Appendix
1.1 Appendix A
Proof of Theorem 3.1
If \(\tau _1=0\), from (4) we get the eigenvalues as
$$\begin{aligned} \begin{aligned} \lambda _1&=-\frac{1}{2}\left\{ (\mu _1+\mu _2+\alpha )+ \sqrt{(\mu _1+\mu _2+\alpha )^2+4(a\alpha -(\mu _1 \mu _2+\mu _2 \alpha ))} \right\} ,\\ \lambda _2&=-\frac{1}{2}\left\{ (\mu _1+\mu _2+\alpha )-\sqrt{(\mu _1+\mu _2+\alpha )^2+4(a\alpha -(\mu _1 \mu _2+\mu _2 \alpha ))} \right\} , \end{aligned} \end{aligned}$$
where \(\mu _1+\mu _2+\alpha >0\). If \(\mathcal{R}_0<1\), then \(a\alpha -(\mu _1 \mu _2+\mu _2 \alpha )<0\) which implies that \(\lambda _1<0\) and \(\lambda _2<0\). Hence \(E_0\) is locally stable. On the other hand, if \(a\alpha -(\mu _1 \mu _2+\mu _2 \alpha )>0\) then \(\lambda _2>0\) and \(E_0\) is unstable. Following the above results, we have Theorem 3.1. \(\square \)
Proof of Theorem 3.2
If \(\tau _1\ne 0\), we assume \(\lambda =i\omega (\omega >0) \) be a root of (4), then from (4), separating real and imaginary parts we get
$$\begin{aligned} \alpha \omega \sin \omega \tau _1-(a\alpha -\mu _2\alpha )\cos \omega \tau _1= & {} \omega ^2-\mu _1\mu _2,\nonumber \\ (a\alpha -\mu _2\alpha )\sin \omega \tau _1+\alpha \omega \cos \omega \tau _1= & {} -\omega (\mu _1+\mu _2). \end{aligned}$$
(A.1)
Now, squaring and adding the equations of system (A.1) we get
$$\begin{aligned} \omega ^4+\omega ^2\left( \mu _1^2+\mu _2^2-2\alpha ^2\right) +\left\{ \mu _1^2\mu _2^2+2\alpha ^2(a-\mu _2)^2\right\} =0. \end{aligned}$$
(A.2)
Now, if
$$\begin{aligned} \mu _1^2+\mu _2^2-2\alpha ^2>0\quad \text {and}\quad \mu _1^2\mu _2^2+2\alpha ^2(a-\mu _2)^2>0, \end{aligned}$$
(A.3)
then all the roots of (A.2) will have negative real parts when \(\tau _1\in [0, \infty )\). Thus, the equilibrium \(E_0\) is locally asymptotically stable for all \(\tau _1\ge 0\).
If any of the above conditions (A.3) becomes less than zero, then equation (A.2) has positive root say \(\omega _0^2\). Therefore from (A.1), eliminating \(\sin \omega \tau _1\) and substituting \(\omega _0^2\), we have
$$\begin{aligned} \tau _{1_n}^{*}=\frac{1}{\omega _0}\cos ^{-1} \left[ \frac{(\omega _0^2-\mu _1\mu _2)(\alpha \mu _2-a\alpha )-\alpha \omega _0^2(\mu _1+\mu _2)}{\alpha ^2(\omega _0^2+a^2-2a\mu _2+\mu _2^2)} \right] +\frac{2n\pi }{\omega _0},\quad n=0, 1, 2,... \end{aligned}$$
(A.4)
Now, we define the function \(\theta (\tau _1)\in [0, 2\pi )\) s.t. \(\cos \theta (\tau _1)\) is given by the right-hand side of (A.4). To get the critical value of \(\tau _1\) where stability occurs we have to solve
$$\begin{aligned} S_n(\tau _1)=\tau _1-\tau _{1_n}^{*}. \end{aligned}$$
(A.5)
If, \(\lambda (\tau _1)\), solution of (4) satisfying \(Re \lambda (\tau _{1_n}^{*})=0\) and Im \(\lambda (\tau _{1_n}^{*})=\omega _0\), implies
$$\begin{aligned} \left[ \frac{\hbox {d}}{\hbox {d}\tau _1}(Re \lambda ) \right] \ne 0. \end{aligned}$$
(A.6)
Thus, we see that Hopf bifurcation occurs at \(\tau _1=\tau _{1_0}^{*}\). Following the above results, we have Theorem 3.2. \(\square \)
1.2 Appendix B
$$\begin{aligned} \begin{aligned} p_1&=2\alpha (\tau _1+\tau _2)+2\gamma _1\tau _1-\frac{2\beta _1S_1^*}{A_1}+\frac{2R_1\tau _2}{A_2} ,\quad p_2=2\alpha \tau _1+2\left( \gamma _2\tau _1+\frac{R_2\tau _2}{B_1}\right) \nonumber \\ \\ p_3&=\frac{2\beta _1I_1^*}{A_1}\tau _1 ,\quad p_4=\frac{2\beta _2}{B_2}\tau _2, \quad p_5=\left( \alpha +\gamma _2+\frac{R_2}{B_1}\right) \tau _1-\mu _1-\mu _2+\frac{\beta _1I_1^*}{A_1}\tau _2 \nonumber \\ \nonumber \\ p_6&=\alpha (\tau _1+\tau _2)+\frac{\beta _1}{A_1}(S_1^*\tau _2-I_1^*\tau _1)-\left( \alpha +\frac{R_1}{A_2}+\gamma _1+\mu _1+\mu _3\right) \nonumber \\ \nonumber \\ p_7&=\frac{\beta _1I_2^*}{B_2}\tau _2+\gamma _1+\frac{R_1}{A_2}-\frac{\beta _1(S_1^*+I_1^*)}{A_1}\tau _1+\gamma _2+\mu _4+\frac{R_2}{B_1}+\frac{\beta _1S_2^*}{B_2}\tau _1-\mu _1 \nonumber \\ \nonumber \\ p_8&=\frac{\beta _1(I_1^*+S_1^*)}{A_1}\tau _1+\alpha +\gamma _2+\frac{R_2}{B_1}-\gamma _1\tau _2-\mu _3\tau _1-\mu _2-\frac{R_1}{A_2}\tau _2 \nonumber \\ \nonumber \\ p_9&=\alpha +\frac{\beta _2S_2^*}{B_2}(\tau _1+2\tau _2)-\gamma _2-\mu _2-\mu _4-\frac{R_2}{B_1} \nonumber \\ \nonumber \\ p_{10}&=\frac{\beta _1(S_1^*+I_1^*)}{A_1}(2\tau _1+\tau _2)-\gamma _2-\mu _4-\frac{R_2}{B_1}+\frac{\beta _2(I_2^*+S_2^*)}{B_2}(\tau _1-\tau _2)-\gamma _1-\mu _3 \end{aligned} \end{aligned}$$
$$\begin{aligned} w_1(z)(t)= & {} \bar{X}^2+\left\{ -\alpha ^2+\alpha \mu _1-\alpha \gamma _1+\frac{\alpha _1 \beta _1}{A_1}(S_1^*+I_1^*)-\frac{\alpha _1 R_1}{A_2}\right\} \int _{t-\tau _1}^{t}\int _{s}^{t}X^2(l)\hbox {d}l \hbox {d}s \\&+\left\{ \frac{R_1\alpha -R_1\mu _1+R_1\gamma _1}{A_2}-\frac{R_1\beta _1}{A_1A_2}(S_1^*+I_1^*)+\frac{R_1^2}{A_2^2}\right\} \int _{t-\tau _2}^{t}\int _{s}^{t}U^2(l)\hbox {d}l \hbox {d}s, \nonumber \\ \end{aligned}$$
