### Appendix 1: Proof of Lemma 4.1

This appendix will show the set *Ω* is positively invariant for system (2.4). That is, we will show that if \(x(0)\in \varOmega \), then \(x(t)\in \varOmega \) for all \(t>0\). From (4.1), *∂Ω* (the boundary of *Ω*) is composed of the following 3*n* sets:

$$\begin{aligned}& \partial \varOmega _{i}^{(1)}=\{x\in \varOmega \mid x_{i}=0\},\qquad \partial \varOmega _{i}^{(2)}=\{x\in \varOmega \mid x_{n+i}=0\}, \\& \partial \varOmega _{i}^{(3)}=\{x\in \varOmega \mid x_{i}+x_{n+i}=1\}, \quad i=1,2,\ldots ,n, \end{aligned}$$

and the respective outer normal vectors are

$$\begin{aligned}& \xi _{i}^{(1)}=(0,\ldots ,0,\overset{i}{-1},0,\ldots ,0), \qquad \xi _{i} ^{(2)}=(0,\ldots ,0,\overset{n+i}{-1},0, \ldots ,0), \\& \xi _{i}^{(3)}=(0,\ldots ,0,\overset{i}{1},0,\ldots ,0, \overset{n+i}{1},0,\ldots ,0). \end{aligned}$$

For an arbitrary compact set *Ω̃*, Nagumo had proved that the set *Ω̃* is invariant for the system \(\mathrm{d}z/\mathrm{d}t=f(z)\), if the vector \(f(z)\) is tangent or pointing into *Ω̃* for every point *z* on *∂Ω̃* [36]. Note that *Ω* is a compact set and, for \(1\leq i\leq n\),

$$\begin{aligned}& \biggl(\frac{\mathrm{d}x}{\mathrm{d}t}\bigg|_{x\in \partial \varOmega _{i}^{(1)}}, \xi _{i}^{(1)} \biggr)=-\lambda (i) (1-x_{n+i})\widetilde{\varTheta }\phi _{i}( \widetilde{\varTheta })\leq 0,\qquad \biggl(\frac{\mathrm{d}x}{\mathrm{d}t}\bigg|_{x\in \partial \varOmega _{i}^{(2)}}, \xi _{i}^{(2)}\biggr)=-\gamma x_{i}\leq 0, \\& \biggl(\frac{\mathrm{d}x}{\mathrm{d}t}\bigg|_{x\in \partial \varOmega _{i}^{(3)}}, \xi _{i}^{(3)} \biggr)= -\mu x_{i}-(\mu +\omega )x_{n+i}\leq 0,\quad \mbox{where } \widetilde{\varTheta } =\frac{1}{\langle k\rangle }\sum _{\substack{k=1 \\ k\neq i}}^{n}\varphi (k)P(k)x_{k}\geq 0. \end{aligned}$$

Hence, through Nagumo’s result, we see that the set *Ω* is positively invariant.

### Appendix 2: Proof of Theorem 5.2

This appendix will show the global attractivity of the endemic equilibrium \(E^{*}\) of system (2.4). In the following, *k* is fixed to be any integer in \(\{1,2,\ldots ,n\}\). From Theorem 4.3, there exist a small enough constant \(\xi _{0}\) (\(0<\xi _{0} \ll 1\)) and a large enough constant \(T_{0}>0\) such that \(I_{k}(t) \geq \xi _{0}\) for \(t>T_{0}\). Thus

$$ \varTheta (t)=\frac{1}{\langle k\rangle }\sum_{k=1}^{n} \varphi (k)P(k)I _{k}(t) \geq \frac{\xi _{0}}{\langle k\rangle }\sum _{k=1}^{n} \varphi (k)P(k) =:\varepsilon _{0}>0. $$

Note that \(\phi _{k}(\varTheta )\leq \phi _{k}(0)=1\) for \(0\leq \varTheta \leq 1\). Then from the first equation of system (2.4), we have

$$ \frac{\mathrm{d}I_{k}(t)}{\mathrm{d}t}\leq \lambda (k)\bigl[1 -I_{k}(t)\bigr]-( \mu + \gamma )I_{k}(t) =\lambda (k) -\bigl[\lambda (k)+\mu +\gamma \bigr]I_{k}(t). $$

