Appendix
Proof of Theorem 3.1
Right hand side of system (1) is continuous and locally Lipschitzian functions on \(\mathbb {C}\) and so there exists an unique solution (S(t), A(t), Q(t), I(t), H(t), R(t)) on \([0,\tau ),\) where \(0<\tau \le +\infty \) [40]. First we prove that, \(S(t)>0, \ \forall \, t\in [0,\tau ).\) If it does not hold, then \(\exists \ t_{1} \in (0,\tau )\) such that \(S(t_{1})=0,\ \dot{S}(t_{1})\le 0\) and \(S(t)>0, \ \forall \, t\in [0,t_{1}).\) Hence we must have \(A(t)\ge 0,\ \forall \, t\in [0,t_{1}).\) If it is not true, then \(\exists \ t_{2} \in (0,t_{1})\) such that \(A(t_{2})=0,\ \dot{A}(t_{2})< 0\) and \(A(t)\ge 0, \ \forall \, t\in [0,t_{2}).\) Next we claim \(Q(t)\ge 0,\ \forall \, t\in [0,t_{2}).\) Suppose it is not true. Then \(\exists \ t_{3} \in (0,t_{2})\) such that \(Q(t_{3})=0,\ \dot{Q}(t_{3})< 0\) and \(Q(t)\ge 0, \ \forall \, t\in [0,t_{3}).\) From third equation of (1), we have
$$\begin{aligned} \frac{dQ}{dt}\bigg |_{t=t_{3}}= pA(t_{3})-(\alpha +\xi +d) Q(t_{3})=pA(t_{3})\ge 0, \end{aligned}$$
which is a contradiction to \(\dot{Q}(t_{3})< 0.\) So, \(Q(t)\ge 0, \ \forall \, t\in [0,t_{2}).\)
Similarly, \(I(t)\ge 0, \ \forall \, t\in [0,t_{2}).\)
Our next claim is \(H(t)\ge 0, \ \forall \, t\in [0,t_{2}).\) Suppose this does not hold, then \(\exists \ t_{4} \in (0,t_{2})\) such that \(H(t_{4})=0,\ \dot{H}(t_{4})< 0\) and \(H(t)\ge 0, \ \forall \, t\in [0,t_{4}).\) Now, from the fifth equation of (1):
$$\begin{aligned} \frac{dH}{dt}\bigg |_{t=t_{4}}= & {} \alpha Q(t_{4})+\omega I(t_{4})-(\phi +d+\delta _{3})H(t_{4})\\= & {} \alpha Q(t_{4})+\omega I(t_{4}) \ge 0, \end{aligned}$$
which is a contradiction to \(\dot{H}(t_{4})< 0.\) Hence, \(H(t)\ge 0, \ \forall \, t\in [0,t_{2}).\)
From the second equation of (1), we have
$$\begin{aligned} \begin{aligned} \frac{dA}{dt}\bigg |_{t=t_{2}}&= (\beta _{1}A(t_{2})+\beta _{2}I(t_{2})+\beta _{3}H(t_{2}))S(t_{2})\\&\quad -(p+\eta +d+\delta _{1})A(t_{2})\\&=(\beta _{2}I(t_{2})+\beta _{3}H(t_{2}))S(t_{2}) \ge 0 \end{aligned} \end{aligned}$$
It is a contradiction to \(\dot{A}(t_{2})< 0.\) So, \(A(t)\ge 0, \ \forall \, t\in [0,t_{1}).\) Hence \(Q(t)\ge 0,\ I(t)\ge 0,\ H(t)\ge 0,\ \ \forall \, t\in [0,t_{1}).\)
