By direct computation we get that if \([\alpha \delta_{1}-(\delta_{0}+ \delta_{1})(\delta_{0}+\delta_{2}+\delta_{3})](I_{*}+a)>\beta \delta _{1}(\delta_{0}+\delta_{1})\), then system (2) has a viral equilibrium \(P_{*}(S_{*}, E_{*}, I_{*}, R_{*}, V_{*})\), where
$$\begin{aligned}& S_{*} =\frac{(\delta_{0}+\delta_{1})(\delta_{0}+\delta_{2}+\delta _{3})(I_{*}+a)(I_{*}+c)+\beta \delta_{1}(\delta_{0}+\delta_{1})(I_{*}+c)}{[ \alpha \delta_{1}-(\delta_{0}+\delta_{1})(\delta_{0}+\delta_{2}+\delta _{3})](I_{*}+a)-\beta \delta_{1}(\delta_{0}+\delta_{1})}, \\& E_{*} =\frac{(\delta_{0}+\delta_{2}+\delta_{3})I_{*}}{\delta_{1}}+\frac{ \beta I_{*}}{\delta_{1}(I_{*}+a)}, \\& R_{*} =\frac{\delta_{2}I_{*}}{\delta_{0}}+\frac{\beta I_{*}}{\delta _{0}(I_{*}+a)}, \\& V_{*} =\frac{\mu [(\delta_{0}+\delta_{1})(\delta_{0}+\delta_{2}+ \delta_{3})(I_{*}+a)(I_{*}+c)+\beta \delta_{1}(\delta_{0}+\delta_{1})(I _{*}+c)]}{(\delta_{0}+\eta)\{[\alpha \delta_{1}-(\delta_{0}+\delta _{1})(\delta_{0}+\delta_{2}+\delta_{3})](I_{*}+a)-\beta \delta_{1}( \delta_{0}+\delta_{1})\}}, \end{aligned}$$
and \(I_{*}\) is the positive root of the equation
$$ a_{3}I^{3}+a_{2}I^{2}+a_{1}I+a_{0}=0, $$
(3)
where
$$\begin{aligned}& a_{0} =\delta_{1}B_{6}c-B_{1}c( \delta_{0}+\eta)+\delta_{1}ac\bigl[B_{7}-B _{2}(\delta_{0}+\eta)\bigr] \\& \hphantom{a_{0}=}{}+Aa\delta_{1}(\delta_{0}+\eta) (B_{3}a-B_{4}), \\& a_{1} =B_{6}a^{2}\delta_{1}-a^{2} \delta_{1}(\delta_{0}+\eta) (B_{1} \delta_{1}+B_{3}B_{5}) \\& \hphantom{a_{1}=}{}+\delta_{1}(a+c) \bigl(B_{7}-B_{2}( \delta_{0}+\eta)\bigr) \\& \hphantom{a_{1}=}{}+a(\delta_{0}+\eta) \bigl(B_{4}B_{5} \delta_{1}-B_{3}\beta (\delta_{0}+ \delta_{1})\bigr) \\& \hphantom{a_{1}=}{}+(\delta_{0}+\eta) \bigl(2AB_{3}a \delta_{1}-AB_{4}\delta_{1}+B_{4}\beta (\delta_{0}+\delta_{1})\bigr), \\& a_{2} =\delta_{1}(2B_{6}a+B_{7})+ \delta_{1}(\delta_{0}+\eta) (AB_{3}+B _{4}B_{5}-B_{2}) \\& \hphantom{a_{2}=}{}-(\delta_{0}+\eta) \bigl(B_{3}\beta ( \delta_{0}+\delta_{1})+\delta_{1}(B _{1}+B_{3}B_{5})\bigr), \\& a_{3} =\delta_{1}B_{6}-\delta_{1}( \delta_{0}+\eta) (B_{1}+B_{3}B_{5}), \end{aligned}$$
with
$$\begin{aligned}& B_{1} =(\delta_{0}+\mu) (\delta_{0}+ \delta_{1}) (\delta_{0}+\delta _{2}+ \delta_{3}), \\& B_{2} =\beta \delta_{1}(\delta_{0}+\mu) ( \delta_{0}+\delta_{1}), \\& B_{3} =\alpha \delta_{1}-(\delta_{0}+ \delta_{1}) (\delta_{0}+\delta _{2}+ \delta_{3}), \\& B_{4} =\beta \delta_{1}(\delta_{0}+ \delta_{1}), \\& B_{5} =\frac{(\delta_{0}+\delta_{1})(\delta_{0}+\delta_{2}+\delta _{3})}{\delta_{1}}, \\& B_{6} =\eta \mu (\delta_{0}+\delta_{1}) ( \delta_{0}+\delta_{2}+\delta _{3}), \\& B_{7} =\eta \mu \beta \delta_{1}(\delta_{0}+ \delta_{1}). \end{aligned}$$
For Eq. (3), we have the following results.
