thus \(G^{\prime }(t)\leq 0\) if \(R_{C}\leq R^{\ast }\) and \(G^{\prime }(t)=0\) if and only if \(L=Q=I_{1}=I_{2}=H=T=0\). Moreover, the greatest compact invariant set in \(\{ (S_{L},S_{H},P,L,Q,I _{1},I_{2},H,T,R ) \in D:G^{\prime }=0 \} \) is the singleton set \(\{ \varepsilon _{0} \} \). According to LaSalle’s invariance principle [28], every solution to system (1) converges to \(\varepsilon _{0}\), as \(t\rightarrow \infty \). Hence the DFE is globally asymptotically stable. □
3.3 Endemic equilibrium and its stability
In this section, the existence of endemic equilibrium (EE) for system (1) (that is, equilibria where the infected classes are taken nonzero) and its stability are established. Let
$$ E_{1}=\bigl(S_{L}^{\ast \ast },S_{H}^{\ast \ast },P^{\ast \ast },L^{\ast \ast }, Q^{\ast \ast },I_{1}^{\ast \ast },I_{2}^{\ast \ast },H^{ \ast \ast },T^{\ast \ast },R^{\ast \ast } \bigr) $$
be EE of model (1). Further, suppose that
$$\begin{aligned} \lambda ^{\ast \ast }=\frac{\beta ( \eta _{1}L^{\ast \ast }+ \eta Q^{\ast \ast }+I_{1}^{\ast \ast }+\eta _{2}I_{2}^{\ast \ast }+\eta _{3}H^{\ast \ast }+\eta _{4}T^{\ast \ast } ) }{N^{\ast \ast }} \end{aligned}$$
denotes the infection force at the steady-state. By simplifying the model at steady-state, we have
$$\begin{aligned} & S_{L}^{\ast \ast } =\frac{\pi -pi p }{\lambda ^{\ast \ast }+\sigma _{L}+\mu }, \\ &S_{H}^{\ast \ast } =\frac{\pi p}{\theta _{H}\lambda ^{\ast \ast }+ \sigma _{H}+\mu }, \\ &P^{\ast \ast } =\pi \frac{ ( ( 1-p ) \theta _{H} \sigma _{L}+p\sigma _{H} ) \lambda ^{\ast \ast }+ ( 1-p ) \sigma _{L} ( \sigma _{H}+\mu ) +p\sigma _{H} ( \sigma _{L}+\mu ) }{Q}, \\ &L^{\ast \ast } =\frac{\lambda ^{\ast \ast } (\pi \theta _{P}\theta _{H}\lambda ^{\ast \ast 2}+m_{1}\lambda ^{\ast \ast }+m_{2} ) }{K _{1}Q}, \\ &Q^{\ast \ast } =\frac{\lambda ^{\ast \ast }\beta _{1} ( \pi \theta _{P}\theta _{H}\lambda ^{\ast \ast 2}+m_{1}\lambda ^{\ast \ast }+m_{2} ) }{K_{1}KQ}, \\ &I_{1}^{\ast \ast } =\frac{\alpha \lambda ^{\ast \ast } ( \pi \theta _{P}\theta _{H}\lambda ^{\ast \ast 2}+m_{1}\lambda ^{\ast \ast }+m _{2} ) }{K_{1}K_{2}Q}, \\ &I_{2}^{\ast \ast } =\frac{\gamma \alpha \lambda ^{\ast \ast } ( \pi \theta _{P}\theta _{H}\lambda ^{\ast \ast 2}+m_{1}\lambda ^{\ast \ast }+m_{2} ) }{K_{1}K_{2}K_{3}Q}, \\ &H^{\ast \ast } =\frac{\alpha \gamma \psi \lambda ^{\ast \ast } ( \pi \theta _{P}\theta _{H}\lambda ^{\ast \ast 2}+m_{1}\lambda ^{\ast \ast }+m_{2} ) }{ K_{1}k_{2}K_{3}K_{4}Q}+\frac{\lambda ^{\ast \ast }\eta \beta _{1} ( \pi \theta _{P}\theta _{H}\lambda ^{\ast \ast 2}+m_{1}\lambda ^{\ast \ast }+m_{2} ) }{K_{1}KK_{4}Q}, \\ &T^{\ast \ast } =\frac{\tau _{1}\alpha \lambda ^{\ast \ast } ( \pi \theta _{P}\theta _{H}\lambda ^{\ast \ast 2}+m_{1}\lambda ^{\ast \ast }+m_{2} ) }{K_{1}K_{2}K_{5}Q}+\frac{ \tau _{2}\gamma \alpha \lambda ^{\ast \ast } ( \pi \theta _{P}\theta _{H}\lambda ^{\ast \ast 2}+m_{1}\lambda ^{\ast \ast }+m_{2} ) }{K_{1}K_{2}K_{3}K _{5}Q}, \end{aligned}$$
(4)
where
$$\begin{aligned} &Q = \bigl( \lambda ^{\ast \ast }+\sigma _{L}+\mu \bigr) \bigl( \theta _{H}\lambda ^{\ast \ast }+\sigma _{H}+\mu \bigr) \bigl( \theta _{P}\lambda ^{\ast \ast }+\mu \bigr), \\ &m_{1} = \pi \bigl\{ ( 1-p ) \bigl[ \mu \theta _{H}+ ( \sigma _{H}+\mu ) \theta _{P} \bigr] +p\theta _{H} \bigl[ \mu + ( \sigma _{L}+\mu ) \theta _{P} \bigr] +\theta _{P}A \bigr\} , \\ &m_{2} = \pi \bigl\{ ( 1-p ) ( \sigma _{H}+\mu ) \mu +p\theta _{H} ( \sigma _{L}+\mu ) \mu +\theta _{P}B \bigr\} , \\ &A = ( 1-p ) \theta _{H}\sigma _{L}+p\sigma _{H}, \\ &B = ( 1-p ) \sigma _{L} ( \sigma _{H}+\mu ) +p \sigma _{H} ( \sigma _{L}+\mu ). \end{aligned}$$