Let us claim \(R(t)\ge 0, \ \forall \, t\in [0,t_{1}).\) Suppose this does not hold, then \(\exists \ t_{5} \in (0,t_{1})\) such that \(R(t_{5})=0,\ \dot{R}(t_{5})< 0\) and \(R(t)\ge 0, \ \forall \, t\in [0,t_{5}).\) From the last equation of (1), we have:
$$\begin{aligned} \frac{dR}{dt}\bigg |_{t=t_{5}}= & {} \kappa S(t_{5})+ \xi Q(t_{5})+\phi H(t_{5})-(d+\psi )R(t_{5})\\= & {} \kappa S(t_{5})+ \xi Q(t_{5})+\phi H(t_{5}) > 0, \end{aligned}$$
which is a contradiction to \(\dot{R}(t_{5})< 0.\) Hence, \(R(t)\ge 0, \ \forall \, t\in [0,t_{1}).\)
From the first equation of (1), we have
$$\begin{aligned} \begin{aligned} \frac{dS}{dt}\bigg |_{t=t_{1}}&= \Lambda - (\beta _{1}A(t_{1})+\beta _{2}I(t_{1})+\beta _{3}H(t_{1}))S(t_{1})\\&\quad -(d+\kappa )S(t_{1})+\psi R(t_{1}) \\&=\Lambda +\psi R(t_{1}) > 0, \end{aligned} \end{aligned}$$
which is a contradiction to \(\dot{S}(t_{1})\le 0.\) So, \(S(t)> 0, \ \forall \, t\in [0,\tau ),\) where \(0<\tau \le +\infty \). Also, by the previous steps we have \(A(t)\ge 0, \ Q(t) \ge 0,\ I(t)\ge 0, \ H(t) \ge 0\) and \(R(t)\ge 0,\ \forall \, t\in [0,\tau ),\) where \(0<\tau \le +\infty .\) Hence it is proved. \(\square \)
Proof of Theorem 3.2
$$\begin{aligned} \begin{aligned} \text {Let},\ N(t)&=S(t)+A(t)+Q(t)+I(t)+H(t)+R(t)\\ \therefore \frac{dN}{dt}&=\Lambda -d(S+A+Q+I+H+R)\\&\quad -\delta _{1}A-\delta _{2}I-\delta _{3}H \\&=\Lambda -dN-(\delta _{1}A+\delta _{2}I+\delta _{3}H) \\&\le \Lambda -dN \\&\Rightarrow 0<N(t)\le N(0)e^{-dt} +\frac{\Lambda }{d}\left( 1-e^{-dt}\right) ,\\&\quad \text {where}\ N(0)=S(0)+A(0)+Q(0)+I(0)\\&\quad +H(0)+R(0). \end{aligned} \end{aligned}$$
As \(t\rightarrow \infty ,\ \displaystyle 0<N(t)\le \frac{\Lambda }{d}.\) Therefore, all solutions of system (1) enter into the region: \(\displaystyle \Omega = \left\{ (S,A,Q,I,H,R)\in \mathbb {R}_{+}^{6}: 0< N(t)\le \frac{\Lambda }{d}\right\} .\) \(\square \)
Proof of Theorem 6.1
Jacobian matrix corresponding to DFE \(\displaystyle E_{0}=\left( \frac{\Lambda (d+\psi )}{d(d+\psi +\kappa )},0,0,0,0,\frac{\Lambda \kappa }{d(d+\psi +\kappa )}\right) \) is given as follows:
$$\begin{aligned} \overline{J}\big |_{E_{0}}= \left( \begin{array}{cccccc} -(d+\kappa ) &{} -\beta _{1}S_{0} &{} 0 &{} -\beta _{2}S_{0} &{} -\beta _{3}S_{0} &{} \psi \\ 0 &{} \beta _{1}S_{0}-\alpha _{0} &{} 0 &{} \beta _{2}S_{0} &{} \beta _{3}S_{0} &{} 0 \\ 0 &{} p &{} -\alpha _{1} &{} 0 &{} 0 &{} 0 \\ 0 &{} \eta &{} 0 &{} -\alpha _{2} &{} 0 &{} 0 \\ 0 &{} 0 &{} \alpha &{} \omega &{} -\alpha _{3} &{} 0 \\ \kappa &{} 0 &{} \xi &{} 0 &{} \phi &{} -(d+\psi ) \end{array} \right) \end{aligned}$$