Lemma 1
If
\(a_{3}=0\), then
-
(1)
if
\(a_{2}=0\)
and
\(a_{0}/a_{1}<0\), then there exists a unique positive root
\(I_{*}=-a_{0}/a_{1}\)
of Eq. (3);
-
(2)
When
\(\Delta >0\), if
\(a_{2}/a_{1}<0\)
and
\(a_{0}/a_{2}>0\), then there exist two positive roots
\(I_{*}^{(1)}=I_{*}^{+}\)
and
\(I_{*}^{(2)}=I _{*}^{-}\); if
\(a_{0}/a_{2}<0\), then there is a unique positive root
\(I_{*}=I_{*}^{+}\)
with
\(a_{1}>0\)
or
\(I_{*}=I_{*}^{-}\)
with
\(a_{2}<0\); if
\(a_{0}=0\)
and
\(a_{1}/a_{2}<0\), then there is a unique positive root
\(I_{*}=-a_{1}/a_{2}\);
-
(3)
if
\(\Delta =0\)
and
\(a_{1}/a_{2}<0\), then there is a unique positive root
\(I_{*}=-a_{1}/(2a_{2})\). Here
\(\Delta =a_{1}^{2}-4a_{2}a_{0}\), \(I_{*}^{+}=(-a_{1}+\sqrt{ \Delta })/(2a_{2})\), and
\(I_{*}^{-}=-(a_{1}+\sqrt{\Delta })/(2a_{2})\).
Lemma 2
For
\(a_{3}\neq 0\), let
\(l_{2}=a_{2}/a_{3}\), \(l_{1}=a_{1}/a_{3}\), and
\(l_{0}=a_{0}/a_{3}\). Then:
-
(1)
if
\(l_{0}<0\), then Eq. (3) has at least one positive root;
-
(2)
if
\(l_{0}\geq 0\)
and
\(l_{2}^{2}-3l_{1}\leq 0\), then Eq. (3) has no positive root;
-
(3)
if
\(l_{0}\geq 0\)
and
\(l_{2}^{2}-3l_{1}>0\), then Eq. (3) has a positive root if and only if
\(\frac{-l _{2}+\sqrt{l_{2}^{2}-3l_{1}}}{3}>0\)
and
\(h (\frac{-l_{2}+\sqrt{l _{2}^{2}-3l_{1}}}{3} )\leq 0\), where
\(h(I)=I^{3}+l_{2}I^{2}+l_{1}I+l _{0}\).