Using (4) in the expression of \(\lambda ^{\ast \ast }\), we get
$$ a_{0}\lambda ^{\ast \ast 3}+b_{0} \lambda ^{\ast \ast 2}+c_{0} \lambda ^{\ast \ast }+d_{0}=0, $$
(5)
where
$$\begin{aligned} a_{0} ={}&A_{1}A_{8}, \\ b_{0} ={}&\pi \theta _{H}\theta _{P} \biggl( 1-\frac{R_{C}}{\varOmega } \biggr) +\pi p\theta _{P} ( 1-\theta _{H} ) +A_{2}A_{8}, \\ c_{0} ={}&\pi ( 1-p ) \bigl( \theta _{P} ( \sigma _{H}+ \mu ) +\mu \theta _{H} \bigr) \biggl\{ 1- \frac{R_{C}}{\varOmega } \biggr\} +\pi p \bigl( \theta _{P} ( \mu + \sigma _{L}+\mu ) \bigr) \biggl\{ 1-\frac{\theta _{H}R _{C}}{\varOmega } \biggr\} \\ &{}+\pi A_{7} \biggl( 1-\frac{\theta _{P}R_{C}}{\varOmega } \biggr) +A_{3}A _{8}, \\ d_{0} ={}&\pi ( 1-p ) ( \sigma _{H}+\mu ) \biggl\{ \mu \biggl( 1-\frac{R_{C}}{\varOmega } \biggr) +\sigma _{L} \biggl( 1-\theta _{P}\frac{R_{C}}{\varOmega } \biggr) \biggr\} \\ &{}+\pi P ( \sigma _{L}+\mu ) \biggl\{ \mu \biggl( 1-\theta _{H}\frac{R_{C}}{\varOmega } \biggr) +\sigma _{H} \biggl( 1-\theta _{P}\frac{R _{C}}{\varOmega } \biggr) \biggr\} , \\ A_{1} ={}&\pi \theta _{H}\theta _{P}, \\ A_{2} ={}&\pi \bigl\{ ( 1-p ) \bigl[ \mu \theta _{H}+ ( \sigma _{H}+\mu ) \theta _{P} \bigr] +p\theta _{H} \bigl[ \mu + ( \sigma _{L}+\mu ) \theta _{P} \bigr] +\theta _{P}A \bigr\} , \\ A_{3} ={}&\pi \bigl\{ ( 1-p ) ( \sigma _{H}+\mu ) \mu +p\theta _{H} ( \sigma _{L}+\mu ) \mu +\theta _{P}B \bigr\} , \\ A_{7} ={}& ( 1-p ) \theta _{H}\sigma _{L}+p\sigma _{H}, \\ A_{8} ={}& \frac{1}{K_{1}}+\frac{\beta _{1}}{K_{1}K}+ \frac{\alpha }{ K _{1}K_{2}}+\frac{\gamma \alpha }{K_{1}K_{2}K_{3}}+\frac{\alpha \gamma \psi }{K_{1}K_{2}K_{3}K_{4}}+ \frac{\eta \beta _{1}}{K_{1}KK_{4}}+\frac{ \tau _{1}\alpha }{K_{1}K_{2}K_{5}} \\ &{}+\frac{\tau _{2}\gamma \alpha }{ K_{1}K_{2}K_{3}K_{5}}+\frac{\gamma \alpha \phi _{I_{2}}}{\mu K_{1}K_{2}K_{3}} +\frac{\phi _{H}\alpha \gamma \psi }{\mu K_{1}K_{2}K_{3}K_{4}}+ \frac{\phi _{H}\eta \beta _{1}}{ \mu K_{1}KK_{4}}+\frac{\phi _{T}\tau _{1}\alpha }{\mu K_{1}K_{2}K_{5}}\frac{ \phi _{T}\tau _{2}\gamma \alpha }{\mu K_{1}K_{2}K_{3}K_{5}}. \end{aligned}$$
It can be easily verified that coefficients of Eq. (5) are positive when \(\frac{R_{C}}{\varOmega }<\frac{1}{\theta _{H}}\), then by Descarte’s rule of sign, there is no positive root. When \(\frac{R_{C}}{ \varOmega }>\frac{1}{\theta _{P}}\), all the coefficients of Eq. (5) are positive other than \(d_{0} \), thus in this case, the sign changes only once. Hence, we conclude the above discussion as follows.
Theorem 3.3
If
\(\frac{R_{C}}{\varOmega }>\frac{1}{\theta _{P}}\), then there exists one and only one EE for system (1). However, in the case of
\(\frac{R_{C}}{\varOmega }<\frac{1}{\theta _{H}}\), no EE exists.
Now, the global stability of EE calculated for model (1) is given for the exceptional situation where the disease-induced fatality is negligible. It is noted that the setting \(\delta =0\) implies \(N\rightarrow \frac{\pi }{\mu }\) as \(t\rightarrow \infty \). Using \(N=\frac{\pi }{\mu }\) gives \(\lambda ={\beta _{2}}(\eta _{1} L+\eta Q+I _{1}+\eta _{2}I_{2}+\eta _{3} H+\eta _{4} T)\), where \(\beta _{2}=\beta \frac{ \mu }{\pi }\). Consider the accompanying change of variables: \(\frac{S_{L}}{S_{L}^{\ast }}=x_{1},\frac{S_{H}}{S_{H}^{\ast }}=x_{2},\frac{P}{P ^{\ast }}=x_{3},\frac{L}{L^{\ast }}=x_{4},\frac{Q}{Q^{\ast }}=x_{5}\), \(\frac{I_{1}}{I_{1}^{\ast }}=x_{6},\frac{I_{2}}{I_{2}^{\ast }}=x_{7},\frac{H}{ H ^{\ast }}=x_{8},\frac{T}{T^{\ast }}=x_{9},\frac{R}{R^{\ast }}=x_{10}\).