$$\begin{aligned} w_2(z)(t)= & {} \bar{Y}^2+\left\{ \alpha ^2-\alpha \mu _2+\alpha \gamma _2+\frac{\alpha R_2}{B_1}\right\} \int _{t-\tau _1}^{t}\int _{s}^{t}X^2(l)\hbox {d}l \hbox {d}s \nonumber \\&+\left\{ \frac{R_2\alpha -R_2\mu _2+R_2^2}{B_1}+\gamma _2 R_2\right\} \int _{t-\tau _2}^{t}\int _{s}^{t}V^2(l)\hbox {d}l \hbox {d}s,\nonumber \\ \\ w_3(z)(t)= & {} \bar{U}^2+\left\{ \frac{R_1(\gamma _1+\mu _3)}{A_2}+\frac{R_1^2}{A_2^2}-\frac{R_1\beta _1(I_1^*+S_1^*)}{A_1A_2} \right\} \int _{t-\tau _2}^{t}\int _{s}^{t}U^2(l)\hbox {d}l \hbox {d}s, \nonumber \\ \nonumber \\ w_4(z)(t)= & {} \bar{V}^2+\left\{ \frac{R_2(\gamma _2+\mu _4)}{B_1}+\frac{R_2^2}{B_1^2}+\frac{R_2\beta _2(I_2^*+S_2^*)}{B_1B_2} \right\} \int _{t-\tau _2}^{t}\int _{s}^{t}V^2(l)\hbox {d}l \hbox {d}s, \nonumber \\ \end{aligned}$$
$$\begin{aligned} w_5(z)(t)= & {} \bar{X}\bar{Y}+\left\{ \frac{\alpha }{2}\left( \gamma _1-\gamma _2-\mu _1-\mu _2-\frac{\beta _1(S_1^*+I_1^*)}{A_1}+\frac{R_1}{A_2}\right) -\frac{\alpha R_2}{B_1}\right\} \int _{t-\tau _1}^{t}\int _{s}^{t}X^2(l)\hbox {d}l \hbox {d}s \nonumber \\&+\left\{ \frac{R_1}{2A_2}(\alpha -\mu _2+\gamma _2)+\frac{R_2^2}{2A_2B_1}\right\} \int _{t-\tau _2}^{t}\int _{s}^{t}U^2(l)\hbox {d}l \hbox {d}s\nonumber \\&+\left\{ \frac{R_2}{2B_1}\left( \alpha -\mu _1+\gamma _1+\frac{R_1}{A_2}-\frac{\beta _1(S_1^*+I_1^*)}{A_1}\right) \right\} \int _{t-\tau _2}^{t}\int _{s}^{t}V^2(l)\hbox {d}l \hbox {d}s,\nonumber \\ \nonumber \\ w_6(z)(t)= & {} \bar{X}\bar{U}+\frac{\alpha }{2}\left( \gamma _1+\mu _3-I_1^*+\frac{R_1}{A_2}-\frac{\beta _1S_1^*}{A_1}\right) \int _{t-\tau _1}^{t}\int _{s}^{t}X^2(l)\hbox {d}l \hbox {d}s \nonumber \\&+\frac{R_1}{2A_2}\left( \mu _1+I_1^*-\mu _3-\alpha -2\gamma _1-\frac{R_1}{A_2}+\frac{2\beta _1S_1^*+\beta _1I_1^*}{A_1} \right) \int _{t-\tau _2}^{t}\int _{s}^{t}U^2(l)\hbox {d}l \hbox {d}s, \nonumber \\ \end{aligned}$$
$$\begin{aligned} w_7(z)(t)= & {} \bar{X}\bar{V}+\frac{\alpha }{2}\left( \gamma _2+\mu _4+\frac{R_2}{B_1}+\frac{\beta _1S_2^*}{B_2}-\frac{\beta _1I_2^*}{B_2}\right) \int _{t-\tau _1}^{t}\int _{s}^{t}X^2(l)\hbox {d}l \hbox {d}s \nonumber \\&+\frac{R_1}{2A_2}\left( \frac{\beta _1I_2^*}{B_2}-\gamma _2-\mu _4-\frac{R_2}{B_1}-\frac{\beta _1S_2^*}{B_2} \right) \int _{t-\tau _2}^{t}\int _{s}^{t}U^2(l)\hbox {d}l \hbox {d}s \nonumber \\&+\frac{R_2}{2B_1}\left( -\alpha +\mu _1+\frac{\beta _1I_1^*}{A_1}-\gamma _1-\frac{R_1}{A_2}+\frac{\beta _1S_1^*}{A_1} \right) \int _{t-\tau _2}^{t}\int _{s}^{t}V^2(l)\hbox {d}l \hbox {d}s, \nonumber \\ \end{aligned}$$
$$\begin{aligned} w_8(z)(t)= & {} \bar{Y}\bar{U}+\frac{\alpha }{2}\left( -\gamma _1-\mu _3-\frac{R_1}{A_2}+\frac{\beta _1S_1^*}{A_1}+\frac{\beta _1I_1^*}{A_1}\right) \int _{t-\tau _1}^{t}\int _{s}^{t}X^2(l)\hbox {d}l \hbox {d}s \nonumber \\&+\frac{R_1}{2A_2}\left( \mu _2-\alpha -\gamma _2-\frac{R_2}{B_1}\right) \int _{t-\tau _2}^{t}\int _{s}^{t}U^2(l)\hbox {d}l \hbox {d}s \nonumber \\&+\frac{R_2}{2B_1}\left( \frac{\beta _1S_1^*}{A_1}-\gamma _1-\mu _3-\frac{R_1}{A_2}-\frac{\beta _1I_1^*}{A_1}\right) \int _{t-\tau _2}^{t}\int _{s}^{t}V^2(l)dl ds, \end{aligned}$$
$$\begin{aligned} w_9(z)(t)= & {} \bar{Y}\bar{V}+\frac{\alpha }{2}\left( -\gamma _2-\mu _4-\frac{R_2}{B_1}+\frac{\beta _2S_2^*}{B_2}+\frac{\beta _2I_2^*}{B_2}\right) \int _{t-\tau _1}^{t}\int _{s}^{t}X^2(l)\hbox {d}l \hbox {d}s \nonumber \\&+\frac{R_2}{2B_1}\left( -2\gamma _2-\mu _4-2\frac{R_2}{B_1}+\frac{\beta _2S_2^*}{B_2}+\frac{\beta _2I_2^*}{B_2}-\alpha +\mu _2 \right) \int _{t-\tau _2}^{t}\int _{s}^{t}V^2(l)\hbox {d}l \hbox {d}s, \nonumber \\ \nonumber \\ w_{10}(z)(t)= & {} \bar{U}\bar{V}+\frac{R_1}{2A_2}\left( \gamma _2+\mu _4+\frac{R_2}{B_1}-2\frac{\beta _2S_2^*}{B_2}\right) \int _{t-\tau _2}^{t}\int _{s}^{t}U^2(l)\hbox {d}l \hbox {d}s \nonumber \\&+\frac{R_2}{2B_1}\left( -\frac{\beta _1I_1^*}{A_1}+\gamma _1+\mu _3+\frac{R_1}{A_2}-\frac{\beta _1S_1^*}{A_1}\right) \int _{t-\tau _2}^{t}\int _{s}^{t}V^2(l)dl ds, \end{aligned}$$
where
$$\begin{aligned} A_1= & {} 1+\alpha _1+\alpha _1I_1^*,\nonumber \\ A_2= & {} 1+k_1+k_1^2I_1^*+2k_1I_1^*,\nonumber \\ B_1= & {} 1+k_2+k_2^2I_2^*+2k_2I_2^*,\nonumber \\ B_2= & {} 1+\alpha _2+\alpha _2I_2^*, \end{aligned}$$