Then, we can obtain the linearization of system (2). Let \(u_{1}(t)=S(t)-S_{*}\), \(u_{2}(t)=E(t)-E_{*}\), \(u_{3}(t)=I(t)-I_{*}\), \(u_{4}(t)=R(t)-R_{*}\), \(u_{5}(t)=V(t)-V_{*}\). We can rewrite system (2) as follows:
$$ \textstyle\begin{cases} \dot{u}_{1}(t)=a_{11}u_{1}(t)+a_{13}u_{3}(t)+a_{15}u_{5}(t)+\sum_{i+j\geq 2}\frac{1}{i!j!}f_{ij}^{(1)}u_{1}^{i}(t)u_{3}^{j}(t), \\ \dot{u}_{2}(t)=a_{21}u_{1}(t)+a_{22}u_{2}(t)+a_{23}u_{3}(t)+\sum_{i+j\geq 2}\frac{1}{i!j!}f_{ij}^{(2)}u_{1}^{i}(t)u_{3}^{j}(t), \\ \dot{u}_{3}(t)=a_{32}u_{2}(t)+a_{33}u_{3}(t)+b_{33}u_{3}(t-\tau)+ \sum_{i\geq 2}\frac{1}{i!}f_{i}^{(3)}u_{3}^{i}(t-\tau), \\ \dot{u}_{4}(t)=a_{44}u_{4}(t)+b_{43}u_{3}(t-\tau)+\sum_{i \geq 2}\frac{1}{i!}f_{i}^{(4)}u_{3}^{i}(t-\tau), \\ \dot{u}_{5}(t)=a_{51}u_{1}(t)+a_{55}u_{5}(t), \end{cases} $$
(4)
where
$$\begin{aligned}& a_{11} =- \biggl[\delta_{0}+\mu +\frac{\alpha I_{*}(I_{*}+c)}{(S_{*}+I _{*}+c)^{2}} \biggr],\qquad a_{13}=- \frac{\alpha S_{*}(S_{*}+c)}{(S_{*}+I_{*}+c)^{2}},\qquad a_{15}=\eta, \\& a_{21} =\frac{\alpha I_{*}(I_{*}+c)}{(S_{*}+I_{*}+c)^{2}},\qquad a_{22}=-( \delta_{0}+\delta_{1}), \qquad a_{23}= \frac{\alpha S_{*}(S_{*}+c)}{(S_{*}+I _{*}+c)^{2}}, \\& a_{32} =\delta_{1},\qquad a_{33}=-( \delta_{0}+\delta_{3}),\qquad b_{33}=- \biggl[ \delta_{2}+\frac{a\beta }{(I_{*}+a)^{2}} \biggr], \\& a_{44} =-\delta_{0},\qquad a_{51}=\mu,\qquad a_{55}=-(\delta_{0}+\eta), \\& b_{43}= \biggl[ \delta_{2}+\frac{a\beta }{(I_{*}+a)^{2}} \biggr], \\& f_{ij}^{(k)} =\frac{\partial^{i+j}f^{(k)}(S_{*}, E_{*}, I_{*}, R_{*}, V_{*})}{\partial u_{1}^{i}(t)\partial u_{3}^{j}(t)}, \\& f_{i}^{(k)} =\frac{\partial^{i}f^{(k)}(S_{*}, E_{*}, I_{*}, R_{*}, V _{*})}{\partial u_{3}^{i}(t-\tau)}, \\& f^{(1)} =A-\delta_{0} u_{1}(t)-\frac{\alpha u_{1}(t)u_{3}(t)}{u_{1}(t)+u _{3}(t)+c}- \mu S(t), \\& f^{(2)} =\frac{\alpha u_{1}(t)u_{3}(t)}{u_{1}(t)+u_{3}(t)+c}-(\delta _{0}+ \delta_{1})u_{2}(t), \\& f^{(3)} =\delta_{1} u_{2}(t)-(\delta_{0}+ \delta_{3}) u_{3}(t)-\delta _{2} u_{3}(t- \tau)-\frac{\beta u_{3}(t-\tau)}{u_{3}(t-\tau)+a}, \\& f^{(4)} =\delta_{2} u_{3}(t-\tau)- \delta_{0} u_{4}(t)+\frac{\beta u _{3}(t-\tau)}{u_{3}(t-\tau)+a}. \end{aligned}$$
Then we obtain the linearized system of system (4)
