We define, firstly, a threshold condition for the full model (1) as follows:
$$ \mathcal{R}_{02}=\frac{(\theta+\sigma\psi+\mu_{h})b^{2} \beta_{mh}\beta_{hm}N_{m}^{*}}{d_{m}(1q)(\mu_{h}+\eta_{h})(\theta +\psi+\mu_{h})N_{h}}. $$
In fact, the value of \(\mathcal{R}_{02}\) determines the existence of a positive equilibrium of model (1).
For model (1), we get two nontrivial diseasefree equilibria, that is, the diseasefree equilibrium without mosquitoes \(E_{01}\) for \(\mathcal{R}_{01}\leq1\), and the diseasefree equilibrium with mosquitoes \(E_{02}\) for \(\mathcal{R}_{01}>1\) and \(\mathcal {R}_{02}<1\), where \(E_{01}\) and \(E_{02}\) are given by
$$ \begin{gathered} E_{01}= \biggl( \frac{(\theta+\mu_{h})N_{h}}{\psi+\theta+\mu_{h}}, \frac{\psi N_{h}}{\psi+\theta+\mu_{h}}, 0, 0, 0, 0 \biggr) ,\\ E_{02}= \biggl( \frac{(\theta+\mu_{h})N_{h}}{\psi+\theta+\mu_{h}}, \frac{\psi N_{h}}{\psi+\theta+\mu_{h}}, 0, 0, N_{m}^{*}, 0 \biggr) . \end{gathered} $$
Further, model (1) admits endemic equilibria \(E^{*}(S_{h(1,2)}^{*}, V_{h(1,2)}^{*}, I_{h(1,2)}^{*}, R_{h(1,2)}^{*}, S_{m(1,2)}^{*}, I_{m(1,2)}^{*})\) for \(\mathcal{R}_{01}>1\) and \(\mathcal {R}_{02}>1\), where
$$ \begin{aligned} &S_{h(1,2)}^{*}= \biggl( \frac{\sigma b^{2}\beta_{mh}\beta _{hm}N_{m}^{*}I_{h(1,2)}^{*}}{\psi N_{h}[d_{m}(1q)N_{h}+b\beta _{hm}I_{h(1,2)}^{*}]}+\frac{\mu_{h}+\theta}{\psi} \biggr) V_{h(1,2)}^{*}, \qquad R_{h(1,2)}^{*}=\frac{\eta_{h}}{\mu _{h}}I_{h(1,2)}^{*}, \\ &I_{m(1,2)}^{*}=\frac{b\beta _{hm}N_{m}^{*}I_{h(1,2)}^{*}}{d_{m}(1q)N_{h}+b\beta_{hm}I_{h(1,2)}^{*}}, \quad\quad S_{m(1,2)}^{*}=N_{m}^{*}I_{m(1,2)}^{*}, \\ &V_{h(1,2)}^{*}=\frac{(\eta_{h}+\mu_{h})\psi I_{h(1,2)}^{*}}{(\sigma M+\theta+\mu_{h}+\sigma\psi)M}, \qquad M=\frac{b^{2}\beta_{mh}\beta _{hm}N_{m}^{*}I_{h(1,2)}^{*}}{N_{h}[d_{m}(1q)N_{h}+b\beta _{hm}I_{h(1,2)}^{*}]}, \end{aligned} $$
and \(I_{h(1,2)}^{*}\) is obtained by the solutions \(I_{h}\) of the following equation:
$$ AI_{h}^{2}+BI_{h}+C=0 $$
(5)
with
$$\begin{aligned}& A=b^{2}\beta_{hm}^{2}( \mu_{h}+\eta_{h}) \bigl[ \bigl(\sigma b\beta _{mh}N_{m}^{*}+(\theta+\mu_{h})N_{h} \bigr) \bigl(b\beta_{mh}N_{m}^{*}+(\psi + \mu_{h})N_{h} \bigr)+\theta\psi N_{h}^{2} \bigr], \\& \begin{aligned} B&= b\beta_{hm}N_{h} \bigl\{ \sigma\mu_{h}b^{3} \beta_{mh}^{2}\beta_{hm}N_{m}^{*2} +2d_{m}(1q)\mu_{h}(\mu_{h}+\eta_{h}) (\psi+\theta+\mu _{h})N_{h}^{2} \\ &\quad{} +b\beta_{mh}N_{m}^{*}N_{h} \bigl[d_{m}(1q) (\mu_{h}+\eta_{h}) \bigl(\theta+ \mu _{h}+\sigma(\psi+\mu_{h}) \bigr) \mu_{h}b \beta_{hm}(\theta+\sigma\psi+\mu_{h}) \bigr] \bigr\} , \end{aligned} \\& \begin{aligned} C&= \mu_{h}d_{m}(1q)N_{h}^{3} \bigl[d_{m}(1q) (\mu_{h}+\eta_{h}) (\psi + \theta+\mu_{h})N_{h} (\theta+\sigma\psi+ \mu_{h})b^{2}\beta_{mh}\beta_{hm}N_{m}^{*} \bigr] \\ & =\mu_{h}d_{m}(1q)N_{h}^{3} \frac{1}{d_{m}(1q)(\mu_{h}+\eta _{h})(\psi+\theta+\mu_{h})N_{h}}(1\mathcal{R}_{02}). \end{aligned} \end{aligned}$$
It is obvious that \(A>0\) for positive parameters, and \(\mathcal {R}_{02}\geq1\) if and only if \(C\leq0\). Further, if \(B>0\) and \(C>0\), there is no positive root of equation (5); if \(B<0\) and \(B^{2}4AC>0\), there are two positive roots of equation (5); if \(C<0\), there is a unique positive root of equation (5). According to the abovementioned discussion, we have a conclusion as follows.
Theorem 2
If
\(\mathcal{R}_{01}\leq1\), then model (1) has a unique diseasefree equilibrium without mosquitoes
\(E_{01}\); if
\(\mathcal {R}_{01}>1\)
and
\(\mathcal{R}_{02}<1\), then model (1) has a unique diseasefree equilibrium with mosquitoes
\(E_{02}\). Furthermore, if
\(\mathcal{R}_{01}>1\), the following statements are valid:

(i)
if
\(C\leq0\), then model (1) has a unique endemic equilibrium;

(ii)
if
\(B<0\)
and
\(B^{2}4AC>0\), then model (1) has two endemic equilibria;

(iii)
if
\(B>0\)
and
\(C\geq0\), then model (1) has no endemic equilibrium.
Noting that \(C\leq0\) if and only if \(\mathcal{R}_{02}\geq1\). It is clear from Theorem 2 (Case (i)) that the model has a unique endemic equilibrium if \(\mathcal{R}_{01}\geq1\) and \(\mathcal {R}_{02}>1\). Further, Case (ii) indicates the possibility of backward bifurcation (where a local asymptotically stable diseasefree equilibrium coexists with a locally asymptotically stable endemic equilibrium) in model (1) for \(\mathcal{R}_{01}\geq1\) and \(\mathcal{R}_{02}<1\). To check for this, the discriminant \(B^{2}4AC\) is set to zero and solved for the critical value of \(\mathcal{R}_{02}\), denoted by \(\mathcal{R}_{02}^{c}\). Thus, backward bifurcation would occur for values of \(\mathcal{R}_{02}\) such that \(\mathcal{R}_{01}\geq 1\) and \(\mathcal{R}_{02}^{c}<\mathcal{R}_{02}<1\).
