In this section, we discuss the occurrence of transcritical bifurcation for system (1.1). As was pointed out in [7], since \(S_{k}(t)+I_{k}(t)=1\), \(k=1, 2, \ldots , n\), we only need to present the bifurcation analysis for the following system:

$$ I_{k}(t)=\lambda k\bigl(1-\beta \varTheta (t)\bigr) \bigl(1-I_{k}(t)\bigr)\varTheta (t)-(\omega +\delta )I_{k}(t)),\quad k=1, 2, \ldots , n. $$

(2.1)

For system (2.1), there exists a disease-free equilibrium \(E_{0}(\underbrace{0,0,\ldots ,0}_{n})\). We have the main result as follows.

### Theorem 2.1

*System* (2.1) *undergoes the transcritical bifurcation at*
\(E_{0}\)*when*
\(R_{0}=1\).

### Proof

The corresponding Jacobian matrix *A* at \(E_{0}\) is as follows:

\left(\begin{array}{cccc}\frac{\lambda \cdot 1\cdot 1\cdot P(1)}{\u3008k\u3009}-(\omega +\delta )& \frac{\lambda \cdot 1\cdot 2\cdot P(2)}{\u3008k\u3009}& \cdots & \frac{\lambda \cdot 1\cdot n\cdot P(n)}{\u3008k\u3009}\\ \frac{\lambda \cdot 2\cdot 1\cdot P(1)}{\u3008k\u3009}& \frac{\lambda \cdot 2\cdot 2\cdot P(2)}{\u3008k\u3009}-(\omega +\delta )& \cdots & \frac{\lambda \cdot 2\cdot n\cdot P(n)}{\u3008k\u3009}\\ \cdots & \cdots & \cdots & \cdots \\ \frac{\lambda \cdot n\cdot 1\cdot P(1)}{\u3008k\u3009}& \frac{\lambda \cdot n\cdot 2\cdot P(2)}{\u3008k\u3009}& \cdots & \frac{\lambda \cdot n\cdot n\cdot P(n)}{\u3008k\u3009}-(\omega +\delta )\end{array}\right).

Here, \(\langle k \rangle =\sum_{k=1}^{n}kP(k)\) is the average degree. As was pointed out in [7], the *n* eigenvalues of *A* are \(\lambda _{1}=\lambda _{2}=\cdots =\lambda _{n-1}=-(\delta + \omega )\) and \(\lambda _{n}=(\delta +\omega )(R_{0}-1)\). One can compute that the corresponding eigenvectors are \(e_{1}= (\underbrace{1,- \tfrac{P(1)}{2P(2)},0,\ldots ,0}_{n} )^{T}\), \(e_{2}= (\underbrace{1,0,- \tfrac{P(1)}{3P(3)},\ldots ,0}_{n} )^{T}\), \(e_{n-1}= (\underbrace{1,0,0,- \tfrac{P(1)}{nP(n)}}_{n} )^{T}\) and \(e_{n}= (1,2,3,\ldots ,n )^{T}\). Let \(\bar{\lambda }=\lambda -\frac{(\delta +\omega ) \langle k \rangle }{\langle k^{2} \rangle }\), \(P=(e_{1},e_{2},\ldots , e_{n})\), and

\left(\begin{array}{c}{I}_{1}\\ {I}_{2}\\ \cdots \\ {I}_{n}\end{array}\right)=P\left(\begin{array}{c}{U}_{1}\\ {U}_{2}\\ \cdots \\ {U}_{n}\end{array}\right).

We obtain the system as follows:

