## Appendix B Global Stability Result Proof
\right.\]  
Hence,  
\[\dot{\mathcal{L}}(x,s,y,v,z) =\left(\lambda-d_{1}x-\frac{k_{1}xv}{x+v}\right)\left(1-(d_{2}+k_ {2})s_{1}\frac{x+v_{1}}{k_{1}xv_{1}}\right)\] \[+\frac{k_{1}xv}{x+v}-(d_{2}+k_{2})s-\frac{s_{1}}{s}\left(\frac{k _{1}xv}{x+v}-(d_{2}+k_{2})s\right)\] \[+\frac{d_{2}+k_{2}}{k_{2}}(k_{2}s-d_{3}y-pyz)-\frac{d_{2}+k_{2}}{ k_{2}}\frac{y_{1}}{y}(k_{2}s-d_{3}y-pyz)\] \[+\frac{d_{3}(d_{2}+k_{2})}{ak_{2}}(ay-d_{4}v)-\frac{d_{3}(d_{2}+k_ {2})}{ak_{2}}\frac{v_{1}}{v}(ay-d_{4}v)\] \[+\frac{p}{c}\frac{d_{2}+k_{2}}{k_{2}}(cyz-bz),\]  
and  
\[\dot{\mathcal{L}}(x,s,y,v,z)= \lambda-d_{1}x-(d_{2}+k_{2})s_{1}\frac{x+v_{1}}{k_{1}xv_{1}}\left( \lambda-d_{1}x-\frac{k_{1}xv}{x+v}\right)\] \[-\frac{s_{1}}{s}\left(\frac{k_{1}xv}{x+v}\right)+(d_{2}+k_{2})s_{1 }+\frac{d_{2}+k_{2}}{k_{2}}py_{1}z\] \[+\frac{(d_{2}+k_{2})d_{3}}{k_{2}}y_{1}-(d_{2}+k_{2})s\frac{y_{1}} {y}-\frac{d_{3}d_{4}(d_{2}+k_{2})}{ak_{2}}v\] \[+\frac{d_{3}d_{4}(d_{2}+k_{2})}{ak_{2}}v_{1}-\frac{d_{3}(d_{2}+k_ {2})}{k_{2}}\frac{v_{1}y}{v}-\frac{d_{2}+k_{2}}{k_{2}}\frac{bp}{c}z.\]Since  
\[\left\{ \begin{array}{l}\lambda-d_{1}x=d_{1}x_{1}+(d_{2}+k_{2})s_{1}-d_{1}x,\\ \lambda-d_{1}x-(d_{2}+k_{2})s_{1}\dfrac{x+v_{1}}{k_{1}xv_{1}}\left(\lambda-d_{1 }x-\dfrac{k_{1}xv}{x+v}\right)=d_{1}x_{1}\left(1-\dfrac{x}{x_{1}}-\dfrac{x_{1}} {x}\dfrac{x+v_{1}}{x_{1}+v_{1}}+\dfrac{x+v_{1}}{x_{1}+v_{1}}\right)\\ \qquad+(d_{2}+k_{2})s_{1}\left(1-\dfrac{x_{1}}{x}\dfrac{x+v_{1}}{x_{1}+v_{1}}+ \dfrac{v}{v_{1}}\dfrac{x+v_{1}}{x+v}\right),\\ -\dfrac{s_{1}}{s}\left(\dfrac{k_{1}xv}{x+v}\right)+(d_{2}+k_{2})s_{1}=-\dfrac{s _{1}}{s}\dfrac{xv}{x_{1}v_{1}}\dfrac{x_{1}+v_{1}}{x+v}(d_{2}+k_{2})s_{1}+(d_{ 2}+k_{2})s_{1},\end{array} \right.\]  
we have  