To obtain the stability of the equilibria of model (1), we take out the variate of \(R_{h}(t)\) and linearize model (1) about equilibria \((S_{h}^{*},V_{h}^{*},I_{h}^{*},S_{m}^{*},I_{m}^{*})\) and we get the following Jacobian matrix:
$$ J= \begin{pmatrix} a_{11}\lambda&\theta&0&0&b\beta_{mh}\frac{S_{h}^{*}}{N_{h}}\\ \psi&a_{22}\lambda&0&0&\sigma b\beta_{mh}\frac{V_{h}^{*}}{N_{h}}\\ b\beta_{mh}\frac{I_{m}^{*}}{N_{h}}&\sigma b\beta_{mh}\frac {I_{m}^{*}}{N_{h}}&(\eta_{h}+\mu_{h})\lambda&0&b\beta_{mh}\frac {S_{h}^{*}+\sigma V_{h}^{*}}{N_{h}}\\ 0&0&b\beta_{hm}\frac{S_{m}^{*}}{N_{h}}&a_{44}\lambda&a_{45}\\ 0&0&b\beta_{hm}\frac{S_{m}^{*}}{N_{h}}&a_{54}&a_{55}\lambda \end{pmatrix} , $$
where
$$\begin{aligned} &a_{11}= \biggl( b\beta_{mh} \frac{I_{m}^{*}}{N_{h}}+\psi+\mu_{h} \biggr) ,\qquad a_{22}= \biggl( \sigma b\beta_{mh}\frac {I_{m}^{*}}{N_{h}}+\theta+ \mu_{h} \biggr) , \\ &a_{45}=r_{m}e^{(d_{j}\tau+\alpha N_{m}^{*})} \bigl[ (1q)e^{\lambda \tau} \alpha N_{m}^{*}+\alpha qI_{m}^{*} \bigr] , \\ &a_{54}=b\beta_{hm}\frac{I_{h}(t)}{N_{h}}\alpha qI_{m}^{*}(t\tau )e^{(d_{j}\tau+\alpha N_{m}^{*}}, \\ &a_{44}=r_{m}e^{(d_{j}\tau+\alpha N_{m}^{*})} \bigl( e^{\lambda\tau } \alpha N_{m}^{*}+\alpha qI_{m}^{*} \bigr) d_{m}, \\ &a_{55}=qr_{m}e^{(d_{j}\tau+\alpha N_{m}^{*})} \bigl( e^{\lambda\tau } \alpha I_{m}^{*} \bigr) d_{m}, \end{aligned}$$
and λ is an eigenvalue. We obtain the characteristic equation about \(E_{01}\) according to the Jacobian matrix of model (1)
$$ \begin{aligned} F(\lambda)&= ( \lambda+\eta_{h}+ \mu_{h} ) \bigl[ \lambda ^{2}+(\theta+\psi+2 \mu_{h})\lambda+(\theta+\psi+\mu_{h})\mu _{h} \bigr] \\ &\quad{} \times \bigl( \lambda+d_{m}qr_{m}e^{(d_{j}+\lambda)\tau} \bigr) \bigl( \lambda+d_{m}r_{m}e^{(d_{j}+\lambda)\tau} \bigr) . \end{aligned} $$
To continue, we recall Theorem 4.7 in [26], which states that \(\lambda=A+Be^{\lambda\tau}\) has a root with positive real part if \(A+B>0\), and has no roots with nonnegative real parts if \(A+B<0\) and \(B\geq A\). By this theorem, we see that all roots of the above characteristic equation have negative real parts for \(\mathcal {R}_{01}<1\). Therefore, \(E_{01}\) is asymptotically stable.
Now, on the globally asymptotically stable diseasefree equilibrium without mosquitoes \(E_{01}\) of model (1), we have Theorem 3.
Theorem 3
If
\(\mathcal{R}_{01}<1\), then model (1) has a unique diseasefree equilibrium without mosquitoes
\(E_{01}\), which is globally asymptotically stable.
Proof
It obvious that \(\lim_{t\rightarrow\infty} S_{m}(t)=\lim_{t\rightarrow\infty} I_{m}(t)=0\) for \(\mathcal{R}_{01}<1\) depending on Theorem 1. So we merely prove that
$$ \lim_{t\rightarrow\infty} S_{h}(t)= \frac{(\theta+\mu _{h})N_{h}}{(\psi+\theta+\mu_{h})}, \qquad\lim_{t\rightarrow \infty} V_{h}(t)= \frac{\psi N_{h}}{(\psi+\theta+\mu_{h})}, $$
(6)
and
$$ \lim_{t\rightarrow\infty}I_{h}(t)=\lim_{t\rightarrow\infty} R_{h}(t)=0. $$
Due to \(\lim_{t\rightarrow\infty} I_{m}(t)=0\), for a small enough positive constant ϵ, there is a constant \(T>0\) such that \(I_{m}(t)<\epsilon\), for all \(t>T\). Then, from the third equation of model (1), we have
$$ \frac{\mathrm{d} I_{h}(t)}{\mathrm{d} t}< b\beta_{mh}\epsilon(\mu _{h}+ \eta_{h})I_{h}(t), \quad\text{for all }t>T. $$
By the comparison theorem and the arbitrariness of ϵ, we have \(\lim_{t\rightarrow\infty} I_{h}(t)=0\). Further, it follows that \(\lim_{t\rightarrow\infty} R_{h}(t)=0\).