$$\begin{aligned} \Lambda _1= & {} p_1\left\{ -2(\alpha +\mu _1)-\frac{2\beta _1I_1^*}{A_1}+\alpha (\alpha +\mu _1)\tau _1+\frac{\alpha _1\beta _1I_1^*\tau _1}{A_1}-\frac{R_1(\alpha +\mu _1)\tau _2}{A_2}-\frac{R_1\beta _1I_1^*}{A_1A_2}\right\} \nonumber \\&+p_1\tau _1\left( -\alpha ^2+\alpha \mu _1-\alpha \gamma _1+\frac{\alpha _1\beta _1(S_1^*+I_1^*)}{A_1}-\frac{\alpha _1R_1}{A_2} \right) +p_2\left( \alpha ^2\tau _1+\frac{\alpha R_2\tau _2}{B_1}\right) \nonumber \\&+p_2\tau _1\left( \alpha ^2-\alpha \mu _2+\alpha \gamma _2+\frac{\alpha R_2}{B_1} \right) -p_3\frac{R_1\beta _1I_1^*\tau _2}{A_1A_2}\nonumber \\&+p_5\left\{ \alpha -\frac{\alpha (\mu _1+\alpha )\tau _1}{2}-\frac{\alpha \beta _1I_1^*\tau _1}{2A_1}-\frac{\alpha ^2\tau _1}{2}+\frac{\alpha R_1\tau _2}{2A_2}-\frac{R_2\tau _2(\alpha +\mu _1)}{2B_1}-\frac{\beta _1R_2\tau _2 I_1^*}{2A_1B_1}\right\} \nonumber \\&+p_5\frac{\alpha \tau _1}{2}\left( \gamma _1-\gamma _2-\mu _1-\mu _2-\frac{\beta _1 (S_1^*+I_1^*)}{A_1}+\frac{R_1}{A_2}-\frac{R_2}{B_1}\right) +p_6\left\{ \frac{\beta _1I_1^*}{A_1}-\frac{\alpha \tau _1 I_1^*}{2}\right\} \nonumber \\&+p_6\left\{ \frac{R_1\tau _2}{2A_2}\left( I_1^*+\alpha +\mu _1+\frac{\beta _1I_1^*}{A_1} \right) +\frac{\alpha \tau _1}{2}\left( \gamma _1+\mu _3-I_1^*+\frac{R_1}{A_2}-\frac{\beta _1S_1^*}{A_1}\right) \right\} \nonumber \\&+p_7\left\{ \frac{R_2\tau _2}{2B_1}\left( \alpha +\mu _1+\frac{\beta _1I_1^*}{A_1}\right) +\frac{\alpha \tau _1}{2}\left( \gamma _2+\mu _4+\frac{R_2}{B_1}+\frac{\beta _1S_2^*}{B_2} \right) -\frac{\alpha \beta _1I_2^*\tau _1}{2B_2}\right\} \nonumber \\&+p_8\left\{ \frac{\alpha \beta _1I_1^*\tau _1}{2A_1}+\frac{R_2\beta _1I_1^*\tau _2}{2A_1B_1}-\frac{\alpha R_1\tau _2}{2A_2}-\frac{\alpha \tau _1}{2}(\gamma _1+\mu _3)-\frac{\alpha R_1\tau _1}{2A_2}+\frac{\alpha \beta _1S_1^*\tau _1}{2A_1}+\frac{\alpha \beta _1\tau _1I_1^*}{2A_1}\right\} \nonumber \\&+p_9\left\{ -\frac{\alpha R_2\tau _2}{2B_1}+\frac{\alpha \tau _1}{2}\left( -\gamma _2-\mu _4-\frac{R_2}{B_1}+\frac{\beta _2(S_2^*+I_2^*)}{B_2} \right) \right\} -p_{10}\frac{R_2\beta _1I_1^*\tau _2}{2A_1B_1}, \end{aligned}$$
$$\begin{aligned} \Lambda _2= & {} p_1\left\{ -\alpha ^2\tau _1+\frac{R_1\alpha \tau _2}{A_2} \right\} -p_2\left( 2\mu _2+\alpha \mu _2\tau _1+\frac{\mu _2R_2\tau _2}{B_1} \right) +p_4\frac{R_2\beta _2I_2^*\tau _2}{B_1B_2}-p_6\frac{\alpha R_1\tau _2}{2A_2}+p_8\frac{\mu _2R_1\tau _2}{2A_2}\nonumber \\&+p_5\left\{ \alpha +\frac{\alpha ^2\tau _1}{2}+\frac{\alpha \mu _2\tau _1}{2}-\frac{R_1\mu _2\tau _2}{2A_2}+\frac{R_2\alpha \tau _2}{2B_1}\right\} +p_7\left( -\frac{\alpha \beta _1 I_2^*\tau _1}{2B_2}+\frac{R_1\beta _1I_2^*\tau _2}{2A_2B_2}-\frac{\alpha R_2\tau _2}{2B_1}\right) \nonumber \\&+p_9\left( \frac{\beta _2I_2^*}{B_2}+\frac{\alpha \beta _2I_2^*\tau _1}{2B_2}+\frac{R_2\beta _2I_2^*\tau _2}{2B_1B_2}+\frac{\mu _2R_2\tau _2}{2B_1} \right) -p_{10}\frac{R_1\beta _2I_2^*\tau _2}{2A_2B_2}, \end{aligned}$$
$$\begin{aligned} \Lambda _3= & {} p_1\left\{ -\alpha \gamma _1\tau _1+\frac{\alpha _1\beta _1S_1^*\tau _1}{A_1}-\frac{\alpha _1R_1\tau _1}{A_2}+\frac{R_1\gamma _1\tau _2}{A_2}-\frac{R_1\beta _1S_1^*\tau _2}{A_1A_2}+\frac{R_1^2\tau _2}{A_2^2}\right\} \nonumber \\&+p_1\tau _2\left( \frac{\alpha R_1}{A_2}-\frac{R_1\mu _1}{A_2}+\frac{R_1\gamma _1}{A_2}-\frac{R_1\beta _1(S_1^*+I_1^*)}{A_1A_2}+\frac{R_1^2}{A_2^2} \right) +p_3\left( -2(\gamma _1+\mu _3)-\frac{2R_1}{A_2}+\frac{2\beta _1S_1^*}{A_1}\right) \nonumber \\&+p_3\left\{ \frac{2R_1\tau _2}{A_2}\left( \gamma _1+\mu _3+\frac{R_1}{A_2}-\frac{\beta _1S_1^*}{A_1}\right) -\frac{\beta _1R_1\tau _2I_1^*}{A_1A_2} \right\} +p_5\left\{ \frac{R_1\tau _2}{2A_2}(\alpha -\mu _2+\gamma _2)+\frac{R_2^2\tau _2}{2A_2B_1}\right\} \nonumber \\&+p_5\left( \frac{\alpha \gamma _1\tau _1}{2}-\frac{\alpha \beta _1S_1^*\tau _1}{2A_1}+\frac{\alpha R_1\tau _1}{2A_2}+\frac{R_2\gamma _1\tau _2}{2B_1}-\frac{R_2\beta _1S_1^*\tau _2}{2A_1B_1}+\frac{R_1R_2\tau _2}{2A_2B_1}\right) \nonumber \\&+p_6\left\{ \frac{\alpha \tau _1}{2}\left( \gamma _1+\mu _3+\frac{R_1}{A_2}-\frac{\beta _1S_1^*}{A_1}\right) +\frac{R_1\tau _2}{2A_2}\left( \frac{3\beta _1S_1^*}{A_1}-4\gamma _1-\alpha -2\mu _3+\mu _1+I_1^*-\frac{2R_1}{A_2}+\frac{\beta _1(S_1^*+I_1^*)}{A_1}\right) \right\} \nonumber \\&+p_7\left\{ -\frac{R_2\tau _2}{2B_1}\left( \gamma _1-\frac{\beta _1S_1^*}{A_1}+\frac{R_1}{A_2}\right) +\frac{R_1\beta _1I_2^*\tau _2}{2A_2B_2}-\frac{R_1\tau _2}{2A_2}\left( \gamma _2+\mu _4+\frac{R_2}{B_1}+\frac{\beta _1S_2^*}{B_2}\right) \right\} \nonumber \\&+p_8\left\{ \frac{R_1}{2A_2}\left( \mu _2\tau _2-\alpha \tau _2-\gamma _2\tau _2-\frac{R_2\tau _2}{B_1}-\alpha \tau _1\right) -\frac{\alpha \tau _1(\gamma _1+\mu _3)}{2}+\frac{\alpha \beta _1S_1^*\tau _1}{2A_1}-\frac{R_2\tau _2}{2B_1}(\gamma _1+\mu _3)\right\} \nonumber \\&+p_8\frac{R_2\tau _2}{2B_1}\left( \frac{R_1}{A_2}-\frac{\beta _1S_1^*}{A_1}\right) +p_{10}\left\{ \frac{R_2}{2B_1}(\gamma _1\tau _2+\mu _3\tau _2) +\frac{R_1R_2\tau _2}{A_2B_1}+\frac{R_1}{2A_2}(\gamma _2\tau _2+\mu _4\tau _2)\right\} \nonumber \\&-p_{10}\left( \frac{R_2\beta _1S_1^*\tau _2}{2A_1B_1}+\frac{R_1\beta _2S_2^*\tau _2}{2A_2B_1}+\frac{R_1\beta _2S_2^*\tau _2}{2A_2B_2} \right) , \end{aligned}$$