$$\begin{aligned}& \begin{aligned}[b] \frac{dU_{k}(t)}{dt}={}&{-}(\omega +\delta )U_{k}(t)-\frac{(k+1)^{2}P(k+1)}{ \langle k \rangle } \biggl(\bar{\lambda }+ \frac{(\omega +\delta )\langle k \rangle }{\langle k^{2} \rangle } \biggr) \sum_{j=2}^{n}(1-j)U _{j-1}U_{n} \\ &{} -\frac{(k+1)^{2}P(k+1)}{P(1)\langle k \rangle } \biggl(\bar{ \lambda }+\frac{(\omega +\delta )\langle k \rangle }{\langle k^{2} \rangle } \biggr) \bigl(-2\bigl\langle k^{2} \bigr\rangle +\bigl\langle k^{3} \bigr\rangle \bigr)U_{n}^{2} \\ &{} -2 \biggl(\bar{\lambda }+\frac{(\omega +\delta )\langle k \rangle }{\langle k^{2} \rangle } \biggr)\frac{\langle k^{2} \rangle }{ \langle k \rangle }U_{1}U_{n} \\ &{}+\frac{(k+1)^{2}P(k+1)\langle k^{2} \rangle \beta }{P(1) {\langle k \rangle }^{2}} \biggl(\bar{\lambda }+\frac{(\omega +\delta ) \langle k \rangle }{\langle k^{2} \rangle } \biggr) \\ &{} \cdot \Biggl(P(1)\sum_{j=2}^{n}(1-j)U_{j-1}U_{n} ^{2}+\bigl\langle k^{3} \bigr\rangle U^{3}_{n} \Biggr) \\ &{}-\frac{(k+1)^{2}P(k+1){\langle k^{2} \rangle }^{2} \beta }{P(1){\langle k \rangle }^{2}} \biggl(\bar{\lambda }+\frac{( \omega +\delta )\langle k \rangle }{\langle k^{2} \rangle } \biggr) \\ &{}\cdot \biggl(-\frac{P(1)}{(k+1)^{2}P(k+1)}U_{1}U_{n}^{2}+(k+1) U^{3}_{n} \biggr),\quad k=1,2,\ldots , n-1, \end{aligned} \end{aligned}$$

(2.2)

$$\begin{aligned}& \begin{aligned}[b] \frac{dU_{n}(t)}{dt}={}& \frac{\langle k^{2} \rangle }{\langle k \rangle }U_{n}\bar{\lambda }+ \biggl(\bar{\lambda }+ \frac{(\omega +\delta ) \langle k \rangle }{\langle k^{2} \rangle } \biggr)\frac{\langle k^{2} \rangle \beta }{{\langle k \rangle }^{2}} \Biggl(P(1)\sum _{j=2} ^{n}(1-j)U_{j-1}U_{n}^{2}+ \bigl\langle k^{3} \bigr\rangle U_{n}^{3} \Biggr) \\ &{} -\frac{1}{\langle k \rangle } \biggl(\bar{\lambda }+\frac{( \omega +\delta )\langle k \rangle }{\langle k^{2} \rangle } \biggr) \\ &{}\cdot \Biggl(P(1)\sum_{j=2}^{n}(1-j)U_{j-1}U_{n}+ \biggl(\frac{\langle k ^{2} \rangle \alpha }{\langle k \rangle }+\bigl\langle k^{3} \bigr\rangle \biggr)U_{n}^{2} \Biggr) \end{aligned} \end{aligned}$$

(2.3)

and \(\frac{d \bar{\lambda }}{dt}=0\). There exists the following center manifold \(W^{c}\):

$$ \bigl\{ (U_{1}, U_{2}, \ldots , U_{n-1})^{T}=h(U_{n}, \bar{\lambda })= \bigl(h_{1}(U_{n}, \bar{\lambda }), h_{2}(U_{n}, \bar{\lambda }), \ldots , h_{n-1}(U_{n}, \bar{\lambda }) \bigr)^{T}\bigr\} . $$

Since \(h(0,0)=0\), \(h_{U_{n}}(0,0)=\frac{\partial h}{\partial {U_{n}}}(0,0)=0\), and \(h_{\bar{\lambda }}(0,0)=\frac{\partial h}{\partial \bar{\lambda }}(0,0)=0\), and by use of the Taylor expansion of \(h(U_{n},\bar{\lambda })\), we can suppose that \(U_{k}=d_{k} U_{n}^{2}+e_{k}U_{n}\bar{\lambda }+f_{k}\bar{ \lambda }^{2}\), \(k=1, 2,\ldots , n-1\). Here, the coefficients \(d_{k}\), \(e_{k}\), \(f_{k}\) will be determined later. Actually, putting the above equalities about \(U_{k}\) into (2.2) and (2.3), through complicated computing, we can obtain