Two eigenvalues of \(\overline{J}\big |_{E_{0}}\) are roots of the equation: \(\lambda ^{2}+(2d+\kappa +\psi )\lambda +d(\kappa +d+\psi )=0\) which imply that they are the roots with negative real parts and other four eigenvalues are roots of the following equation: \(\lambda ^{4}+M_{1}\lambda ^{3}+M_{2}\lambda ^{2}+M_{3}\lambda +M_{4}=0,\) where \(M_{1}=\alpha _{0}+\alpha _{1}+\alpha _{2}+\alpha _{3}-\beta _{1}S_{0}\), \(M_{2}=\alpha _{0}\alpha _{2}+\alpha _{1}\alpha _{3}+(\alpha _{0}+\alpha _{2})(\alpha _{1}+\alpha _{3})-\beta _{1}S_{0}(\alpha _{1}+\alpha _{2}+\alpha _{3})-\beta _{2}S_{0}\eta \), \(M_{3}=\alpha _{0}\alpha _{2}(\alpha _{1}+\alpha _{3})+\alpha _{1}\alpha _{3}(\alpha _{0}+\alpha _{2})-\beta _{1}S_{0}\{\alpha _{1}\alpha _{3}+\alpha _{2}(\alpha _{1}+\alpha _{3})\}-\beta _{2}S_{0}\eta (\alpha _{1}+\alpha _{3})-\beta _{3}S_{0}(p\alpha +\omega \eta )\) and \(M_{4}=\alpha _{0}\alpha _{1}\alpha _{2}\alpha _{3}(1-R_{c})\). So, \(M_{4}>0\) for \(R_{c}<1\). Therefore the characteristic equation has roots with negative real parts only when \(M_{1},\ M_{2},\ M_{3}>0\) when \(R_{c}<1\). \(\square \)
Proof of Theorem 6.2
The Jacobian matrix at endemic equilibrium point \(E^{*}\) is given as:
$$\begin{aligned} \overline{J}\big |_{E^{*}}=\left( \begin{array}{cccccc} a_{11} &{} a_{12} &{} 0 &{} a_{14} &{} a_{15} &{} a_{16} \\ a_{21} &{} a_{22} &{} 0 &{} a_{24} &{} a_{25} &{} 0 \\ 0 &{} a_{32} &{} a_{33} &{} 0 &{} 0 &{} 0 \\ 0 &{} a_{42} &{} 0 &{} a_{44} &{} 0 &{} 0 \\ 0 &{} 0 &{} a_{53} &{} a_{54} &{} a_{55} &{} 0 \\ a_{61} &{} 0 &{} a_{63} &{} 0 &{} a_{65} &{} a_{66} \end{array} \right) \end{aligned}$$
where \(a_{11}=-(\beta _{1}A^{*}+\beta _{2}I^{*}+\beta _{3}H^{*})-(d+\kappa ),\ a_{12}=-\beta _{1}S^{*},\ a_{14}=-\beta _{2}S^{*},\ a_{15}=-\beta _{3}S^{*},\ a_{16}=\psi ,\ a_{21}=\beta _{1}A^{*}+\beta _{2}I^{*}+\beta _{3}H^{*},\ a_{22}=\beta _{1}S^{*}-\alpha _{0},\ a_{24}=\beta _{2}S^{*},\ a_{25}=\beta _{3}S^{*},\ a_{32}=p,\ a_{33}=-\alpha _{1},\ a_{42}=\eta ,\ a_{44}=-\alpha _{2},\ a_{53}=\alpha ,\ a_{54}=\omega ,\ a_{55}=-\alpha _{3},\ a_{61}=\kappa ,\ a_{63}=\xi ,\ a_{65}=\phi ,\ a_{66}=-(d+\psi )\).