$$ \textstyle\begin{cases} \dot{u}_{1}(t)=a_{11}u_{1}(t)+a_{13}u_{3}(t)+a_{15}u_{5}(t), \\ \dot{u}_{2}(t)=a_{21}u_{1}(t)+a_{22}u_{2}(t)+a_{23}u_{3}(t), \\ \dot{u}_{3}(t)=a_{32}u_{2}(t)+a_{33}u_{3}(t)+b_{33}u_{3}(t-\tau), \\ \dot{u}_{4}(t)=a_{44}u_{4}(t)+b_{43}u_{3}(t-\tau), \\ \dot{u}_{5}(t)=a_{51}u_{1}(t)+a_{55}u_{5}(t). \end{cases} $$
(5)
The characteristic equation is
$$\begin{aligned} P(\lambda) =&\lambda^{5}+p_{4} \lambda^{4}+p_{3}\lambda^{3}+p_{2}\lambda ^{2}+p_{1}\lambda +p_{0}+\bigl(q_{4} \lambda^{4}+q_{3}\lambda^{3}+q_{2} \lambda^{2}+q_{1}\lambda +q_{0} \bigr)e^{-\lambda \tau } \\ =&0, \end{aligned}$$
(6)
where
$$\begin{aligned}& p_{0} =a_{44}(a_{22}a_{33}-a_{23}a_{32}) (a_{15}a_{51}-a_{11}a_{55})-a _{13}a_{21}a_{32}a_{44}a_{55}, \\& p_{1} =a_{55}\bigl(a_{11}a_{22}(a_{33}+a_{44})+a_{33}a_{44}(a_{11}+a_{22}) \bigr)+a _{11}a_{22}a_{33}a_{44} \\& \hphantom{p_{1}=}{}-a_{23}a_{32}(a_{11}a_{44}+a_{11}a_{55}+a_{44}a_{55})+a_{15}a_{23}a _{32}a_{51} \\& \hphantom{p_{1}=}{}-a_{15}a_{51}(a_{22}a_{33}+a_{22}a_{44}+a_{33}a_{44})+a_{13}a_{21}a _{32}(a_{44}+a_{55}), \\& p_{2} =a_{23}a_{32}(a_{11}+a_{44}+a_{55})+a_{15}a_{51}(a_{22}+a_{33}+a _{44})-a_{13}a_{21}a_{32} \\& \hphantom{p_{2}=}{}-\bigl(a_{11}a_{22}(a_{33}+a_{44})+a_{33}a_{44}(a_{11}+a_{22}) \bigr) \\& \hphantom{p_{2}=}{}-a_{55}\bigl(a_{11}a_{22}+a_{33}a_{44}+(a_{11}+a_{22}) (a_{33}+a_{44})\bigr), \\& p_{3} =a_{11}a_{22}+a_{33}a_{44}+(a_{11}+a_{22}) (a_{33}+a_{44})-a _{23}a_{32}-a_{15}a_{51} \\& \hphantom{p_{3}=}{}+a_{55}(a_{11}+a_{22}+a_{33}+a_{44}), \\& p_{4} =-(a_{11}+a_{22}+a_{33}+a_{44}+a_{55}),\qquad q_{0}=a_{22}a_{44}b _{33}(a_{15}a_{51}-a_{11}a_{55}), \\& q_{1} =a_{11}a_{22}b_{33}(a_{44}+a_{55})+a_{44}a_{55}b_{33}(a_{11}+a _{22})-a_{15}a_{51}b_{33}(a_{22}+a_{44}), \\& q_{2} =a_{15}a_{51}b_{33}-b_{33} \bigl(a_{11}a_{22}+a_{44}a_{55}+(a_{11}+a _{22}) (a_{44}+a_{55})\bigr), \\& q_{3} =b_{33}(a_{11}+a_{22}+a_{44}+a_{55}), \qquad q_{4}=-b_{33}. \end{aligned}$$
When \(\tau =0\), Eq. (3) reduces to
$$ \lambda^{5}+p_{04}\lambda^{4}+p_{03} \lambda^{3}+p_{02}\lambda^{2}+p _{01} \lambda +p_{00}=0 $$
(7)
with
$$ \begin{aligned}&p_{00}=p_{0}+q_{0},\qquad p_{01}=p_{1}+q_{1},\qquad p_{02}=p_{2}+q_{2}, \\ &p_{03}=p _{3}+q_{3},\qquad p_{04}=p_{4}+q_{4}. \end{aligned} $$
Obviously,
$$ p_{04}=\mu +5\delta_{0}+\delta_{1}+ \delta_{2}+\delta_{3}+\frac{\alpha I_{*}(I_{*}+c)}{(S_{*}+I_{*}+c)^{2}}+\frac{a\beta }{(I_{*}+a)^{2}}>0. $$