The Lyapunov function for the sub-framework comprising the initial nine equations of (1) is as follows:
$$\begin{aligned} \mathtt{L(t)} ={}&a_{1} ( x_{1}-1-\log x_{1} ) +a_{2} ( x_{2}-1-\log x_{2} ) +a_{3} ( x_{3}-1-\log x_{3} )\\ &{} +a _{4} ( x_{4}-1-\log x_{4} )+a_{5} ( x_{5}-1-\log x_{5} ) +a_{6} ( x_{6}-1- \log x_{6} ) \\ &{}+a_{7} ( x_{7}-1-\log x_{7} ) +a_{8} ( x_{8}-1-\log x_{8} ) +a_{9} ( x_{9}-1-\log x_{9} ), \end{aligned}$$
where \(a_{i}\ (i=1,2,\ldots,9)\) are constants and their values are found later. Now, differentiating L w.r.t. \(time\) along the solutions of (1), we have
$$\begin{aligned} \mathtt{L}^{\prime } ={}&a_{1}\frac{\pi }{S_{L}^{\ast }} ( 1-p ) \biggl( 2-\frac{1}{x_{1}}-x_{1} \biggr) +a_{1} \beta _{2} \eta _{1}L^{\ast } ( x_{1}+x_{4}-x_{1}x_{4}-1 ) \\ &{}+a_{1}\beta _{2}I_{1}^{\ast } ( x_{1}+x_{6}-x_{1}x_{6}-1 ) + a_{1}\beta _{2}\eta _{2}I_{2}^{\ast } ( x_{1}+x_{7}-x_{1}x_{7}-1 ) \\ &{}+a_{1}\beta _{2}\eta Q^{\ast } ( x_{1}+x_{5}-x_{1}x_{5}-1 ) +a_{1}\beta _{2}\eta _{3}H^{\ast } ( x_{1}+x_{8}-x_{1}x_{8}-1 ) \\ &{}+a_{1}\beta _{2}\eta _{4}T^{\ast } ( x_{1}+x_{9}-x_{1}x_{9}-1 ) +a_{2}\frac{\pi p}{S_{H}^{\ast }} \biggl( 2-\frac{1}{x_{2}}-x_{2} \biggr)\\ &{} +a_{2}\beta _{2}\eta _{1}L^{ \ast } \theta _{H} ( x_{2}+x_{4}-x_{2}x_{4}-1 ) \\ &{} +a_{2}\beta _{2}I_{1}^{\ast } \theta _{H} ( x_{2}+x_{6}-x_{2}x _{6}-1 ) +a_{2}\beta _{2}\eta _{2}I_{2}^{\ast }\theta _{H} ( x _{2}+x_{7}-x_{2}x_{7}-1 ) \\ &{} +a_{2}\beta _{2}\eta Q^{\ast }\theta _{H} ( x_{2}+x_{5}-x_{2}x _{5}-1 ) +a_{2}\beta _{2}\eta _{3}H^{\ast }\theta _{H} ( x _{2}+x_{8}-x_{2}x_{8}-1 ) \\ &{} +a_{2}\beta _{2}\eta _{4}T^{\ast } \theta _{H} ( x_{2}+x_{9}-x _{2}x_{9}-1 ) +a_{3}\sigma _{H}\frac{S_{H}^{\ast }}{P^{\ast }} \biggl( x_{2}-x_{3}- \frac{x_{2}}{x_{3}}+1 \biggr) \\ &{}+a_{3}\sigma _{L} \frac{S _{L}^{\ast }}{P^{\ast }} \biggl( x_{1}-x_{3}- \frac{x_{1}}{x_{3}}+1 \biggr) \\ &{} +a_{3}\beta _{2}\eta _{1}L^{\ast } \theta _{P} ( x_{3}+x_{4}-x _{3}x_{4}-1 ) +a_{3}\beta _{2}I_{1}^{\ast }\theta _{P} ( x _{3}+x_{6}-x_{3}x_{6}-1 ) \\ &{} +a_{3}\beta _{2}\eta _{2}I_{2}^{\ast } \theta _{P} ( x_{3}+x_{7}-x _{3}x_{7}-1 ) +a_{3}\beta _{2}\eta Q^{\ast }\theta _{P} ( x _{3}+x_{5}-x_{3}x_{5}-1 ) \\ &{} +a_{3}\beta _{2}\eta _{3}H^{\ast } \theta _{P} ( x_{3}+x_{8}-x _{3}x_{8}-1 ) +a_{3}\beta _{2}\eta _{4}T^{\ast }\theta _{P} ( x_{3}+x_{9}-x_{3}x_{9}-1 ) \\ &{} +a_{4}\beta _{2}\eta _{1}S_{L}^{\ast } ( x_{1}x_{4}-x_{4}-x_{1}+1 ) +a_{4}\beta _{2}\eta _{1}\theta _{P} P^{\ast } ( x_{3}x_{4}-x_{4}-x_{3}+1 ) \\ & {}+a_{4}\beta _{2}\eta _{1}\theta _{H}S_{H}^{\ast } ( x_{2}x_{4}-x _{4}-x_{2}+1 ) +a_{4}\beta _{2}I_{1}^{\ast }\frac{S_{L}^{\ast }}{L ^{\ast }} \biggl( x_{1}x_{6}-x_{4}-\frac{x_{1}}{x_{4}}x_{6}+1 \biggr) \\ & {}+a_{4}\beta _{2}\theta _{H}I_{1}^{\ast } \frac{S_{H}^{\ast }}{L^{ \ast }} \biggl( x_{2}x_{6}-x_{4}- \frac{x_{2}}{x_{4}}x_{6}+1 \biggr) +a _{4}\beta _{2}\eta _{2}I_{2}^{\ast } \frac{S_{L}^{\ast }}{L^{\ast }} \biggl( x_{1}x_{7}-x_{4}- \frac{x_{1}}{x_{4}}x_{7}+1 \biggr) \\ & {}+a_{4}\beta _{2}\theta _{P} \frac{I_{1}^{\ast }}{L^{\ast }}P^{\ast } \biggl( x_{3}x_{6}-x_{4}- \frac{x_{3}}{x_{4}}x_{6}+1 \biggr) +a_{4} \eta \beta _{2}\frac{S_{L}^{\ast }}{L^{\ast }}Q^{\ast } \biggl( x_{1}x _{5}-x_{4}-\frac{x_{1}}{x_{4}}x_{5}+1 \biggr) \\ & {}+a_{4}\beta _{2}\eta _{3}S_{L}^{\ast } \frac{H^{\ast }}{L^{\ast }} \biggl( x_{1}x_{8}-x_{4}- \frac{x_{1}}{x_{4}}x_{8}+1 \biggr) +a_{4} \beta _{2}\eta _{4}\frac{S_{L}^{\ast }}{L^{\ast }}T^{\ast } \biggl( x _{1}x_{9}-x_{4}- \frac{x_{1}}{x_{4}}x_{9}+1 \biggr) \\ & {}+a_{4}\eta \beta _{2}\theta _{H} \frac{S_{H}^{\ast }}{L^{\ast }}Q^{ \ast } \biggl( x_{2}x_{5}-x_{4}- \frac{x_{2}}{x_{4}}x_{5}+1 \biggr) \\ &{}+a _{4}\beta _{2}\eta _{3} \theta _{H}S_{H}^{\ast } \frac{H^{\ast }}{L^{\ast }} \biggl( x_{2}x_{8}-x _{4}-\frac{x_{2}}{x_{4}}x_{8}+1 \biggr) \\ &{} +a_{4}\beta _{2}\eta _{4}\theta _{H}\frac{S_{H}^{\ast }}{L^{\ast }}T ^{\ast } \biggl( x_{2}x_{9}-x_{4}-\frac{x_{2}}{x_{4}}x_{9}+1 \biggr) \\ &{}+a_{4}\beta _{2}\eta _{2}\theta _{H}I_{2}^{\ast }\frac{S_{H}^{\ast }}{L ^{\ast }} \biggl( x_{2}x_{7}-x_{4}-\frac{x_{2}}{x_{4}}x_{7}+1 \biggr) \\ &{} +a_{4}\eta \beta _{2}\frac{\theta _{P}}{L^{\ast }} P^{\ast }Q^{\ast } \biggl( x_{3}x_{5}-x_{4}- \frac{x_{3}}{x_{4}}x_{5}+1 \biggr) \\ &{}+ a_{4}\beta _{2} \eta _{3}\theta _{P} \frac{H^{\ast }}{L^{\ast }} P^{\ast } \biggl( x_{3}x_{8}-x_{4}- \frac{x_{3}}{x_{4}}x_{8}+1 \biggr) \\ &{} +a_{4}\beta _{2}\eta _{4} \frac{\theta _{P}}{L^{\ast }}P^{\ast } T^{\ast } \biggl( x_{3}x_{9}-x_{4}-\frac{x_{3}}{x_{4}}x_{9}+1 \biggr)\\ &{} +a _{4} \beta _{2}\eta _{2}\theta _{P}\frac{I_{2}^{\ast }}{L^{\ast }}P^{\ast } \biggl( x_{3}x_{7}-x_{4}-\frac{x_{3}}{x_{4}}x_{7}+1 \biggr) \\ &{} +a_{5}\beta _{1}\frac{L^{\ast }}{Q^{\ast }} \biggl[ x_{4}-x_{5}-\frac{x _{4}}{x_{5}}+1 \biggr] +a_{6}\frac{\alpha }{I_{1}^{\ast }}L^{\ast } \biggl[ x_{4}-x_{6}-\frac{x_{4}}{x_{6}}+1 \biggr] \\ &{}+a_{7}\gamma \frac{I _{1}^{\ast }}{I_{2}^{\ast }} \biggl[ x_{6}-x_{7}- \frac{x_{6}}{x_{7}}+1 \biggr] \\ &{} +a_{8} \biggl[ \frac{\eta }{H^{\ast }}Q^{\ast } \biggl( x_{5}-x_{8}-\frac{x _{5}}{x_{8}}+1 \biggr) +\psi \frac{I_{2}^{\ast }}{H^{\ast }} \biggl( x _{7}-x_{8}- \frac{x_{7}}{x_{8}}+1 \biggr) \biggr] \\ &{} +a_{9} \biggl[ \tau _{1}\frac{I_{1}^{\ast }}{T^{\ast }} \biggl( x_{6}-x _{9}-\frac{x_{6}}{x_{9}}+1 \biggr) + \tau _{2}\frac{I_{2}^{\ast }}{T ^{\ast }} \biggl( x_{7}-x_{9}- \frac{x_{7}}{x_{9}}+1 \biggr) \biggr]. \end{aligned}$$
In order to find the values of \(a_{i}\ (i=1,2,\ldots,9)\), putting the coefficients of \(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6}, x_{7},x_{8}, x_{9},x_{1}x_{4},x_{1}x_{5},x_{1}x_{6},x_{1}x_{7},x_{1}x_{8},x_{1}x _{9},x_{2}x_{4},x_{2}x_{5},x_{2}x_{6},x_{2}x_{7},x_{2}x_{8},x_{2}x _{9},x_{3}x_{4},x_{3}x_{5},x_{3}x_{6}, x_{3} x_{7}, x_{3}x_{8},x_{3}x _{9} \) equal to zero, we get
$$\begin{aligned} &a_{1} = a_{4}\frac{S_{L}^{\ast }}{L^{\ast }},\qquad a_{2}=a_{4} \frac{S_{H} ^{\ast }}{L^{\ast }},\qquad a_{3}=a_{4} \frac{P^{\ast }}{L^{\ast }}, \\ &a_{5} = \frac{Q^{\ast }}{\beta _{1}L^{\ast }} \biggl( a_{1}\beta _{2} \eta Q^{\ast }+a_{2}\beta _{2}\eta Q^{\ast }\theta _{H}+a_{3} \beta _{2} \eta Q^{\ast }\theta _{P}+a_{8} \frac{\eta }{H^{\ast }}Q^{\ast } \biggr), \\ &a_{6} = \frac{I_{1}^{\ast }}{\alpha L^{\ast }} \biggl( a_{1}\beta _{2}I _{1}^{\ast }+a_{2}\beta _{2}I_{1}^{\ast }\theta _{H}+a_{3} \beta _{2}I _{1}^{\ast }\theta _{P}+a_{7}\gamma \frac{I_{1}^{\ast }}{I_{2}^{\ast }}+a_{9} \tau _{1}\frac{I_{1}^{\ast }}{T ^{\ast }} \biggr), \\ &a_{7} = \frac{I_{2}^{\ast }}{\gamma I_{1}^{\ast }} \biggl( a_{1}\beta _{2} \eta _{2}I_{2}^{\ast }+a_{2} \beta _{2}\eta _{2}I_{2}^{\ast } \theta _{H}+a _{3}\beta _{2}\eta _{2}I_{2}^{\ast }\theta _{P}+a_{8} \psi \frac{I_{2} ^{\ast }}{H^{\ast }}+a_{9}\tau _{2} \frac{I_{2}^{\ast }}{T^{\ast }} \biggr), \\ &a_{8} = \frac{1}{ ( \frac{\eta }{H^{\ast }}Q^{\ast }+\psi \frac{I _{2}^{\ast }}{H^{\ast }} ) } \bigl( a_{1}\beta _{2}\eta _{3}H^{ \ast }+a_{2}\beta _{2}\eta _{3}H^{\ast }\theta _{H}+a_{3}\beta _{2}\eta _{3}H ^{\ast }\theta _{P} \bigr), \\ &a_{9} = \frac{1}{ ( \tau _{1}\frac{I_{1}^{\ast }}{T^{\ast }}+\tau _{2}\frac{I_{2}^{\ast }}{T^{\ast }} ) } \bigl( a_{1}\beta _{2} \eta _{4}T^{\ast }+a_{2}\beta _{2}\eta _{4}T^{\ast }\theta _{H}+a_{3}\beta _{2} \eta _{4}T^{\ast }\theta _{P} \bigr). \end{aligned}$$
Now
$$\begin{aligned} \mathtt{L}^{\prime } ={}&a_{1}\frac{\pi }{S_{L}^{\ast }} ( 1-p ) \biggl( 1-\frac{1}{x_{1}} \biggr) +a_{1} ( \mu + \sigma _{L} ) +a_{2}\frac{\pi p}{S_{H}^{\ast }} \biggl( 1- \frac{1}{x _{2}} \biggr) +a_{2} ( \sigma _{H}+\mu ) \\ &{}+a_{3}\sigma _{H}\frac{S_{H}^{\ast }}{P^{\ast }} \biggl( 1- \frac{x _{2}}{x_{3}} \biggr) +a_{3}\sigma _{L} \frac{S_{L}^{\ast }}{P^{\ast }} \biggl( 1-\frac{x_{1}}{x_{3}} \biggr) +a_{3} \mu +a_{4}\beta _{2}\eta _{1}S_{L}^{\ast }+a_{4} \beta _{2}\eta _{1}\theta _{P} P^{\ast } \\ &{}+a_{4}\beta _{2}\eta _{1}\theta _{H}S_{H}^{\ast }+a_{4}\beta _{2}I_{1} ^{\ast }\frac{S_{L}^{\ast }}{L^{\ast }} \biggl( 1-\frac{x_{1}}{x_{4}}x _{6} \biggr) +a_{4}\beta _{2}\theta _{H}I_{1}^{\ast } \frac{S_{H}^{ \ast }}{L^{\ast }} \biggl( 1-\frac{x_{2}}{x_{4}}x_{6} \biggr) \\ &{}+a_{4}\beta _{2}\eta _{2}I_{2}^{\ast } \frac{S_{L}^{\ast }}{L^{\ast }} \biggl( 1-\frac{x_{1}}{x_{4}}x_{7} \biggr) +a_{4}\beta _{2}\theta _{P} \frac{I _{1}^{\ast }}{L^{\ast }}P^{\ast } \biggl( 1-\frac{x_{3}}{x_{4}}x_{6} \biggr) +a_{4}\eta \beta _{2}\frac{S_{L} ^{\ast }}{L^{\ast }}Q^{\ast } \biggl( 1-\frac{x_{1}}{x_{4}}x_{5} \biggr) \\ &{}+a_{4}\beta _{2}\eta _{3}S_{L}^{\ast } \frac{H^{\ast }}{L^{\ast }} \biggl( 1-\frac{x_{1}}{x_{4}}x_{8} \biggr) +a_{4}\beta _{2}\eta _{4} \frac{S _{L}^{\ast }}{L^{\ast }}T^{\ast } \biggl( 1-\frac{x_{1}}{x_{4}}x_{9} \biggr) \\ &{}+a_{4}\eta \beta _{2}\theta _{H} \frac{S_{H}^{\ast }}{L^{\ast }}Q^{ \ast } \biggl( 1-\frac{x_{2}}{x_{4}}x_{5} \biggr) +a_{4}\beta _{2}\eta _{3} \theta _{H}S_{H}^{\ast }\frac{H^{\ast }}{L^{\ast }} \biggl( 1-\frac{x _{2}}{x_{4}}x_{8} \biggr) \\ &{}+a_{4}\beta _{2}\eta _{4}\theta _{H}\frac{S_{H}^{\ast }}{L^{\ast }}T ^{\ast } \biggl( 1- \frac{x_{2}}{x_{4}}x_{9} \biggr) +a_{4}\beta _{2}\eta _{2}\theta _{H}I_{2}^{\ast } \frac{S_{H}^{\ast }}{L ^{\ast }} \biggl( 1-\frac{x_{2}}{x_{4}}x_{7} \biggr) \\ &{}+a_{4}\eta \beta _{2}\frac{\theta _{P}}{L^{\ast }} P^{\ast }Q^{\ast } \biggl( 1-\frac{x_{3}}{x_{4}}x_{5} \biggr) \\ &{}+ a_{4}\beta _{2}\eta _{3}\theta _{P}\frac{H^{\ast }}{L^{\ast }} P^{\ast } \biggl( 1- \frac{x_{3}}{x_{4}}x_{8} \biggr) + a_{4}\beta _{2}\eta _{4}\frac{\theta _{P}}{L^{\ast }}P^{\ast } T^{\ast } \biggl( 1-\frac{x_{3}}{x_{4}}x_{9} \biggr) \\ &{}+a_{4} \beta _{2}\eta _{2}\theta _{P}\frac{I_{2}^{\ast }}{L^{\ast }}P^{\ast } \biggl( 1- \frac{x_{3}}{x_{4}}x_{7} \biggr) \\ &{}+a_{5}\beta _{1}\frac{L^{\ast }}{Q^{\ast }} \biggl[ 1- \frac{x_{4}}{x _{5}} \biggr] +a_{6}\frac{\alpha }{I_{1}^{\ast }}L^{\ast } \biggl[ 1-\frac{x _{4}}{x_{6}} \biggr] +a_{7}\gamma \frac{I_{1}^{\ast }}{I_{2}^{\ast }} \biggl[ 1-\frac{x_{6}}{x_{7}} \biggr] \\ &{}+a_{8} \biggl[ \frac{\eta }{H^{\ast }}Q^{\ast } \biggl( 1- \frac{x_{5}}{x _{8}} \biggr) +\psi \frac{I_{2}^{\ast }}{H^{\ast }} \biggl( 1- \frac{x _{7}}{x_{8}} \biggr) \biggr] \\ &{}+a_{9} \biggl[ \tau _{1}\frac{I_{1}^{ \ast }}{T^{\ast }} \biggl( 1-\frac{x_{6}}{x_{9}} \biggr) + \tau _{2}\frac{I _{2}^{\ast }}{T^{\ast }} \biggl( 1-\frac{x_{7}}{x_{9}} \biggr) \biggr] \\ ={}&P ( x_{1},x_{2},\ldots,x_{9} ). \end{aligned}$$
Let us construct the following function to show that \(\mathtt{L}^{ \prime }\leq 0\) in D, which is a positively invariant region:
$$\begin{aligned} \mathcal{U} ( x_{1},x_{2},\ldots,x_{9} ) = \sum_{k=1}^{15}U _{k} (x_{1},x_{2},\ldots,x_{9} ), \end{aligned}$$
where
$$\begin{aligned} &U_{1} = d_{1} \biggl( 3-\frac{1}{x_{1}}- \frac{x_{1}}{x_{4}}x_{6}-\frac{x _{4}}{x_{6}} \biggr), \\ &U_{2} = d_{2}\biggl(4-\frac{1}{x_{2}}- \frac{x_{2}}{x_{3}}-\frac{x_{3}}{x _{4}}x_{6}- \frac{x_{4}}{x_{6}}\biggr), \\ &U_{3} = d_{3}\biggl(4-\frac{x_{1}}{x_{3}}- \frac{1}{x_{1}}-\frac{x_{3}}{x _{4}}x_{5}- \frac{x_{4}}{x_{5}}\biggr), \\ &U_{4} = d_{4} \biggl( 4-\frac{x_{1}}{x_{4}}x_{9}- \frac{1}{x_{1}}-\frac{x _{4}}{x_{6}}-\frac{x_{6}}{x_{9}} \biggr), \\ &U_{5} = d_{5} \biggl( 4-\frac{x_{1}}{x_{4}}x_{7}- \frac{1}{x_{1}}-\frac{x _{4}}{x_{6}}-\frac{x_{6}}{x_{7}} \biggr), \\ &U_{6} = d_{6} \biggl( 3-\frac{x_{2}}{x_{4}}x_{6}- \frac{1}{x_{2}}-\frac{x _{4}}{x_{6}} \biggr), \\ &U_{7} = d_{7} \biggl( 3-\frac{x_{1}}{x_{4}}x_{5}- \frac{1}{x_{1}}-\frac{x _{4}}{x_{5}} \biggr), \\ &U_{8} = d_{8} \biggl( 4-\frac{x_{1}}{x_{4}}x_{8}- \frac{1}{x_{1}}-\frac{x _{5}}{x_{8}}-\frac{x_{4}}{x_{5}} \biggr), \\ &U_{9} = d_{9} \biggl( 3-\frac{x_{2}}{x_{4}}x_{5}- \frac{1}{x_{2}}-\frac{x _{4}}{x_{5}} \biggr), \\ &U_{10} = d_{10}\biggl(5-\frac{x_{2}}{x_{4}}x_{8}- \frac{x_{7}}{x_{8}}-\frac{1}{x _{2}}-\frac{x_{6}}{x_{7}}- \frac{x_{4}}{x_{6}}\biggr), \\ &U_{11} = d_{11}\biggl(6-\frac{x_{3}}{x_{4}}x_{8}- \frac{x_{2}}{x_{3}}-\frac{1}{x _{2}}-\frac{x_{7}}{x_{8}}- \frac{x_{6}}{x_{7}}-\frac{x_{4}}{x_{6}}\biggr), \\ &U_{12} = d_{12}\biggl(6-\frac{x_{3}}{x_{4}}x_{9}- \frac{x_{2}}{x_{3}}-\frac{1}{x _{2}}-\frac{x_{7}}{x_{9}}- \frac{x_{6}}{x_{7}}-\frac{x_{4}}{x_{6}}\biggr), \\ &U_{13} = d_{13}\biggl(5-\frac{x_{3}}{x_{4}}x_{7}- \frac{x_{2}}{x_{3}}-\frac{1}{x _{2}}-\frac{x_{6}}{x_{7}}- \frac{x_{4}}{x_{6}}\biggr), \\ &U_{14} = d_{14}\biggl(5-\frac{x_{2}}{x_{4}}x_{9}- \frac{1}{x_{2}}-\frac{x _{7}}{x_{9}}-\frac{x_{6}}{x_{7}}- \frac{x_{4}}{x_{6}}\biggr), \\ &U_{15} = d_{15} \biggl( 4-\frac{x_{2}}{x_{4}}x_{7}- \frac{1}{x_{2}}-\frac{x _{6}}{x_{7}}-\frac{x_{4}}{x_{6}} \biggr). \end{aligned}$$
Comparison of the same terms between \(P (x_{1},x_{2},x_{3},x_{4},x _{5},x_{6},x_{7},x_{8},x_{9} )\) and