By Lemma 2.1 in Ref. [38], we derive that \(\limsup_{t\rightarrow \infty }I_{k}(t)\leq \frac{\lambda (k)}{ \lambda (k)+\mu +\gamma }\). Then, for any given small constant \(0<\varepsilon _{1}^{(1)}<\frac{\mu +\gamma }{2[\lambda (k)+\mu + \gamma ]}\), by the comparison theorem, there exists a \(T_{1}^{(1)}>T_{0}\) such that \(I_{k}(t)\leq X_{k}^{(1)}-\varepsilon _{1}^{(1)}\) for \(t>T_{1}^{(1)}\), where

$$ X_{k}^{(1)}:=\frac{\lambda (k)}{\lambda (k)+\mu +\gamma }+2\varepsilon _{1}^{(1)}< 1. $$

(B.1)

It then follows from the second equation of system (2.4) that

$$ \frac{\mathrm{d}R_{k}(t)}{\mathrm{d}t}=\gamma \bigl[1 -S_{k}(t)-R_{k}(t) \bigr]-( \mu +\omega )R_{k}(t) \leq \gamma -(\gamma +\mu +\omega )R_{k}(t). $$

Similarly, for any given small constant \(0<\varepsilon _{1}^{(2)}< \min \{\frac{1}{2},\varepsilon _{1}^{(1)}, \frac{\mu +\omega }{2( \mu +\gamma +\omega )}\}\), there exists a \(T_{1}^{(2)}>T _{1}^{(1)}\) such that \(R_{k}(t)\leq Y_{k}^{(1)}-\varepsilon _{1}^{(2)}\) for \(t>T_{1}^{(2)}\), where

$$ Y_{k}^{(1)}:=\frac{\gamma }{\gamma +\mu +\omega }+2\varepsilon _{1} ^{(1)}< 1 . $$

(B.2)

Since \(\varTheta (t)\leq \frac{1}{\langle k\rangle }\sum_{k=1}^{n}\varphi (k)P(k)=\frac{\langle \varphi (k)\rangle }{\langle k\rangle }=:\beta \), \(\phi _{k}(\beta )\leq \phi _{k}(\varTheta )\). Substituting (B.2) into the first equation of system (2.4) gives

$$ \frac{\mathrm{d}I_{k}(t)}{\mathrm{d}t}\geq \lambda (k) \bigl(1 -Y_{k}^{(1)} \bigr) \varepsilon _{0}\phi _{k}(\beta )-\bigl[\lambda (k)+ \mu +\gamma \bigr]I_{k}(t),\quad \mbox{for } t>T_{1}^{(4)}. $$

By Lemma 2.1 in Ref. [38], we derive that \(\liminf_{t\rightarrow \infty }I_{k}(t) \geq \frac{\lambda (k)(1 -Y_{k}^{(1)})\varepsilon _{0}\phi _{k}(\beta )}{\lambda (k)+\mu +\gamma }\). That is, for any given small constant \(0<\varepsilon _{1}^{(3)}< \min \{\frac{1}{3},\varepsilon _{1}^{(2)}, \frac{\lambda (k)(1 -Y_{k} ^{(1)})\varepsilon _{0}\phi _{k}(\beta )}{2[\lambda (k)+\mu +\gamma ]} \}\), there exists a \(T_{1}^{(3)}>T_{1}^{(2)}\) such that \(I_{k}(t)\geq x_{k}^{(1)}+\varepsilon _{1}^{(3)}\) for \(t>T_{1}^{(2)}\), where

$$ x_{k}^{(1)}:=\frac{\lambda (k)(1 -Y_{k}^{(1)})\varepsilon _{0}\phi _{k}( \beta )}{\lambda (k)+\mu +\gamma } -2\varepsilon _{1}^{(3)}>0. $$

(B.3)