An application of the Routh–Hurwitz criterion gives \(\operatorname{Re}( \lambda)<0\) if and only if condition (\(H_{1}\)) is satisfied, that is, if the following inequalities hold:
$$\begin{aligned}& \operatorname{det}_{2} = \left \vert \begin{matrix} p_{04} &{1} \\ p_{02} &p_{03} \end{matrix} \right \vert >0, \end{aligned}$$
(8)
$$\begin{aligned}& \operatorname{det}_{3} =\left \vert \begin{matrix} p_{04} &{1} &{0} \\ p_{02} &p_{03} &p_{04} \\ {0} &p_{01} &p_{02} \end{matrix} \right \vert >0, \end{aligned}$$
(9)
$$\begin{aligned}& \operatorname{det}_{4} =\left \vert \begin{matrix} p_{04} &{1} &{0} &{0} \\ p_{02} &p_{03} &p_{04} &{1} \\ p_{00} &p_{01} &p_{02} &p_{03} \\ {0} &{0} &p_{00} &p_{01} \end{matrix} \right \vert >0, \end{aligned}$$
(10)
$$\begin{aligned}& \operatorname{det}_{5} =\left \vert \begin{matrix} p_{04} &{1} &{0} &{0} &{0} \\ p_{02} &p_{03} &p_{04} &{1} &{0} \\ p_{00} &p_{01} &p_{02} &p_{03} &p_{04} \\ {0} &{0} &p_{00} &p_{01} &p_{02} \\ {0} &{0} &{0} &{0} &p_{00} \end{matrix} \right \vert >0. \end{aligned}$$
(11)
For \(\tau >0\), we assume that \(\lambda =i\omega \) (\(\omega >0\)) is a root of Eq. (6). Then
$$ \textstyle\begin{cases} (q_{1}\omega -q_{3}\omega^{3})\sin \tau \omega +(q_{4}\omega^{4}-q _{2}\omega^{2}+q_{0})\cos \tau \omega =p_{2}\omega^{2}-p_{4}\omega ^{4}-p_{0}, \\ (q_{1}\omega -q_{3}\omega^{3})\cos \tau \omega -(q_{4}\omega^{4}-q _{2}\omega^{2}+q_{0})\sin \tau \omega =p_{3}\omega^{3}-\omega^{5}-p _{1}\omega. \end{cases} $$
Thus
$$ \omega^{10}+e_{4}\omega^{8}+e_{3} \omega^{6}+e_{2}\omega^{4}+e_{1} \omega^{2}+e_{0}=0, $$
(12)
where
$$\begin{aligned}& e_{0} =p_{0}^{2}-q_{0}^{2}, \qquad e_{1}=p_{1}^{2}-2p_{0}p_{2}+2q_{0}q_{2}-q _{1}^{2}, \\& e_{2} =p_{2}^{2}-2p_{1}p_{3}+2p_{0}p_{4}-q_{2}^{2}-2q_{1}q_{3}, \\& e_{3} =p_{3}^{2}+2p_{1}-2p_{2}p_{4}+2q_{2}q_{4}-q_{3}^{2}, \\& e_{4} =p_{4}^{2}-2p_{3}-q_{4}^{2}. \end{aligned}$$
Let \(v=\omega^{2}\). Then Eq. (12) becomes
$$ v^{5}+e_{4}v^{4}+e_{3}v^{3}+e_{2}v^{2}+e_{1}v+e_{0}=0. $$
(13)
Based on the discussion about the distribution of the roots of Eq. (13) in [23] and considering that all the values of parameters in system (2) are given, we can obtain all the roots of Eq. (13). Thus we make the following assumption:
- (\(H_{2}\)):
-
Equation (13) has at least one positive root \(v_{0}\).