$$\begin{aligned} \Lambda _4= & {} p_1\left\{ -\alpha ^2\tau _1+\frac{R_1\alpha \tau _2}{A_2} \right\} +p_2\left\{ (\alpha \tau _1+2R_2\tau _2)\left( \gamma _2+\frac{R_2}{B_1}\right) +\frac{R_2\tau _2}{B_1}(\alpha -\mu _2) \right\} \nonumber \\&+p_4\left\{ \left( \gamma _2+\mu _4+\frac{R_2}{B_1}+\frac{\beta _2S_2^*}{B_2}\right) \left( \frac{R_2\tau _2}{B_1}-2\right) +\frac{R_2\beta _2(S_2^*+I_2^*)}{B_1B_2}+\frac{R_2}{B_1}(\gamma _2+\mu _4)+\frac{R_2^2}{B_1^2} \right\} \nonumber \\&+p_5\left\{ \frac{\alpha ^2\tau _1}{2}-\frac{\alpha \gamma _2\tau _1}{2}-\frac{R_2\alpha \tau _1}{B_1}+\frac{R_1\gamma _2\tau _2}{2A_2}+\frac{R_2^2\tau _2}{2A_2B_1}+\frac{\alpha R_2\tau _2}{2B_1} \right\} \nonumber \\&+p_5\frac{R_2\tau _2}{2B_1}\left( \alpha -\mu _1+\gamma _1-\frac{\beta _1(S_1^*+I_1^*)}{A_1}+\frac{R_1}{A_2}\right) -p_6\frac{\alpha R_1\tau _2}{2A_2}+p_7\left( \alpha -\frac{3\alpha R_2 \tau _2}{2B_1} \right) \nonumber \\&+p_7\left\{ \left( \frac{\alpha \tau _1}{2}-\frac{R_1\tau _2}{2A_2}\right) \left( \gamma _2+\mu _4+\frac{R_2}{B_1}+\frac{\beta _1S_2^*}{B_2} \right) +\frac{R_2\tau _2}{2B_1}\left( \alpha +\mu _1+\frac{\beta _1I_1^*}{A_1}+\gamma _1+\frac{R_1}{A_2}-\frac{\beta _1S_1^*}{A_1}\right) \right\} \nonumber \\&-p_8\left\{ \frac{R_1\tau _2}{2A_2}\left( \gamma _2+\frac{R_2}{B_1}\right) +\frac{R_2\tau _2}{2B_1}\left( \gamma _1+\mu _3+\frac{R_1}{A_2}-\frac{\beta _1(S_1^*+I_1^*)}{A_1}\right) \right\} +p_9\left( \gamma _2+\frac{R_2}{B_1}+\frac{\beta _2R_2S_2^*\tau _2}{B_1B_2} \right) \nonumber \\&+p_9\left\{ -\frac{\alpha \tau _1(\gamma _2+\mu _4)}{2}-\frac{1}{2B_1}\left( \alpha \tau _1 R_2+4\gamma _2\tau _2 R_2+2\mu _4\tau _2 R_2+\alpha R_2\tau _2-\mu _2 R_2\tau _2 \right) +\frac{\alpha \beta _2 S_2^*\tau _1}{2B_2} \right\} \nonumber \\&+p_9\left( -\frac{2R_2^2\tau _2}{B_1^2}+\frac{\beta _2 R_2I_2^*\tau _2}{2B_1B_2}\right) +p_{10}\left( \frac{R_1\tau _2(\gamma _2+\mu _4)}{2A_2}+\frac{R_2\tau _2(\gamma _1+\mu _3)}{2B_1}-\frac{R_2\beta _1\tau _2(S_1^*+I_1^*)}{2A_1B_1} \right) \nonumber \\&+p_{10}\left( \frac{R_1R_2\tau _2}{A_2B_1}-\frac{R_1\beta _2S_2^*\tau _2}{2A_2B_2}\right) . \end{aligned}$$
1.3 Appendix C
$$\begin{aligned} A_{11}= & {} a_{11}+a_{22}+a_{33}+a_{44} ,\nonumber \\ A_{12}= & {} a_{11}a_{22}-a_{13}a_{31}+a_{11}a_{33}+a_{22}a_{33}-a_{24}a_{42}+a_{11}a_{44}+a_{22}a_{44}+a_{33}a_{44} ,\nonumber \\ A_{13}= & {} a_{13}a_{22}a_{31}-a_{11}a_{22}a_{33}+a_{11}a_{24}a_{42}+a_{24}a_{33}a_{42}-a_{11}a_{22}a_{44}+a_{13}a_{31}a_{44}-a_{11}a_{33}a_{44}-a_{22}a_{33}a_{44} ,\nonumber \\ A_{14}= & {} a_{13}a_{24}a_{31}a_{42}-a_{11}a_{24}a_{33}a_{42}-a_{13}a_{22}a_{31}a_{44}+a_{11}a_{22}a_{33}a_{44}, \end{aligned}$$
$$\begin{aligned} B_{11}= & {} -\alpha ,\nonumber \\ B_{12}= & {} -a_{12}\alpha -a_{22}\alpha -a_{33}\alpha -a_{44}\alpha ,\nonumber \\ B_{13}= & {} a_{12}a_{33}\alpha +a_{22}a_{33}\alpha -a_{12}a_{42}\alpha -a_{24}a_{42}\alpha +a_{12}a_{44}\alpha +a_{22}a_{44}\alpha +a_{33}a_{44}\alpha ,\nonumber \\ B_{14}= & {} a_{12}a_{33}a_{42}\alpha +a_{24}a_{33}a_{42}\alpha -a_{12}a_{33}a_{44}\alpha -a_{22}a_{33}a_{44}\alpha . \end{aligned}$$
$$\begin{aligned} C_{11}= & {} \frac{R_2}{B_1}+\frac{R_1}{A_2} ,\nonumber \\ C_{12}= & {} -\frac{R_2}{B_1}\left( a_{11}+a_{22}+a_{33}+a_{42}\right) -\frac{R_1}{A_2}\left( a_{11}+a_{22}+a_{31}+a_{44} \right) ,\nonumber \\ C_{13}= & {} \frac{R_2}{B_1}\left( a_{11}a_{22}-a_{13}a_{31} +a_{11}a_{33}+a_{22}a_{33}+a_{11}a_{42}+a_{33}a_{42}\right) \nonumber \\&+\frac{R_1}{A_2}\left( a_{11}a_{22}+a_{22}a_{31}-a_{24}a_{42}+a_{11}a_{44}+a_{22}a_{44} +a_{31}a_{44}\right) ,\nonumber \\ C_{14}= & {} \frac{R_1}{A_2}\left( a_{11}a_{24}a_{42}+a_{24}a_{31}a_{42}- a_{11}a_{22}a_{44}-a_{22}a_{31}a_{44}\right) \nonumber \\&+\frac{R_2}{B_1}\left( a_{13}a_{22}a_{31}-a_{11}a_{22}a_{33}+a_{13}a_{31}a_{42}-a_{11}a_{33}a_{42} \right) . \end{aligned}$$
$$\begin{aligned} D_{11}= & {} \frac{R_1R_2}{A_2B_1} ,\nonumber \\ D_{12}= & {} -\frac{R_1R_2}{A_2B_1}\left( a_{11}+a_{22}+a_{31}+a_{42}\right) ,\nonumber \\ D_{13}= & {} \frac{R_1R_2}{A_2B_1}\left( a_{11}a_{22}+a_{22}a_{31}+a_{11}a_{42}+a_{31}a_{42}\right) . \end{aligned}$$