From the first and second equations of model (1), we have
$$ \begin{gathered} \mu_{h}N_{h}+\theta V_{h}(t) \biggl( b \beta_{mh}\frac{\epsilon }{N_{h}}+\psi+\mu_{h} \biggr) S_{h}(t)\\ \quad \leq\frac{\mathrm{d}S_{h}(t)}{\mathrm{d}t}\leq\mu_{h}N_{h}+ \theta V_{h}(t)(\psi+\mu_{h})S_{h}(t) \end{gathered} $$
and
$$ \psi S_{h}(t) \biggl( \sigma b\beta_{mh} \frac{\epsilon}{N_{h}}+\theta +\mu_{h} \biggr) V_{h}(t)\leq \frac{\mathrm{d}V_{h}(t)}{\mathrm {d}t}\leq\psi S_{h}(t)(\theta+\mu_{h})V_{h}(t). $$
Then it is easy to see that (6) is valid, that is, \(E_{01}\) is globally attractive. This completes the proof. □
Finally, we give a conclusion on the global attractiveness of the diseasefree equilibrium with mosquitoes \(E_{02}\) of model (1).
Theorem 4
Supposing that
\(\mathcal{R}_{01}>1\). If
$$ \mathcal{R}_{02}^{*}:=\frac{b^{2}\beta_{mh}\beta_{hm}N_{m}^{*}}{d_{m}(1 qe^{d_{m}\tau})(\mu_{h}+\eta_{h})N_{h}}< 1, $$
then model (1) has a unique diseasefree equilibrium with mosquitoes
\(E_{02}\), which is globally attractive.
Proof
From the expressions of \(\mathcal{R}_{02}\) and \(\mathcal{R}_{02}^{*}\), we get \(\mathcal{R}_{02}<1\) for \(\mathcal{R}_{02}^{*}<1\). Therefore, model (1) has a unique diseasefree equilibrium with mosquitoes \(E_{02}\) for \(\mathcal{R}_{02}^{*}<1\) and \(\mathcal {R}_{01}>1\). From the sixth equation of model (1) we get
$$ \frac{\mathrm{d} I_{m}(t)}{\mathrm{d} t}\geqd_{m}I_{m}(t). $$
By integrating the above inequality from \(t\tau\) to t, we obtain \(I_{m}(t\tau)\leq e^{d_{m}\tau}I_{m}(t)\). Then
$$ \textstyle\begin{cases} \frac{\mathrm{d} I_{m}(t)}{\mathrm{d} t}\leq d_{m} ( qe^{d_{m}\tau }1 ) I_{m}(t)+b\beta_{hm} \frac{N_{m}^{*}}{N_{h}}I_{h}(t), \\ \frac{\mathrm{d} I_{h}(t)}{\mathrm{d} t}\leq b\beta _{mh}I_{m}(t)( \eta_{h}+\mu_{h})I_{h}(t). \end{cases} $$
Consider the following auxiliary system:
$$ \textstyle\begin{cases} \frac{\mathrm{d} u(t)}{\mathrm{d} t}=d_{m} ( qe^{d_{m}\tau }1 ) u(t)+b\beta_{hm}\frac{N_{m}^{*}}{N_{h}}v(t), \\ \frac{\mathrm{d} v(t)}{\mathrm{d} t}=b\beta_{mh}u(t)(\eta_{h}+ \mu_{h})v(t). \end{cases} $$
(7)
It is obvious that the equilibrium \((0, 0)\) always exists. The characteristic equation of model (7) about \((0, 0)\) is
$$ I(\lambda)=\lambda^{2}+(b_{2}a_{1}) \lambda(b_{2}a_{1}+a_{2}b_{1})=0, $$
(8)
where \(a_{1}=d_{m} ( qe^{d_{m}\tau}1 ) \), \(a_{2}=b\beta _{hm}N_{m}^{*}/N_{h}\), \(b_{1}=b\beta_{mh}\) and \(b_{2}=\eta_{h}+\mu _{h}\). To obtain two negative solutions about (8), it is required that
$$ \lambda_{1}+\lambda_{2}=a_{1}b_{2}< 0, \quad \quad \lambda_{1}\cdot\lambda_{2}=(b_{2}a_{1}+a_{2}b_{1})>0. $$
So we see that the equilibrium \((0, 0)\) of model (7) is globally asymptotically stable for \({\mathcal{R}_{02}^{*}<1}\).