$$ \sum_{k=1}^{15}U_{k} ( x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x _{8},x_{9} ) $$
yields the following equations:
$$\begin{aligned} &d_{1}=a_{4}\beta _{2}I_{1}^{\ast } \frac{S_{L}^{\ast }}{L^{\ast }},\qquad d _{2}=a_{4}\beta _{2}\theta _{P}\frac{I_{1}^{\ast }}{L^{\ast }}P^{\ast },\qquad d _{3}=a_{4}\eta \beta _{2} \frac{\theta _{P}}{L^{\ast }} P^{\ast }Q^{\ast },\\ & d_{4}=a_{4} \beta _{2}\eta _{4}\frac{S_{L}^{\ast }}{L ^{\ast }}T^{\ast },\qquad d_{5}=a_{4} \beta _{2}\eta _{2}I_{2}^{\ast } \frac{S _{L}^{\ast }}{L^{\ast }}, \\ &d_{6}=a_{4}\beta _{2}\theta _{H}I_{1}^{\ast }\frac{S_{H}^{\ast }}{L^{ \ast }},\qquad d_{7}=a_{4}\eta \beta _{2} \frac{S_{L}^{\ast }}{L^{\ast }}Q ^{\ast },\qquad d_{8}=a_{4} \beta _{2}\eta _{3}S_{L}^{\ast } \frac{H^{\ast }}{L ^{\ast }},\\ & d_{9}=a_{4}\eta \beta _{2}\theta _{H}\frac{S_{H}^{\ast }}{L ^{\ast }}Q^{\ast },\qquad d_{10}=a_{4} \beta _{2}\eta _{3} \theta _{H}S_{H}^{\ast } \frac{H^{\ast }}{L^{\ast }}, \\ &d_{11}=a_{4}\beta _{2}\eta _{3}\theta _{P}\frac{H^{\ast }}{L^{\ast }} P^{\ast },\qquad d_{12}=a_{4}\beta _{2}\eta _{4}\frac{\theta _{P}}{L^{\ast }}P ^{\ast } T^{\ast },\qquad d_{13}=a_{4} \beta _{2}\eta _{2}\theta _{P} \frac{I_{2}^{\ast }}{L^{\ast }}P^{\ast },\\ & d _{14}=a_{4}\beta _{2}\eta _{4}\theta _{H} \frac{S_{H}^{\ast }}{L^{\ast }}T ^{\ast }, \qquad d_{15}=a_{4}\beta _{2}\eta _{2}\theta _{H}I_{2}^{\ast } \frac{S_{H}^{ \ast }}{L^{\ast }}. \end{aligned}$$
Thus
$$\begin{aligned} \mathtt{L}^{\prime } ={}&a_{4}\beta _{2}I_{1}^{\ast } \frac{S_{L}^{\ast }}{L ^{\ast }} \biggl( 3-\frac{1}{x_{1}}-\frac{x_{1}}{x_{4}}x_{6}- \frac{x _{4}}{x_{6}} \biggr)+ a_{4}\beta _{2}\theta _{P}\frac{I_{1}^{\ast }}{L ^{\ast }}P^{\ast }\biggl(4- \frac{1}{x_{2}}-\frac{x_{2}}{x_{3}}-\frac{x_{3}}{x _{4}}x_{6}- \frac{x_{4}}{x_{6}}\biggr) \\ &{}+ a_{4}\eta \beta _{2}\frac{\theta _{P}}{L^{\ast }} P^{\ast }Q^{\ast }\biggl(4-\frac{x_{1}}{x_{3}}- \frac{1}{x_{1}}-\frac{x_{3}}{x _{4}}x_{5}- \frac{x_{4}}{x_{5}}\biggr)\\ &{}+ a_{4}\beta _{2}\eta _{4}\frac{S_{L} ^{\ast }}{L^{\ast }} T^{\ast } \biggl( 4- \frac{x_{1}}{x_{4}}x_{9}-\frac{1}{x _{1}}- \frac{x_{4}}{x_{6}} -\frac{x_{6}}{x_{9}} \biggr) \\ &{}+ a_{4}\beta _{2}\eta _{2}I_{2}^{\ast } \frac{S_{L}^{\ast }}{ L^{ \ast }} \biggl( 4-\frac{x_{1}}{x_{4}}x_{7}- \frac{1}{x_{1}}-\frac{x_{4}}{x _{6}} -\frac{x_{6}}{x_{7}} \biggr)+ a_{4}\beta _{2}\theta _{H}I_{1}^{ \ast } \frac{S_{H}^{\ast }}{ L^{\ast }} \biggl( 3-\frac{x_{2}}{x_{4}}x _{6}- \frac{1}{x_{2}}-\frac{x_{4}}{x_{6}} \biggr) \\ &{}+ a_{4}\eta \beta _{2}\frac{S_{L}^{\ast }}{L^{\ast }}Q^{\ast } \biggl( 3-\frac{x_{1}}{x_{4}}x_{5}-\frac{1}{x_{1}}- \frac{x_{4}}{x_{5}} \biggr)+ a_{4}\beta _{2}\eta _{3}S_{L}^{\ast }\frac{H^{\ast }}{L^{\ast }} \biggl( 4-\frac{x_{1}}{x_{4}}x_{8}-\frac{1}{x_{1}}- \frac{x_{5}}{x_{8}} -\frac{x _{4}}{x_{5}} \biggr) \\ &{}+a_{4}\eta \beta _{2}\theta _{H} \frac{S_{H}^{\ast }}{L^{\ast }}Q^{ \ast } \biggl( 3-\frac{x_{2}}{x_{4}}x_{5}- \frac{1}{x_{2}}-\frac{x_{4}}{x _{5}} \biggr)\\ &{}+ a_{4}\beta _{2}\eta _{3} \theta _{H}S_{H}^{\ast } \frac{ H^{\ast }}{L^{\ast }}\biggl(5-\frac{x_{2}}{x _{4}}x_{8}- \frac{x_{7}}{x_{8}}-\frac{1}{x_{2}}- \frac{x_{6}}{x_{7}}- \frac{x _{4}}{x_{6}}\biggr) \\ &{}+ a_{4}\beta _{2}\eta _{3}\theta _{P}\frac{ H^{\ast }}{L^{\ast }} P^{\ast }\biggl(6- \frac{x_{3}}{x_{4}}x_{8}-\frac{x_{2}}{x_{3}}- \frac{1}{x _{2}}- \frac{x_{7}}{x_{8}}-\frac{x_{6}}{x_{7}}- \frac{x_{4}}{x_{6}}\biggr)\\ &{}+ a _{4}\beta _{2}\eta _{4} \frac{\theta _{P}}{L^{\ast }}P^{\ast } T^{\ast } \biggl(6-\frac{x_{3}}{x_{4}}x_{9}-\frac{x_{2}}{x_{3}}- -\frac{1}{x_{2}}- \frac{x_{7}}{x_{9}}-\frac{x_{6}}{x_{7}}- \frac{x _{4}}{x_{6}}\biggr)\\ &{}+ a_{4} \beta _{2}\eta _{2}\theta _{P}\frac{I_{2}^{\ast }}{ L^{\ast }}P^{\ast } \biggl(5-\frac{x _{3}}{x_{4}}x_{7}-\frac{x_{2}}{x_{3}}- \frac{1}{x_{2}}- \frac{x_{6}}{x _{7}}-\frac{x_{4}}{x_{6}}\biggr)\\ &{}+ a_{4}\beta _{2}\eta _{4}\theta _{H}\frac{S _{H}^{\ast }}{L^{\ast }} T^{\ast }\biggl(5-\frac{x_{2}}{x_{4}}x_{9}-\frac{1}{x_{2}}- \frac{x_{7}}{x_{9}}- \frac{x _{6}}{x_{7}}-\frac{x_{4}}{x_{6}}\biggr)\\ &{}+ a_{4}\beta _{2}\eta _{2}\theta _{H}I _{2}^{\ast }\frac{ S_{H}^{\ast }}{L^{\ast }} \biggl( 4-\frac{x_{2}}{x _{4}}x_{7}-\frac{1}{x_{2}}- \frac{x_{6}}{x_{7} }-\frac{x_{4}}{x_{6}} \biggr). \end{aligned}$$
Since the arithmetic mean is greater than or equal to the geometric mean, we have \(\frac{1}{x_{1}}+\frac{x_{1}}{x_{4}}x_{6}+\frac{x_{4}}{x_{6}}\geq 3, \frac{1}{x _{2}}+\frac{x_{2}}{x_{3}}+\frac{x_{3}}{x_{4}}x_{6}+ \frac{x_{4}}{x _{6}}\geq 4, \frac{x_{1}}{x_{3}}+\frac{1}{x_{1}}+\frac{x_{3}}{x_{4}}x _{5}+ \frac{x_{4}}{x_{5}}\geq 4, \frac{x_{1}}{x_{4}}x_{9}+\frac{1}{x _{1}}+\frac{x_{4}}{x_{6}}+\frac{x_{6}}{x_{9}}\geq 4, \frac{x_{1}}{x _{4}}x_{7}+\frac{1}{x_{1}}+\frac{x_{4}}{x_{6}}+\frac{x_{6}}{x_{7}} \geq 4, \frac{x_{2}}{x_{4}}x_{6}+\frac{1}{x_{2}}+\frac{x_{4}}{x_{6}} \geq 3, \frac{x_{1}}{x_{4}}x_{5}+\frac{1}{x_{1}}+\frac{x_{4}}{x_{5}} \geq 3, \frac{x_{1}}{x_{4}}x_{8}+\frac{1}{x_{1}}+\frac{x_{5}}{x_{8}}+\frac{x _{4}}{x_{5}}\geq 4, \frac{x_{2}}{x_{4}}x_{5}+\frac{1}{x_{2}}+\frac{x _{4}}{x_{5}}\geq 3, \frac{x_{2}}{x_{4}}x_{8}+\frac{x_{7}}{x_{8}}+\frac{1}{x _{2}}+ \frac{x_{6}}{x_{7}}+\frac{x_{4}}{x_{6}}\geq 5, \frac{x_{3}}{x _{4}}x_{8}+\frac{x_{2}}{x_{3}}+\frac{1}{x_{2}}+ \frac{x_{7}}{x_{8}}+\frac{x _{6}}{x_{7}}+\frac{x_{4}}{x_{6}}\geq 6, \frac{x_{3}}{x_{4}}x_{9}+\frac{x _{2}}{x_{3}}+\frac{1}{x_{2}}+ \frac{x_{7}}{x_{9}}+\frac{x_{6}}{x_{7}}+\frac{x _{4}}{x_{6}}\geq 6, \frac{x_{3}}{x_{4}}x_{7}+\frac{x_{2}}{x_{3}}+\frac{1}{x _{2}}+ \frac{x_{6}}{x_{7}}+\frac{x_{4}}{x_{6}}\geq 5, \frac{x_{2}}{x _{4}}x_{9}+\frac{1}{x_{2}}+\frac{x_{7}}{x_{9}}+ \frac{x_{6}}{x_{7}}+\frac{x _{4}}{x_{6}}\geq 5, \frac{x_{2}}{x_{4}}x_{7}+\frac{1}{x_{2}}+\frac{x _{6}}{x_{7}}+\frac{x_{4}}{x_{6}}\geq 4\).
In this manner, we have \(\mathtt{L}^{\prime }\leq 0\) in D. The equality \(\mathtt{L}^{\prime }=0\) exists if and only if \(\{x_{i}=1,i=1,2,\ldots,9 \}\). That is, \(S_{L}=S_{L}^{\ast }, S_{H}=S_{H}^{\ast },P=P^{\ast },L=L ^{\ast },Q=Q^{\ast },I_{1}=I_{1}^{\ast },I_{2}=I_{2}^{\ast },H=H^{ \ast },T=T^{\ast }\) in D. The maximal compact invariant set in \(\{ ( S_{L},S_{H},P,L,Q,I_{1},I_{2},H,T,R ) \in D: \mathtt{L}^{\prime }=0 \} \) is “\(E_{1}\)” whenever \(R_{C}>1\). By LaSalle’s invariance principle [28], “\(E_{1}\)” is globally asymptotically stable for \(R_{C}>1\).