$$\begin{aligned} E_{11}= & {} \alpha \left( \frac{R_1}{A_2}+\frac{R_2}{B_1}\right) ,\nonumber \\ E_{12}= & {} -\frac{R_1\alpha }{A_2}\left( a_{12}+a_{22}+a_{44}\right) -\frac{R_2\alpha }{B_1}\left( a_{12}+a_{22}+a_{33}-a_{42}\right) ,\nonumber \\ E_{13}= & {} \frac{R_1\alpha }{A_2}\left( a_{22}a_{44}+a_{12}a_{44}-a_{24}a_{42}-a_{12}a_{42} \right) +\frac{R_2\alpha }{B_1}\left( a_{12}a_{33}+a_{22}a_{33}+a_{33}a_{42}\right) ,\nonumber \\ F_{11}= & {} \frac{R_1R_2\alpha }{A_2B_1} ,\nonumber \\ F_{12}= & {} -\frac{R_1R_2\alpha }{A_2B_1}\left( a_{12}+a_{22}+a_{42}\right) . \end{aligned}$$
Proof of Theorem 4.1
For \(\tau _1\ne 0\) but \(\tau _2=0\). Equation (9) can be written as:
$$\begin{aligned} \left\{ \lambda ^4+\mathcal {M}_1\lambda ^3+\mathcal {M}_2\lambda ^2+\mathcal {M}_3\lambda +\mathcal {M}_4\right\} +e^{-\lambda \tau _1}\left\{ \mathcal {N}_1\lambda ^3+\mathcal {N}_2\lambda ^2+\mathcal {N}_3\lambda +\mathcal {N}_4\right\} =0, \end{aligned}$$
(C.1)
where \(\mathcal {M}_1= A_{11}+C_{11}\), \(\mathcal {M}_2=A_{12}+C_{12}+D_{11} \), \(\mathcal {M}_3=A_{13}+C_{13}+D_{12}\), \(\mathcal {M}_4=A_{14}+C_{14}+D_{13}\), and \(\mathcal {N}_1=B_{11}\), \(\mathcal {N}_2=B_{12}+E_{11}\), \(\mathcal {N}_3=B_{13}+E_{12}+F_{11}\), \(\mathcal {N}_4=B_{14}+E_{13}+F_{12}\). Let \(\lambda =i\omega >0\), then from (C.1) separating real and imaginary part we get
$$\begin{aligned} \omega ^4-\mathcal {M}_2\omega ^2+\mathcal {M}_4=(\mathcal {N}_2\omega ^2-\mathcal {N}_4)\cos \omega \tau _1+(\mathcal {N}_1\omega ^3-\mathcal {N_3}\omega )\sin \omega \tau _1, \end{aligned}$$
(C.2)
$$\begin{aligned} \mathcal {M}_3\omega -\mathcal {M}_1\omega ^3=(\mathcal {N}_1\omega ^3-\mathcal {N}_3\omega )\cos \omega \tau _1-(\mathcal {N}_2\omega ^2-\mathcal {N}_4)\sin \omega \tau _1. \end{aligned}$$
(C.3)
Squaring (C.2) and (C.3) and then adding yields
$$\begin{aligned} \Omega ^4+\bar{\mathcal {E}}_{13}\Omega ^3+\bar{\mathcal {E}}_{12}\Omega ^2++\bar{\mathcal {E}}_{11}\Omega +\bar{\mathcal {E}}_{10}=0, \end{aligned}$$
(C.4)
where \(\Omega =\omega ^2\), \(\bar{\mathcal {E}}_{10}=\mathcal {M}_4^2-\mathcal {N}_4^2\), \(\bar{\mathcal {E}}_{11}=\mathcal {M}_3^2+2\mathcal {N}_2\mathcal {N}_4-\mathcal {N}_3^2-2\mathcal {M}_2\mathcal {M}_4\), \(\bar{\mathcal {E}}_{12}=\mathcal {M}_2^2+2\mathcal {M}_4-\mathcal {N}_2^2-2\mathcal {M}_1\mathcal {M}_3+2\mathcal {N}_1\mathcal {N}_3\) and \(\bar{\mathcal {E}}_{13}=\mathcal {M}_1^2-\mathcal {N}_1^2-2\mathcal {M}_2\).
\(H_{11}\): Assume that the equation (C.4) has a positive root say, \(\Omega _0\). Eliminating \(\sin \omega \tau _1\) from (C.2) and (C.3) and substituting \(\omega =\omega _0=\sqrt{\Omega _0}\), where \(\Omega _0\) is a positive root of (C.4), we get
$$\begin{aligned}&\cos \omega _0\tau _1=\frac{(\omega _0^4-\mathcal {M}_2\omega _0^2+\mathcal {M}_4)(\mathcal {N}_2\omega _0^2-\mathcal {N}_4)+(\mathcal {M}_3\omega _0-\mathcal {M}_1\omega _0^3)(\mathcal {N}_1\omega _0^3-\mathcal {N}_3\omega _0)}{(\mathcal {N}_2\omega _0^2-\mathcal {N}_4)^2+(\mathcal {N}_1\omega _0^3-\mathcal {N}_3\omega _0)^2}\nonumber \\&\implies \tau _{1_n}=\frac{1}{\omega _0}\times \left\{ \arccos \left[ \frac{\mathcal {F}_1(\omega _0)}{\mathcal {F}_2(\omega _0)} \right] \right\} +\frac{2n\pi }{\omega _0}, \quad n=0, 1, 2, ... \end{aligned}$$
(C.5)
with \(\mathcal {F}_1(\omega _0)=(\omega _0^4-\mathcal {M}_2\omega _0^2+\mathcal {M}_4)(\mathcal {N}_2\omega _0^2-\mathcal {N}_4)+(\mathcal {M}_3\omega _0-\mathcal {M}_1\omega _0^3)(\mathcal {N}_1\omega _0^3-\mathcal {N}_3\omega _0)\),
\(\mathcal {F}_2(\omega _0)=(\mathcal {N}_2\omega _0^2-\mathcal {N}_4)^2+(\mathcal {N}_1\omega _0^3-\mathcal {N}_3\omega _0)^2\).
We define the function \(\theta (\tau _1)\in [0, 2\pi )\) such that \(\cos \theta (\tau _1)\) is given by the right-hand side of (C.5). Then solving
$$\begin{aligned} \bar{S}_n(\tau _1)=\tau _1-\tau _{1_n}, \end{aligned}$$
we get the \(\tau _1\), at which stability occurs. Differentiating (C.1) with respect to \(\tau _1\) and substituting \(\lambda =i\omega _0\), and simplifying, one obtains
$$\begin{aligned} Re\left[ \frac{\hbox {d}\lambda }{\hbox {d}\tau _1}\right] ^{-1}_{\lambda =i\omega _0}=\frac{F_1^{'}(\Omega _0)}{\mathcal {F}_2(\omega _0^2)}. \end{aligned}$$
(C.6)
Therefore, \(Re\left[ \frac{\hbox {d}\lambda }{\hbox {d}\tau _1}\right] ^{-1}_{\lambda =i\omega _0}\ne 0\) if the condition \(H_{12}\): \(F_1^{'}(\Omega _0)=\frac{\hbox {d}F_1(\Omega )}{\hbox {d}\Omega }|_{\Omega =\Omega _0}\ne 0\) holds,
where \(F_1(\Omega )=\Omega ^4+\bar{\mathcal {E}}_{13}\Omega ^3+\bar{\mathcal {E}}_{12}\Omega ^2++\bar{\mathcal {E}}_{11}\Omega +\bar{\mathcal {E}}_{10}\).
Thus, with the help of Hopf bifurcation theorem [44], we obtain the result of Theorem 4.1 if the conditions \(H_{11}\): and \(H_{12}\): hold. \(\square \)
Proof of Theorem 4.2
For \(\tau _1=0\) but \(\tau _2\ne 0\).