\left(\begin{array}{c}{U}_{1}\\ {U}_{2}\\ \cdots \\ {U}_{n-1}\end{array}\right)=\left(\begin{array}{c}\frac{{2}^{2}P(2)(\delta +\omega )(2\u3008{k}^{2}\u3009-\u3008{k}^{3}\u3009)}{P(1)\u3008{k}^{3}\u3009}{U}_{n}^{2}+o({U}_{n}^{3})\\ \frac{{3}^{2}P(3)(\delta +\omega )(3\u3008{k}^{2}\u3009-\u3008{k}^{3}\u3009)}{P(1)\u3008{k}^{3}\u3009}{U}_{n}^{2}+o({U}_{n}^{3})\\ \cdots \\ \frac{{n}^{2}P(n)(\delta +\omega )(n\u3008{k}^{2}\u3009-\u3008{k}^{3}\u3009)}{P(1)\u3008{k}^{3}\u3009}{U}_{n}^{2}+o({U}_{n}^{3})\end{array}\right).

(2.4)

As a result, from (2.3) and (2.4), system (2.1) reduced on the center manifold reads as follows:

$$ \frac{dU_{n}(t)}{dt}=\frac{\langle k^{2} \rangle }{\langle k \rangle }U_{n} \bar{\lambda }- (\omega +\delta ) \biggl(\frac{\beta }{ \langle k \rangle }+ \frac{\langle k^{3} \rangle }{\langle k^{2} \rangle } \biggr)U_{n}^{2}+o \bigl(U_{n}^{3}\bigr)\triangleq G(U_{n},\bar{ \lambda }). $$

(2.5)

Since \(G(0,0)=0\), \(G_{U_{n}}(0,0)=0\), \(G_{\bar{\lambda }}(0,0)=0\), and \(G_{U_{n},\bar{\lambda }}^{2}-G_{U_{n}, U_{n}}G_{\bar{\lambda }, \bar{ \lambda }}=\frac{{\langle k^{2} \rangle }^{2}}{{\langle k \rangle } ^{2}}\neq 0\). Thus the proof is completed. □

### Remark 2.1

We also can determine the direction of the bifurcation in Theorem 1 as follows. For system (2.1), the endemic equilibrium should satisfy the following equation:

$$ \frac{1}{\langle k \rangle }\sum _{h=1}^{n}\frac{\lambda h^{2}p(h)}{\frac{ \delta +\omega }{1-\beta \varTheta }+\lambda h \varTheta } =1. $$

(2.6)

Since \(R_{0}=\frac{\lambda \langle k^{2} \rangle }{(\delta +\omega ) \langle k \rangle }\), we multiply the denominator and numerator of (2.6) by \(\frac{\langle k^{2} \rangle }{(\delta +\omega )\langle k \rangle }\). Then we can obtain the following equality:

$$ \frac{1}{\langle k \rangle }\sum _{h=1}^{n}\frac{R_{0} h^{2}p(h)}{\frac{ \langle k^{2} \rangle }{\langle k \rangle (1-\beta \varTheta )}+R_{0} h \varTheta } =1. $$

(2.7)

Consequently, according to (2.7), the derivative of *Θ* with respect to \(R_{0}\) at the critical value \((R_{0}, \varTheta )=(1,0)\) is \(\frac{\partial \varTheta }{\partial R_{0}}|_{(R_{0}, \varTheta )=(1,0)}=\frac{\langle k^{2} \rangle ^{2}}{\delta \langle k^{2} \rangle ^{2}+{\langle k \rangle }^{2}\langle k^{3} \rangle }>0\). So the endemic equilibrium curve bifurcates forward. In other words, system (2.1) exhibits forward bifurcation at \(R_{0}=1\).

### Remark 2.2

In system (2.5), let \(\bar{\lambda }=0\), we have

$$ \frac{dU_{n}(t)}{dt}=- (\omega +\delta ) \biggl( \frac{\beta }{ \langle k \rangle }+\frac{\langle k^{3} \rangle }{\langle k^{2} \rangle } \biggr)U_{n}^{2}+o \bigl(U_{n}^{3}\bigr). $$

(2.8)

So \(E_{0}\) is stable when \(R_{0}=1\).

### Remark 2.3

For the epidemic model on networks, most existing literature works only investigate the two cases, i.e., \(R_{0}>1\) and \(R_{0}<1\). In Theorem 2.1, Remark 2.1, and Remark 2.2, we pay attention to the critical case \(R_{0}=1\) and present the strict theoretical proof for the existence of the transcritical bifurcation. Thus, the above results improve and supplement the corresponding results in [7] and [29, 30].