According to the above discussion and the comparison theorem of differential equations, we know that \(\lim_{t\rightarrow\infty} I_{m}(t)=0\) and \(\lim_{t\rightarrow\infty} I_{h}(t)=0\) for \(\mathcal {R}_{01}>1\) and \(\mathcal{R}_{02}^{*}<1\). Finally, in the light of Theorem 1, we get \(\lim_{t\rightarrow\infty }(S_{h}(t),I_{h}(t),V_{h}(t),R_{h}(t),S_{m}(t),I_{m}(t))=E_{02}\). This completes the proof. □
Remark 1
Obviously, \(qe^{d_{m}\tau}\approx q\) due to the small vertical transmission probability q according to existing literature, therefore \(\mathcal{R}_{02}\approx\mathcal{R}_{02}^{*}\).
To discuss the stability of the endemic equilibrium \(E^{*}\), we write the corresponding characteristic equation for \(E^{*}\) as follows:
$$\begin{aligned} H(\lambda)&= \bigl[ \lambda+d_{m} \bigl( 1+\alpha N_{m}^{*}e^{\lambda \tau} \bigr) \bigr] \biggl\{ (\lambda+\mu_{h}) \biggl\{ \biggl[ (\lambda+ \mu_{h}+\eta_{h}) \biggl( \lambda+d_{m} \bigl( 1qe^{\lambda\tau} \bigr) \\ &\quad{} +b\beta_{hm}\frac{I_{h(1,2)}^{*}}{N_{h}} \biggr) b^{2} \beta_{mh}\beta_{hm}\frac{S_{m(1,2)}^{*}(S_{h(1,2)}^{*}+\sigma V_{h(1,2)}^{*})}{N_{h}^{2}} \biggr] \biggl[ \lambda^{2}+ \biggl( (1+\sigma )b\beta_{mh} \frac{I_{m(1,2)}^{*}}{N_{h}} \\ &\quad{} +\psi+\theta+\mu_{h} \biggr) \lambda+b \beta_{mh} \frac {I_{m(1,2)}^{*}}{N_{h}} \biggl( \sigma b\beta_{mh} \frac {I_{m(1,2)}^{*}}{N_{h}}+\theta+ \sigma\psi+\mu_{h} \biggr) \biggr] \\ &\quad{} +\sigma b^{3}\beta_{mh}^{2} \beta_{hm}\frac {V_{h(1,2)}^{*}I_{m(1,2)}^{*}S_{m(1,2)}^{*}}{N_{h}^{3}} \biggl[ \sigma \biggl( \lambda +b \beta_{mh}\frac{I_{m(1,2)}^{*}}{N_{h}} +\psi \biggr) +\theta+\mu_{h} \biggr] \biggr\} \\ &\quad{} \mu_{h}b\beta_{mh}\frac{I_{m(1,2)}^{*}}{N_{h}}(\lambda+\mu _{h}+\eta_{h}) \biggl[ \lambda +d_{m} \bigl( 1qe^{\lambda\tau} \bigr) +b\beta_{hm}\frac {I_{h(1,2)}^{*}}{N_{h}} \biggr] \\ &\quad{} \times \biggl[ \lambda+\sigma b\beta_{mh}\frac {I_{m(1,2)}^{*}}{N_{h}}+ \theta+ \sigma\psi+\mu_{h} \biggr] \biggr\} =0. \end{aligned}$$
(9)
Nevertheless, the study of solving this transcendental equation (9) is very difficult. And though we get the conditions by math software, it is not difficult to imagine that the conditions are very complex. Of course, it is very difficult to make a rational interpretation on biology. So the solving of (9) is insignificant, and we omit it.