From the second equation of system (2.4), we derive that \(\frac{\mathrm{d}R_{k}(t)}{\mathrm{d}t}\geq \gamma x_{k}^{(1)}-( \mu +\omega )R_{k}(t)\), \(t>T_{1}^{(3)}\). Similarly, for any given small constant \(0<\varepsilon _{1}^{(4)}<\min \{\frac{1}{4},\varepsilon _{1}^{(3)}, \frac{\gamma x_{k}^{(1)}}{2(\mu +\omega )}\}\), there exists a \(T_{1}^{(4)}>T_{1}^{(3)}\) such that \(R_{k}(t) \geq y_{k}^{(1)}+\varepsilon _{1}^{(4)}\) for \(t>T_{1}^{(4)}\), where

$$ y_{k}^{(1)}:=\frac{\gamma x_{k}^{(1)}}{\mu +\omega }-2\varepsilon _{1} ^{(4)}>0. $$

(B.4)

Note that \(\xi _{0}\) is a small enough constant, then \(0< x_{k}^{(1)} \ll 1\) and \(0< y_{k}^{(1)}\ll 1\). Then, from the discussion above, one obtains \(0< x_{k}^{(1)}< X_{k}^{(1)}<1 \) and \(0< y_{k}^{(1)}< Y_{k}^{(1)}<1 \) for \(t>T_{1}^{(4)}\). Accordingly, it follows from (2.2) that

$$ \begin{aligned} &0< m_{1}< \varTheta (t)< M_{1}< \beta < 1, \\ &\phi _{k}(\beta )\leq \phi _{k}(M_{1})\leq \phi _{k}\bigl(\varTheta (t)\bigr)\leq \phi _{k}(m_{1}) \leq 1, \end{aligned} $$

(B.5)

where \(m_{1}=\frac{1}{\langle k\rangle }\sum_{k=1}^{n}\varphi (k)P(k)x _{k}^{(1)}\) and \(M_{1}=\frac{1}{\langle k\rangle }\sum_{k=1}^{n} \varphi (k)P(k)X_{k}^{(1)}\). Again, from the first equation of system (2.4), we have

$$ \frac{\mathrm{d}I_{k}(t)}{\mathrm{d}t}\leq \lambda (k)\bigl[1 -I_{k}(t)-y _{k}^{(1)}\bigr]M_{1}\phi _{k}(m_{1}) -(\mu +\gamma )I_{k}(t),\quad t>T_{1}^{(4)}. $$

So, for any given constant \(0<\varepsilon _{2}^{(1)}<\min \{ \frac{1}{5},\varepsilon _{1}^{(4)}, \frac{\mu +\gamma +\lambda (k)y _{k}^{(1)}M_{1}\phi _{k}(m_{1})}{\lambda (k) M_{1}\phi _{k}(m_{1})+ \mu +\gamma }\}\), there exists a \(T_{2}^{(1)}>T_{1}^{(4)}\) such that

$$ I_{k}(t)\leq X_{k}^{(2)}:=\frac{\lambda (k)(1 -y_{k}^{(1)})M_{1}\phi _{k}(m_{1})}{\lambda (k) M_{1}\phi _{k}(m_{1})+\mu +\gamma }+ \varepsilon _{2}^{(1)}< 1 . $$

(B.6)

From the second equation of system (2.4) one then obtains \(\frac{\mathrm{d}R_{k}(t)}{\mathrm{d}t}\leq \gamma X_{k}^{(2)}-( \mu +\omega )R_{k}(t)\), \(t>T_{2}^{(1)}\). Hence, for any given constant \(0<\varepsilon _{2}^{(2)}<\min \{\frac{1}{6},\varepsilon _{2} ^{(1)}\}\), there exists a \(T_{2}^{(2)}>T_{2}^{(1)}\) such that

$$ R_{k}(t)\leq Y_{k}^{(2)}:=\min \biggl\{ Y_{k}^{(1)}-\varepsilon _{1}^{(2)}, \frac{\gamma X_{k}^{(2)}}{\mu +\omega }+\varepsilon _{2}^{(2)} \biggr\} ,\quad t>T_{2}^{(2)}. $$

(B.7)

It follows from (B.1), (B.6) and (B.7) that \(X_{k}^{(2)}< X_{k}^{(1)}\) and \(Y_{k}^{(2)}< Y_{k}^{(1)}\).