If condition (\(H_{2}\)) holds, then there exists \(v_{0}>0\) such that Eq. (6) has a pair of purely imaginary roots \(\pm i \omega_{0}=\pm i\sqrt{v_{0}}\). For \(\omega_{0}\), we have
$$ \tau_{0}=\frac{1}{\omega_{0}}\times \biggl\{ \frac{g_{1}(\omega_{0})}{g _{2}(\omega_{0})} \biggr\} , $$
where
$$\begin{aligned}& g_{1}(\omega_{0}) =(q_{3}-p_{4}q_{4}) \omega_{0}^{8}+(p_{3}q_{3}-q _{1}+p_{2}q_{4}+p_{4}q2) \omega_{0}^{6} \\& \hphantom{g_{1}(\omega_{0}) =}{}+(p_{1}q_{3}+p_{3}q_{1}-p_{0}q_{4}-p_{2}q_{2}-p_{4}q_{0}) \omega_{0} ^{4} \\& \hphantom{g_{1}(\omega_{0}) =}{}+(p_{0}q_{2}+p_{2}q_{0}-p_{1}q_{1}) \omega_{0}^{2}-p_{0}q_{0}, \\& g_{2}(\omega_{0}) =q_{4}^{2} \omega_{0}^{8}+\bigl(q_{3}^{2}-2q_{2}q_{4} \bigr) \omega_{0}^{6}+\bigl(q_{2}^{2}+2q_{0}q_{4}+2q_{1}q_{3} \bigr)\omega_{0}^{4} \\& \hphantom{g_{2}(\omega_{0}) =}{}+\bigl(q_{1}^{2}-2q_{0}q_{2} \bigr)\omega_{0}^{2}+q_{0}^{2}. \end{aligned}$$
Next, differentiating Eq. (6) with respect to τ, we obtain
$$\begin{aligned} \biggl[\frac{d\lambda }{d\tau } \biggr]^{-1} =&-\frac{5\lambda^{4}+4p_{4} \lambda^{3}+3p_{3}\lambda^{2}+2p_{2}\lambda +p_{1}}{\lambda (\lambda ^{5}+p_{4}\lambda^{4}+p_{3}\lambda^{3}+p_{2}\lambda^{2}+p_{1}\lambda +p_{0})} \\ &{}+\frac{4q_{4}\lambda^{3}+3q_{3}\lambda^{2}+2q_{2}\lambda +q _{1}}{\lambda (q_{4}\lambda^{4}+q_{3}\lambda^{3}+q_{2}\lambda^{2}+q _{1}\lambda +q_{0})}-\frac{\tau }{\lambda }. \end{aligned}$$
Further, we have
$$ \operatorname{Re}\biggl[\frac{d\lambda }{d\tau } \biggr]_{\tau =\tau_{0}}^{-1}= \frac{f ^{\prime }(v_{0})}{(q_{1}\omega_{0}-q_{3}\omega_{0}^{3})^{2}+(q_{4} \omega_{0}^{4}-q_{2}\omega_{0}^{2}+q_{0})^{2}}, $$
where \(v_{0}=\omega_{0}^{2}\) and \(f(v)=v^{5}+e_{4}v^{4}+e_{3}v^{3}+e _{2}v^{2}+e_{1}v+e_{0}\).
Therefore, if condition (\(H_{3}\)): \(f^{\prime }(v_{0})\neq 0\) holds, then \(\operatorname{Re}[\frac{d\lambda }{d\tau }]_{\tau =\tau_{0}}\neq 0\). Based on the previous discussion and the Hopf bifurcation theorem in [24], we have the following:
Theorem 1
Suppose that the conditions (\(H_{1}\)), (\(H_{2}\)), and (\(H_{3}\)) hold for system (2). The viral equilibrium
\(P_{*}(S_{*}, E_{*}, I _{*}, R_{*}, V_{*})\)
is locally asymptotically stable when
\(\tau \in [0, \tau_{0})\); a Hopf bifurcation occurs at the viral equilibrium
\(P_{*}(S_{*}, E_{*}, I_{*}, R_{*}, V_{*})\)
when
\(\tau =\tau_{0}\), and a family of periodic solutions bifurcate from the viral equilibrium
\(P_{*}(S_{*}, E_{*}, I_{*}, R_{*}, V_{*})\)
near
\(\tau =\tau_{0}\).