\[\dot{\mathcal{L}} =d_{1}x_{1}\left(1-\dfrac{x}{x_{1}}-\dfrac{x_{1}}{x}\dfrac{x+v_{1 }}{x_{1}+v_{1}}+\dfrac{x+v_{1}}{x_{1}+v_{1}}\right)\] \[+(d_{2}+k_{2})s_{1}\left(1-\dfrac{x_{1}}{x}\dfrac{x+v_{1}}{x_{1}+ v_{1}}+\dfrac{v}{v_{1}}\dfrac{x+v_{1}}{x+v}\right)\] \[+(d_{2}+k_{2})s_{1}\left(1-\dfrac{s_{1}}{s}\dfrac{xv}{x_{1}v_{1} }\dfrac{x_{1}+v_{1}}{x+v}\right)\] \[+(d_{2}+k_{2})s_{1}\left(1-\dfrac{sy_{1}}{s_{1}y}-\dfrac{v}{v_{1 }}\right)\] \[+(d_{2}+k_{2})s_{1}\left(1-\dfrac{v_{1}y}{y_{1}v}\right)+pz\dfrac {d_{2}+k_{2}}{k_{2}}(y_{1}-\dfrac{b}{c}).\]  
Therefore,  
\[\dot{\mathcal{L}} =-\dfrac{d_{1}v_{1}}{x(x_{1}+v_{1})}(x-x_{1})^{2}\] \[+(d_{2}+k_{2})s_{1}\left(-1-\dfrac{v}{v_{1}}+\dfrac{v}{v_{1}} \dfrac{x+v_{1}}{x+v}+\dfrac{x+v}{x+v_{1}}\right)\] \[+(d_{2}+k_{2})s_{1}\left(5-\dfrac{x_{1}}{x}\dfrac{x+v_{1}}{x_{1} +v_{1}}-\dfrac{s_{1}}{s}\dfrac{xv}{x_{1}v_{1}}\dfrac{x_{1}+v_{1}}{x+v}-\dfrac{ sy_{1}}{s_{1}y}-\dfrac{yv_{1}}{y_{1}v}-\dfrac{x+v}{x+v_{1}}\right)\] \[+pz\dfrac{d_{2}+k_{2}}{k_{2}}(y_{1}-\dfrac{b}{c}),\]  
which implies that  
\[\dot{\mathcal{L}} =-\dfrac{d_{1}v_{1}}{x(x_{1}+v_{1})}(x-x_{1})^{2}\] \[+(d_{2}+k_{2})s_{1}\left(1-\dfrac{x_{1}}{x}\dfrac{x+v_{1}}{x_{1}+ v_{1}}+\dfrac{v}{v_{1}}\dfrac{x+v_{1}}{x+v}\right)\]0 \[+(d_{2}+k_{2})s_{1}\left(1-\dfrac{x_{1}}{x}\dfrac{x+v_{1}}{x_{1}+ v_{1}}+\dfrac{v}{v_{1}}\dfrac{x+v_{1}}{x+v}\right)\]1 \[+(d_{2}+k_{2})s_{1}\left(1-\dfrac{x_{1}}{x}\dfrac{x+v_{1}}{x_{1}+ v_{1}}+\dfrac{v}{v_{1}}\dfrac{x+v_{1}}{x+v}\right)\]2  
since the arithmetic mean is greater than or equal to the geometric mean, it follows that  
\[+(d_{2}+k_{2})s_{1}\left(1-\dfrac{x_{1}}{x}\dfrac{x+v_{1}}{x_{1}+ v_{1}}+\dfrac{v}{v_{1}}\dfrac{x+v_{1}}{x+v}\right)\]3  
In addition, when \(R_{CTL}<1\), we will have \(y_{1}-\dfrac{b}{c}\leq 0\), which means that \(\dot{L}\leq 0\), and the equality hods when \(x=x_{1}\), \(y=y_{1}\), \(s=s_{1}\) and \(v=v_{1}\). By the LaSalle invariance principle, the endemic point \(E_{1}\) is globally stable when \(R_{0}>1\). Finally, the global stability proof of the endemic point \(E_{2}\) (Proposition 5) is given as follows.