Turning back to system (2.4), it can be seen that

$$ \frac{\mathrm{d}I_{k}(t)}{\mathrm{d}t}\geq \lambda (k)\bigl[1 -I_{k}(t)-Y _{k}^{(2)}\bigr]m_{1}\phi _{k}(M_{1}) -(\mu +\gamma )I_{k}(t),\quad t>T_{2}^{(2)}. $$

So, for any given constant \(0<\varepsilon _{2}^{(3)}<\min \{ \frac{1}{7},\varepsilon _{2}^{(2)}, \frac{\lambda (k)(1 -Y_{k}^{(2)})m _{1}\phi _{k}(M_{1})}{\lambda (k) m_{1}\phi _{k}(M_{1})+\mu +\gamma }\}\), there exists a \(T_{2}^{(3)}>T_{2}^{(2)}\) such that \(I_{k}(t)\geq x_{k}^{(2)}\), where

$$ x_{k}^{(2)}:=\max \biggl\{ x_{k}^{(1)}+ \varepsilon _{1}^{(2)}, \frac{ \lambda (k)(1 -Y_{k}^{(2)})m_{1}\phi _{k}(M_{1})}{\lambda (k) m_{1}\phi _{k}(M_{1})+\mu +\gamma }-\varepsilon _{2}^{(3)} \biggr\} ,\quad t>T_{2}^{(3)}. $$

(B.8)

Therefore, by (2.4), one has \(\frac{\mathrm{d}R_{k}(t)}{ \mathrm{d}t}\geq \gamma x_{k}^{(2)}-(\mu +\omega )R_{k}(t)\), \(t>T_{2}^{(3)}\). Then, for any given constant \(0<\varepsilon _{2}^{(4)}< \min \{\frac{1}{8},\varepsilon _{2}^{(3)}, \frac{\gamma x_{k}^{(2)}}{ \mu +\omega }\}\), there exists a \(T_{2}^{(4)}>T_{2}^{(3)}\) such that

$$ R_{k}(t)\geq y_{k}^{(2)}:=\frac{\gamma x_{k}^{(2)}}{\mu +\omega }- \varepsilon _{2}^{(4)}, \quad t>T_{2}^{(4)}. $$

(B.9)

Repeating the above procedure, we get four sequences: \(X_{k}^{(i)}\), \(Y _{k}^{(i)}\), \(x_{k}^{(i)}\), \(y_{k}^{(i)}\), \(i=1,2,\ldots \) . By induction, we know that the first two are monotone decreasing sequences and the last two are monotone increasing sequences. Then there exists a sufficiently large positive integer *N* such that, with \(n\geq N\),

$$ \begin{aligned} &X_{k}^{(n)}=\frac{\lambda (k)(1 -y_{k}^{n-1})M_{n-1}\phi _{k}(m_{n-1})}{ \lambda (k)M_{n-1}\phi _{k}(m_{n-1})+\mu +\gamma }+ \varepsilon _{n}^{(1)},\qquad Y_{k}^{(n)}= \frac{\gamma X_{k}^{(n)}}{\mu +\omega }+\varepsilon _{n} ^{(2)}, \\ &x_{k}^{(n)}=\frac{\lambda (k)(1 -Y_{k}^{n})m_{n-1}\phi _{k}(M_{n-1})}{ \lambda (k)m_{n-1}\phi _{k}(M_{n-1})+\mu +\gamma }-\varepsilon _{n}^{(3)},\qquad y_{k}^{(n)}= \frac{\gamma x_{k}^{(n)}}{\mu +\omega }-\varepsilon _{n} ^{(4)}. \end{aligned} $$

(B.10)