From (9), we get
$$\begin{aligned} \left\{ \lambda ^4+\bar{\mathcal {M}_1}\lambda ^3+\bar{\mathcal {M}_2}\lambda ^2+\bar{\mathcal {M}_3}\lambda +\bar{\mathcal {M}_4}\right\}+ & {} e^{-\lambda \tau _2}\left\{ \bar{\mathcal {N}_1}\lambda ^3+\bar{\mathcal {N}_2}\lambda ^2+\bar{\mathcal {N}_3}\lambda +\bar{\mathcal {N}_4}\right\} \nonumber \\+ & {} e^{-2\lambda \tau _2}\left\{ \bar{\mathcal {P}_1}\lambda ^2+\bar{\mathcal {P}_2}\lambda +\bar{\mathcal {P}_3}\right\} =0, \end{aligned}$$
(C.7)
where \(\bar{\mathcal {M}_1}=A_{11}+B_{11}\), \(\bar{\mathcal {M}_2}=A_{12}+B_{12}\), \(\bar{\mathcal {M}_3}=A_{13}+B_{13}\), \(\bar{\mathcal {M}_4}=A_{14}+B_{14}\), \(\bar{\mathcal {N}_1}=C_{11}\), \(\bar{\mathcal {N}_2}=C_{12}+E_{11}\), \(\bar{\mathcal {N}_3}=C_{13}+E_{12}\), \(\bar{\mathcal {N}_4}=C_{14}+E_{13}\) and \(\bar{\mathcal {P}_1}=D_{11}\), \(\bar{\mathcal {P}_2}=D_{12}+F_{11}\), \(\bar{\mathcal {P}_3}=D_{13}+F_{12}\).
Multiplying \(e^{\lambda \tau _2}\) on both sides of (C.7), we find
$$\begin{aligned} \left\{ \lambda ^4+\bar{\mathcal {M}_1}\lambda ^3+\bar{\mathcal {M}_2}\lambda ^2+\bar{\mathcal {M}_3}\lambda +\bar{\mathcal {M}_4}\right\} e^{\lambda \tau _2}+ & {} \left\{ \bar{\mathcal {N}_1}\lambda ^3+\bar{\mathcal {N}_2}\lambda ^2+\bar{\mathcal {N}_3}\lambda +\bar{\mathcal {N}_4}\right\} \nonumber \\+ & {} e^{-\lambda \tau _2}\left\{ \bar{\mathcal {P}_1}\lambda ^2+\bar{\mathcal {P}_2}\lambda +\bar{\mathcal {P}_3}\right\} =0. \end{aligned}$$
(C.8)
Let \(\lambda =i\omega (\omega >0)\) be a root of Eq. (C.8), then
$$\begin{aligned} G_1(\omega )\cos \omega \tau _2+G_2(\omega )\sin \omega \tau _2= & {} G_3(\omega ),\nonumber \\ G_4(\omega )\sin \omega \tau _2-G_5(\omega )\cos \omega \tau _2= & {} G_6(\omega ), \end{aligned}$$
(C.9)
where
$$\begin{aligned} G_1(\omega )= & {} \omega ^4-(\bar{\mathcal {M}_2}+\bar{\mathcal {P}_1})\omega ^2+(\bar{\mathcal {M}_4}+\bar{\mathcal {P}_3}), \nonumber \\ G_2(\omega )= & {} \bar{\mathcal {M}_1}\omega ^3-(\bar{\mathcal {M}_3}-\bar{\mathcal {P}_2})\omega , \nonumber \\ G_3(\omega )= & {} \bar{\mathcal {N}_2}\omega ^2-\bar{\mathcal {N}_4}, \nonumber \\ G_4(\omega )= & {} \omega ^4-(\bar{\mathcal {M}_2}-\bar{\mathcal {P}_1})\omega ^2+(\bar{\mathcal {M}_4}-\bar{\mathcal {P}_3}), \nonumber \\ G_5(\omega )= & {} \bar{\mathcal {M}_1}\omega ^3-(\bar{\mathcal {M}_3}+\bar{\mathcal {P}_2})\omega , \nonumber \\ G_6(\omega )= & {} \bar{\mathcal {N}_1}\omega ^3-\bar{\mathcal {N}_3}\omega . \end{aligned}$$
Solving (C.9) we get
$$\begin{aligned} \cos \omega \tau _2=\frac{G_{11}(\omega )}{G_{12}(\omega )},\quad \sin \omega \tau _2=\frac{G_{13}(\omega )}{G_{14}(\omega )}, \end{aligned}$$
where
$$\begin{aligned} G_{11}(\omega )= & {} \left( \bar{\mathcal {N}_4}-\bar{\mathcal {N}_2}\omega ^2 \right) \left( \bar{\mathcal {M}_4}-\bar{\mathcal {P}_3}-\bar{\mathcal {M}_2}\omega ^2+\bar{\mathcal {P}_1}\omega ^2+\omega ^4\right) -\left( \bar{\mathcal {N}_3}\omega -\bar{\mathcal {N}_1}\omega ^3\right) \left( \bar{\mathcal {M}_1}\omega ^3+\bar{\mathcal {P}_2}\omega -\bar{\mathcal {M}_3}\omega \right) ,\nonumber \\ G_{12}(\omega )= & {} \left( \bar{\mathcal {M}_3}\omega +\bar{\mathcal {P}_2}\omega -\bar{\mathcal {M}_1}\omega ^3\right) \left( \bar{\mathcal {P}_2}\omega -\bar{\mathcal {M}_3}\omega +\bar{\mathcal {M}_1}\omega ^3\right) \nonumber \\&-\left( \bar{\mathcal {M}_4}-\bar{\mathcal {P}_3}-\bar{\mathcal {M}_2}\omega ^2+\bar{\mathcal {P}_1}\omega ^2+\omega ^4\right) \left( \bar{\mathcal {M}_4}+\bar{\mathcal {P}_3}-\bar{\mathcal {M}_2}\omega ^2-\bar{\mathcal {P}_1}\omega ^2+\omega ^4\right) ,\nonumber \\ G_{13}(\omega )= & {} \bar{\mathcal {N}_1}\omega ^6+\omega ^4\left( \bar{\mathcal {M}_1}\bar{\mathcal {N}_2}-\bar{\mathcal {N}_3}-\bar{\mathcal {N}_1}\bar{\mathcal {P}_1}-\bar{\mathcal {M}_2}\bar{\mathcal {N}_1}\right) +\left( \bar{\mathcal {M}_3}\bar{\mathcal {N}_4}+\bar{\mathcal {N}_4}\bar{\mathcal {P}_2}-\bar{\mathcal {N}_3}\bar{\mathcal {P}_3}-\bar{\mathcal {M}_4}\bar{\mathcal {N}_3}\right) \nonumber \\&+\omega ^2\left( \bar{\mathcal {N}_1}\bar{\mathcal {P}_3}-\bar{\mathcal {N}_2}\bar{\mathcal {P}_2}+\bar{\mathcal {N}_3}\bar{\mathcal {P}_1}-\bar{\mathcal {M}_1}\bar{\mathcal {N}_4}+\bar{\mathcal {M}_2}\bar{\mathcal {N}_3}-\bar{\mathcal {M}_3}\bar{\mathcal {N}_2}+\bar{\mathcal {M}_1}\bar{\mathcal {N}_1}\right) ,\nonumber \\ G_{14}(\omega )= & {} \omega ^8+\omega ^6\left( \bar{\mathcal {M}_1}-2\bar{\mathcal {M}_2}\right) +\omega ^4\left( \bar{\mathcal {M}_2}^2-2\bar{\mathcal {M}_1}\bar{\mathcal {M}_3}+2\bar{\mathcal {M}_4}-\bar{\mathcal {P}_1}^2\right) \nonumber \\&+\omega ^2\left( 2\bar{\mathcal {P}_1}\bar{\mathcal {P}_3}-\bar{\mathcal {P}_2}^2-2\bar{\mathcal {M}_2}\bar{\mathcal {M}_4}+\bar{\mathcal {M}_3}^2\right) +\left( \bar{\mathcal {M}_4}^2-\bar{\mathcal {P}_3}^2\right) . \end{aligned}$$
From the relation
$$\begin{aligned} \sin ^2\omega \tau _2+\cos ^2\omega \tau _2=1, \end{aligned}$$
we have
$$\begin{aligned} G_{11}^2G_{14}^2+G_{12}^2G_{13}^2-G_{12}^2G_{14}^2=0. \end{aligned}$$
(C.10)
We consider \(H_{13}:\) Eq. (C.10) has a positive root \(\omega _0\).