Obviously, one has

$$ 0< x_{k}^{(n)}\leq I_{k}(t)\leq X_{k}^{(n)}< 1 ,\qquad 0< y_{k}^{(n)}\leq R_{k}(t)\leq Y_{k}^{(n)}< 1 ,\quad t>T_{n}^{(4)}. $$

(B.11)

Because the sequential limits of (B.10) exist, let \(\lim_{n\rightarrow \infty }H_{k}^{(n)}=H_{k}\), where \(H_{k}^{(n)}=(X _{k}^{(n)}, Y_{k}^{(n)}, x_{k}^{(n)}, y_{k}^{(n)}, M_{n}, m_{n})\) and \(H_{k}=(\bar{X}_{k}, \bar{Y}_{k}, \bar{x}_{k}, \bar{y}_{k}, M, m)\). Note that \(0<\varepsilon _{n}^{(i)}<\frac{1}{4n+i-4}\) (\(i=1,2,3,4\), \(n>1\)), then \(\varepsilon _{n}^{(i)}\rightarrow 0\) as \(n\rightarrow \infty \). Therefore, taking \(n\rightarrow \infty \), it follows from (B.10) that

$$ \begin{aligned} &\bar{X}_{k}=\frac{\lambda (k)(1 -\bar{y}_{k})M\phi _{k}(m)}{\lambda (k)M\phi _{k}(m)+\mu +\gamma },\qquad \bar{Y}_{k}=\frac{\gamma \bar{X}_{k}}{\mu +\omega }, \\ &\bar{x}_{k}=\frac{\lambda (k)(1 -\bar{Y}_{k})m\phi _{k}(M)}{\lambda (k)m\phi _{k}(M)+\mu +\gamma },\qquad \bar{y}_{k}= \frac{\gamma \bar{x}_{k}}{\mu +\omega }, \end{aligned} $$

(B.12)

where \(M=\frac{1}{\langle k\rangle }\sum_{k=1}^{n}\varphi (k)P(k) \bar{X}_{k}\), \(m=\frac{1}{\langle k\rangle }\sum_{k=1}^{n}\varphi (k)P(k) \bar{x}_{k}\), \(0< m\leq M<1\). Furthermore, we obtain from (B.12)

$$ \begin{aligned} &D_{k}\bar{X}_{k}=\lambda (k) M \biggl[\lambda (k)m\phi _{k}(M)\phi _{k}(m)+ \phi _{k}(m) (\mu +\gamma ) -\frac{\lambda (k)m\phi _{k}(M)\phi _{k}(m) \gamma }{\mu +\omega } \biggr], \\ &D_{k}\bar{x}_{k}=\lambda (k) m \biggl[\lambda (k)M\phi _{k}(M)\phi _{k}(m)+ \phi _{k}(M) (\mu +\gamma ) -\frac{\lambda (k)M\phi _{k}(M)\phi _{k}(m) \gamma }{\mu +\omega } \biggr], \end{aligned} $$

(B.13)

where \(D_{k}=[\lambda (k)M\phi _{k}(m)+\mu +\gamma ][\lambda (k)m\phi _{k}(M)+\mu +\gamma ]- \frac{[\lambda (k)]^{2} Mm\phi _{k}(M)\phi _{k}(m) \gamma ^{2}}{(\mu +\omega )^{2}}\). Here, we claim that \(D_{k}\neq 0\). Note that \(\bar{X}_{k}\) is the unique nonzero value determined by (B.12). If \(D_{k}=0\), then \(\lambda (k)m\phi _{k}(M)+(\mu +\gamma )=\frac{ \lambda (k)m\phi _{k}(M)\gamma }{\mu +\omega }\). By t symmetry, we have \(\lambda (k)M\phi _{k}(m)+(\mu +\gamma )=\frac{\lambda (k)M\phi _{k}(m) \gamma }{\mu +\omega }\). Obviously, it follows that \(\lambda (k)[m\phi _{k}(M)-M\phi _{k}(m)]=\frac{\lambda (k)[m\phi _{k}(M)-M\phi _{k}(m)] \gamma }{\mu +\omega }\), i.e. \(\mu +\omega =\gamma \). This is inconsistent with the assumed conditions of the theorem. Therefore, \(D_{k}\neq 0\).