If the condition \(H_{13}\) holds, then we obtain
$$\begin{aligned} \tau _{2_n}=\frac{1}{\omega _0}\times \arccos \left[ \frac{G_{11}(\omega _0)}{G_{12}(\omega _0)}\right] +\frac{2n\pi }{\omega _0}, \quad n=0, 1, 2,... \end{aligned}$$
(C.11)
Define the function \(\theta (\tau _2)\in [0, 2\pi )\) such that \(\cos \theta (\tau _2)\) is given by the right-hand side of (C.11). Then solving
$$\begin{aligned} \bar{S}_{1n}(\tau _2)=\tau _2-\tau _{2_n}, \end{aligned}$$
we get the \(\tau _2\), at which stability occurs. Differentiating (C.7) with respect to \(\tau _2\) and substituting \(\lambda =i\omega _0\) yields
$$\begin{aligned} Re\left[ \frac{\hbox {d}\lambda }{\hbox {d}\tau _2}\right] ^{-1}_{\lambda =i\omega _0}=\frac{G_{11}^{'}(\omega _0)}{G_{12}(\omega _0^2)}. \end{aligned}$$
(C.12)
Therefore, \(Re\left[ \frac{\hbox {d}\lambda }{\hbox {d}\tau _2}\right] ^{-1}_{\lambda =i\omega _0}\ne 0\) if the condition \(H_{14}\): \(G_{11}^{'}(\Omega _0)=\frac{\hbox {d}G_{11}(\omega )}{\hbox {d}\omega }|_{\omega =\Omega _0}\ne 0\) holds.
Thus, using Hopf bifurcation theorem [44], we obtain the result of Theorem 4.2 if the conditions \(H_{13}\): and \(H_{14}\): hold. \(\square \)
Proof of Theorem 4.3
For \(\tau _1\ne 0\) and \(\tau _2\ne 0\).
Let us assume \(\tau _1=\tau _2=\tau \). Then, the characteristic equation (9) becomes-
$$\begin{aligned}&\left( \lambda ^4+A_{11}\lambda ^3+A_{12}\lambda ^2+A_{13}\lambda +A_{14}\right) \nonumber \\&\quad +e^{-\lambda \tau }\left\{ (B_{11}+C_{11})\lambda ^3+(B_{12}+C_{12})\lambda ^2+(B_{13}+C_{13}) \lambda +(B_{14}+C_{14})\right\} \nonumber \\&\quad +e^{-2\lambda \tau }\left\{ (D_{11}+E_{11})\lambda ^2+(D_{12}+E_{12})\lambda +(D_{13}+E_{13})\right\} +e^{-3\lambda \tau }\left( F_{11}\lambda +F_{12}\right) =0. \end{aligned}$$
(C.13)
Multiplying \(e^{2\lambda \tau }\) on both sides of Eq. (C.13), we find
$$\begin{aligned} \lambda ^4+\mathbb {P}_{11}\lambda ^3+\mathbb {P}_{12}\lambda ^2+\mathbb {P}_{13}\lambda +\mathbb {P}_{14}+e^{\lambda \tau }\left( \mathbb {Q}_{11}\lambda ^3+\mathbb {Q}_{12}\lambda ^2+\mathbb {Q}_{13}\lambda +\mathbb {Q}_{14}\right) +e^{-\lambda \tau }\left( F_{11}\lambda +F_{12}\right) =0, \end{aligned}$$
(C.14)
where
$$\begin{aligned} \mathbb {P}_{11}= & {} A_{11}, \nonumber \\ \mathbb {P}_{12}= & {} A_{12}+D_{11}+E_{11},\nonumber \\ \mathbb {P}_{13}= & {} A_{13}+D_{12}+E_{12}, \nonumber \\ \mathbb {P}_{14}= & {} A_{14}+D_{13}+E_{13},\nonumber \\ \mathbb {Q}_{11}= & {} B_{11}+C_{11},\nonumber \\ \mathbb {Q}_{12}= & {} B_{12}+C_{12},\nonumber \\ \mathbb {Q}_{13}= & {} B_{13}+C_{13},\nonumber \\ \mathbb {Q}_{14}= & {} B_{14}+C_{14}. \end{aligned}$$
Let \(\lambda =i\omega (\omega >0)\) be a root of (C.14), then
$$\begin{aligned} \mathbb {M}_{11}(\omega )\cos \omega \tau -\mathbb {M}_{12}(\omega ) \sin \omega \tau= & {} \mathbb {M}_{13}(\omega ),\nonumber \\ \mathbb {M}_{14}(\omega )\sin \omega \tau +\mathbb {M}_{15}(\omega ) \cos \omega \tau= & {} \mathbb {M}_{16}(\omega ), \end{aligned}$$
(C.15)
where
$$\begin{aligned} \mathbb {M}_{11}= & {} -\mathbb {Q}_{12}\omega ^2+\mathbb {Q}_{14}+F_{12},\nonumber \\ \mathbb {M}_{12}= & {} -\mathbb {Q}_{11}\omega ^2+(\mathbb {Q}_{13}-F_{11})\omega ,\nonumber \\ \mathbb {M}_{13}= & {} \mathbb {P}_{12}\omega ^2-\omega ^4-\mathbb {P}_{14},\nonumber \\ \mathbb {M}_{14}= & {} -\mathbb {Q}_{12}\omega ^2+\mathbb {Q}_{14}-F_{12},\nonumber \\ \mathbb {M}_{15}= & {} -\mathbb {Q}_{11}\omega ^3+(\mathbb {Q}_{13}+F_{11})\omega ,\nonumber \\ \mathbb {M}_{16}= & {} \mathbb {Q}_{11}\omega ^3-\mathbb {P}_{13}\omega . \end{aligned}$$
Then we can obtain the expressions of \(\cos \omega \tau \) and \(\sin \omega \tau \) as follows:
$$\begin{aligned} \cos \omega \tau= & {} \frac{\mathbb {M}_{17}(\omega )}{\mathbb {M}_{19}(\omega )}, \nonumber \\ \sin \omega \tau= & {} \frac{\mathbb {M}_{18}(\omega )}{\mathbb {M}_{19}(\omega )}, \end{aligned}$$
(C.16)
where
$$\begin{aligned} \mathbb {M}_{17}(\omega )= & {} \mathbb {M}_{12}\mathbb {M}_{16}+\mathbb {M}_{13}\mathbb {M}_{14}, \nonumber \\ \mathbb {M}_{18}(\omega )= & {} \mathbb {M}_{11}\mathbb {M}_{16}-\mathbb {M}_{13}\mathbb {M}_{15}, \nonumber \\ \mathbb {M}_{19}(\omega )= & {} \mathbb {M}_{11}\mathbb {M}_{14}+\mathbb {M}_{12}\mathbb {M}_{15}. \end{aligned}$$
Thus, squaring and adding the equations of (C.16), we obtain the following equation with respect to ‘\(\omega \)’
$$\begin{aligned} \mathbb {M}_{17}^2(\omega )+\mathbb {M}_{18}^2(\omega )-\mathbb {M}_{19}^2(\omega )=0. \end{aligned}$$
(C.17)
Consider \(\mathbf {H_{15}:}\) Eq. (C.17) has a positive root \(\omega _0\). Then, we obtain
$$\begin{aligned} \tau _n^*=\frac{1}{\omega _0}\arccos \left[ \frac{\mathbb {M}_{17}(\omega _0)}{\mathbb {M}_{19}(\omega _0)}\right] +\frac{2n\pi }{\omega _0}; \quad n=0, 1, 2, ... \end{aligned}$$
(C.18)
Hence, for finite number of positive \(\omega _0\) we get finite number of \(\tau _0^i\), \(i=1, 2, 3, ...\) Let
$$\begin{aligned} \tau _0^*=\min \left\{ \tau _0^i, i=1, 2, 3, ... \right\} \end{aligned}$$
Define the function \(\theta (\tau )\in [0, 2\pi )\) such that \(\cos \theta (\tau )\) is given by the right-hand side of (C.18). Then, solving
$$\begin{aligned} \bar{S}_{2n}(\tau )=\tau -\tau _n^*, \end{aligned}$$
we get the critical value of ‘\(\tau \)’, at which stability switches occurs. Differentiating (C.13) with respect to \(\tau \) and substituting \(\lambda =i\omega _0\), one obtains
$$\begin{aligned} Re\left[ \frac{\hbox {d}\lambda }{\hbox {d}\tau }\right] ^{-1}_{\lambda =i\omega _0}=\frac{\mathbb {M}_{17}^{'}(\omega _0)}{\mathbb {M}_{19}(\omega _0^2)}. \end{aligned}$$
(C.19)
Therefore, \(Re\left[ \frac{\hbox {d}\lambda }{\hbox {d}\tau }\right] ^{-1}_{\lambda =i\omega _0}\ne 0\) if the condition
\(\mathbf {H_{16}:}\) \(\mathbb {M}_{17}^{'}(\omega _0)=\frac{\hbox {d}\mathbb (M)_{17}(\omega )}{\hbox {d}\omega }|_{\omega =\omega _0}\ne 0\) holds.