Combining (B.13) with the expressions of *M* and *m*, one has

$$ \begin{aligned} &\begin{aligned} 1&=\frac{1}{\langle k\rangle }\sum _{k=1}^{n}\varphi (k)P(k)\frac{ \lambda (k)}{D_{k}}\biggl[ \lambda (k)m\phi _{k}(M)\phi _{k}(m) +\phi _{k}(m) ( \mu +\gamma ) \\ &\quad {}-\frac{\lambda (k)m\phi _{k}(M)\phi _{k}(m)\gamma }{\mu +\omega }\biggr], \end{aligned} \\ &\begin{aligned} 1&=\frac{1}{\langle k\rangle }\sum_{k=1}^{n} \varphi (k)P(k)\frac{ \lambda (k)}{D_{k}}\biggl[\lambda (k)M\phi _{k}(M)\phi _{k}(m) +\phi _{k}(M) ( \mu +\gamma ) \\ &\quad {}-\frac{\lambda (k)M\phi _{k}(M)\phi _{k}(m)\gamma }{\mu +\omega }\biggr]. \end{aligned} \end{aligned} $$

(B.14)

From (B.14), a direct computation leads to

$$\begin{aligned} &\frac{1}{\langle k\rangle }\sum_{k=1}^{n}\varphi (k)P(k)\frac{ \lambda (k)}{D_{k}} \biggl\{ \lambda (k)\phi _{k}(M)\phi _{k}(m) (M-m) \biggl(1-\frac{ \gamma }{\mu +\omega }\biggr) \\ &\quad {}+(\mu +\gamma ) \bigl[\phi _{k}(M)-\phi _{k}(m) \bigr] \biggr\} =0. \end{aligned}$$

(B.15)

Now we want to show that \(m=M\). Suppose it is not true, then there exists \(\tau _{0}\in (m,M)\) such that \(\phi _{k}(M)-\phi _{k}(m)=\phi _{k}'(\tau _{0})(M-m)\). Hence, it follows from (B.15) that

$$\begin{aligned} &\frac{1}{\langle k\rangle }\sum_{k=1}^{n}\varphi (k)P(k)\frac{ \lambda (k)}{D_{k}}(M-m) \biggl[\lambda (k)\phi _{k}(M)\phi _{k}(m) \biggl(1-\frac{ \gamma }{\mu +\omega }\biggr) \\ &\quad {}+(\mu +\gamma )\phi _{k}'(\tau _{0}) \biggr]=0. \end{aligned}$$

(B.16)

Since \(\phi _{k}'(\varTheta )\leq 0\) (\(0\leq \varTheta \leq 1\)) and \(\mu +\omega <\gamma \), each item on the left side of (B.16) is negative. This is apparently a contradiction. Consequently, \(m=M\). Then we have \(\frac{1}{\langle k\rangle }\sum_{k=1}^{n}\varphi (k)P(k)( \bar{X}_{k}-\bar{x}_{k})=0\), which implies that \(\bar{X}_{k}=\bar{x} _{k}\) for \(k=1,2,\ldots ,n\). From (B.11) and (B.12), we arrive at \(\lim_{t\rightarrow \infty }I_{k}(t)=\bar{X}_{k}=\bar{x}_{k}\) and \(\lim_{t\rightarrow \infty }R_{k}(t)=\bar{Y}_{k}=\bar{y}_{k}\). Note that Eq. (3.5) has a unique positive solution \(\varTheta ^{*}\) if \(\mathcal{R}_{0}>1\). Then, substituting \(M=m\) and \(\bar{X}_{k}= \bar{x}_{k}\) into (B.13), by virtue of (3.6) and (B.12), one can obtain \(\bar{X}_{k}=I_{k}^{*}\) and \(\bar{Y}_{k}=R_{k}^{*}\). As a result, the endemic equilibrium \(E^{*}\) of system (2.4) is globally attractive if \(\mathcal{R}_{0}>1\) and \(\gamma >\mu +\omega \).