Thus, using Hopf bifurcation theorem [44], we obtain the result of Theorem 4.3 if the conditions \(H_{15}\): and \(H_{16}\): hold. \(\square \)
Proof of Theorem 4.4
For \(\tau _2>0\) and \(\tau _1\in (0, \tau _{1_0})\), here we consider \(\tau _2\) as bifurcation parameter and \(\tau _1\) is in its stable interval. Let \(\lambda =i\omega \) be a root of the characteristic equation (9). Then, we have
$$\begin{aligned} \mathbb {N}_{11}(\omega , \tau _1)\cos \omega \tau _2+\mathbb {N}_{12}(\omega , \tau _1)\sin \omega \tau _2= & {} \mathbb {N}_{13}(\omega , \tau _1),\nonumber \\ \mathbb {N}_{12}(\omega , \tau _1)\cos \omega \tau _2-\mathbb {N}_{11}(\omega , \tau _1)\sin \omega \tau _2= & {} \mathbb {N}_{14}(\omega , \tau _1), \end{aligned}$$
(C.20)
where
$$\begin{aligned} \mathbb {N}_{11}(\omega , \tau _1)= & {} B_{14}-B_{12}\omega ^2-E_{11}\omega ^2\cos \omega \tau _1+E_{13}\cos \omega \tau _1+E_{12}\omega \sin \omega \tau _1+F_{12}\cos 2\omega \tau _1\nonumber \\&+F_{11}\omega \sin 2\omega \tau _1,\nonumber \\ \mathbb {N}_{12}(\omega , \tau _1)= & {} B_{13}\omega -B_{11}\omega ^3+E_{12}\omega \cos \omega \tau _1+E_{11}\omega ^2\sin \omega \tau _1-E_{13}\sin \omega \tau _1+F_{11}\omega \cos 2\omega \tau _1\nonumber \\&-F_{12}\sin 2\omega \tau _1,\nonumber \\ \mathbb {N}_{13}(\omega , \tau _1)= & {} C_{12}\omega ^2\cos \omega \tau _1-C_{14}\cos \omega \tau _1+C_{11}\omega ^3\sin \omega \tau _1+D_{11}\omega ^2\cos 2\omega \tau _1-D_{13}\cos 2\omega \tau _1\nonumber \\&-D_{12}\omega \sin 2\omega \tau _1-\omega ^4+A_{12}\omega ^2-A_{14},\nonumber \\ \mathbb {N}_{14}(\omega , \tau _1)= & {} C_{11}\omega ^3\cos \omega \tau _1-C_{13}\omega \cos \omega \tau _1-C_{12}\omega ^2\sin \omega \tau _1+C_{13}\omega \sin \omega \tau _1+C_{14}\sin \omega \tau _1\nonumber \\&-D_{12}\omega \cos 2\omega \tau _1-D_{11}\omega ^2\sin 2\omega \tau _1+D_{13}\sin 2\omega \tau _1+A_{11}\omega ^3-A_{13}\omega . \end{aligned}$$
Solving the equations of (C.20), we obtain the following expression of \(\cos \omega \tau _2\) and \(\sin \omega \tau _2\) as follows
$$\begin{aligned} \cos \omega \tau _2= & {} \frac{\mathbb {N}_{15}(\omega , \tau _1)}{\mathbb {N}_{17}(\omega , \tau _1)} ,\nonumber \\ \sin \omega \tau _2= & {} \frac{\mathbb {N}_{16}(\omega , \tau _1)}{\mathbb {N}_{17}(\omega , \tau _1)}, \end{aligned}$$
(C.21)
where
$$\begin{aligned} \mathbb {N}_{15}(\omega , \tau _1)= & {} \mathbb {N}_{13}(\omega , \tau _1)\mathbb {N}_{11}(\omega , \tau _1)+\mathbb {N}_{14}(\omega , \tau _1)\mathbb {N}_{12}(\omega , \tau _1),\nonumber \\ \mathbb {N}_{16}(\omega , \tau _1)= & {} \mathbb {N}_{12}(\omega , \tau _1)\mathbb {N}_{13}(\omega , \tau _1)-\mathbb {N}_{11}(\omega , \tau _1)\mathbb {N}_{14}(\omega , \tau _1),\nonumber \\ \mathbb {N}_{17}(\omega , \tau _1)= & {} \mathbb {N}_{11}^2(\omega , \tau _1)+\mathbb {N}_{12}^2(\omega , \tau _1). \end{aligned}$$
Thus, squaring and adding the equations of (C.21), we obtain the following equation with respect to ‘\(\omega \)’ and \(\tau _1\)
$$\begin{aligned} \mathbb {N}_{15}^2(\omega , \tau _1)+\mathbb {N}_{16}^2(\omega , \tau _1)-\mathbb {N}_{17}^2(\omega , \tau _1)=0. \end{aligned}$$
(C.22)
Consider \(\mathbf {H_{17}:}\) Eq. (C.22) has a positive root \(\omega _0\). Then, we obtain
$$\begin{aligned} \tau _{2_n}^*=\frac{1}{\omega _0}\arccos \left[ \frac{\mathbb {N}_{15}(\omega , \tau _1)}{\mathbb {N}_{17}(\omega , \tau _1)}\right] +\frac{2n\pi }{\omega _0}; \quad n=0, 1, 2, ... \end{aligned}$$
(C.23)
Hence, for finite number of positive \(\omega _0\) and \(n=0\) we get finite number of \(\tau _{2_0}^i\), \(i=1, 2, 3, ...\)
Let
$$\begin{aligned} \tau _{2_0}^*=\min \left\{ \tau _{2_0}^i, i=1, 2, 3, ... \right\} \end{aligned}$$
Define the function \(\theta (\tau _2)\in [0, 2\pi )\) such that \(\cos \theta (\tau _2)\) is given by the right-hand side of (C.23). Then, solving
$$\begin{aligned} \bar{S}_{3n}(\tau _2)=\tau _2-\tau _{2_n}^*, \end{aligned}$$
we get the critical value of ‘\(\tau _2\)’, at which stability switches occurs. Differentiating (9) with respect to \(\tau _2\) and substituting \(\lambda =i\omega _0\), one gets
$$\begin{aligned} Re\left[ \frac{\hbox {d}\lambda }{\hbox {d}\tau _2}\right] ^{-1}_{\lambda =i\omega _0}=\frac{\mathbb {N}_{15}^{'}(\omega , \tau _1)}{\mathbb {N}_{17}(\omega , \tau _1)}. \end{aligned}$$
(C.24)
Therefore, \(Re\left[ \frac{\hbox {d}\lambda }{\hbox {d}\tau _2}\right] ^{-1}_{\lambda =i\omega _0}\ne 0\) if the condition
\(\mathbf {H_{18}:}\) \(\mathbb {N}_{15}^{'}(\omega , \tau _1)=\frac{\hbox {d}\mathbb (N)_{15}(\omega , \tau _1)}{\hbox {d}\omega }|_{\omega =\omega _0}\ne 0\) holds. Thus, using Hopf bifurcation theorem [44], we get the result of Theorem 4.4 if the conditions \(H_{17}\): and \(H_{18}\): hold. \(\square \)