Characteristic equation of \(\overline{J}\big |_{E^{*}}\) is \(\lambda ^{6}+B_{1}\lambda ^{5}+B_{2}\lambda ^{4}+B_{3}\lambda ^{3}+B_{4}\lambda ^{2}+B_{5}\lambda +B_{6}=0,\) where
$$\begin{aligned} \begin{aligned} B_{1}&=-(a_{11}+a_{22}+a_{33}+a_{44}+a_{55}+a_{66})\\&=\frac{(\alpha _{0}\alpha _{2}R_{c}^{2}-\beta _{1}\eta S_{0}^{2})}{\eta S_{0}R_{c}}I^{*}+\alpha _{0}+\alpha _{1}\\&\quad +\alpha _{2}+\alpha _{3}+(2d+\kappa +\psi ),\\ B_{2}&= a_{11}(a_{22}+a_{33}+a_{44}+a_{55}+a_{66})-a_{12}a_{21}\\&\quad -a_{16}a_{61}+a_{22}(a_{33}+a_{44}+a_{55}+a_{66})\\&\quad -a_{24}a_{42}+a_{33}(a_{44}+a_{55}+a_{66})\\&\quad +a_{44}(a_{55}+a_{66})+a_{55}a_{66},\\ B_{3}&= -\,a_{11}a_{22}(a_{33}+a_{44}+a_{55}+a_{66})+a_{24}a_{42}a_{11}\\&\quad -a_{11}a_{33}(a_{44}+a_{55}+a_{66})-a_{11}a_{44}(a_{55}+a_{66})\\&\quad -a_{11}a_{55}a_{66}+a_{12}a_{21}(a_{33}+a_{44}+a_{55}+a_{66})\\&\quad -a_{14}a_{42}a_{21}+a_{16}a_{61}(a_{22}+a_{33}+a_{44}+a_{55}) \\&\quad -a_{22}a_{33}(a_{44}+a_{55}+a_{66})-a_{22}a_{44}(a_{55}+a_{66})\\&\quad -a_{22}a_{55}a_{66}+a_{24}a_{42}(a_{33}+a_{55}+a_{66})-a_{53}a_{32}a_{25}\\&\quad a_{54}a_{42}a_{25}-a_{33}a_{44}(a_{55}+a_{66})-(a_{33}+a_{44})a_{55}a_{66}, \end{aligned} \end{aligned}$$
$$\begin{aligned} B_{4}=&a_{11}a_{22}a_{33}(a_{44}+a_{55}+a_{66})+a_{11}a_{22}a_{44}(a_{55}+a_{66})\\&\quad +a_{11}a_{22}a_{55}a_{66}-a_{11}a_{24}a_{42}(a_{33}+a_{55}+a_{66})\\&\quad a_{11}a_{25}(a_{32}a_{53}+a_{42}a_{54})+a_{11}a_{33}a_{44}(a_{55}+a_{66})\\&\quad +a_{11}(a_{33}+a_{44})a_{55}a_{66}-a_{12}a_{21}a_{33}(a_{44}+a_{55}+a_{66})\\&\quad -a_{12}a_{21}a_{44}(a_{55}+a_{66})-a_{12}a_{21}a_{55}a_{66}\\&\quad +a_{14}a_{21}a_{42}(a_{33}+a_{55}+a_{66})-a_{15}a_{21}(a_{33}a_{53}+a_{42}a_{54})\\&\quad -a_{16}a_{21}a_{32}a_{63}-a_{16}a_{22}a_{61}(a_{33}+a_{44}+a_{55})\\&\quad +a_{16}a_{24}a_{42}a_{61}-a_{16}a_{61}a_{33}(a_{44}+a_{55})-a_{16}a_{61}a_{44}a_{55}\\&\quad +a_{22}a_{33}a_{44}(a_{55}+a_{66})+a_{22}a_{55}a_{66}(a_{33}+a_{44})\\&\quad -a_{24}a_{42}a_{33}(a_{55}+a_{66})-a_{24}a_{42}a_{55}a_{66}\\&\quad +a_{25}a_{32}a_{53}(a_{44}+a_{66})+a_{25}a_{42}a_{54}(a_{33}+a_{66})\\&\quad +a_{33}a_{44}a_{55}a_{66},\\ B_{5}&=-a_{11}a_{22}a_{33}a_{44}(a_{55}+a_{66})-a_{11}a_{22}(a_{33}+a_{44})a_{55}a_{66}\\&\quad +a_{11}a_{33}a_{24}a_{42}(a_{55}+a_{66})+a_{11}a_{24}a_{42}a_{55}a_{66}\\&\quad -a_{11}a_{25}a_{32}a_{53}(a_{44}+a_{66})-a_{11}a_{25}a_{42}a_{54}(a_{33}\\&\quad +a_{66})-a_{11}a_{33}a_{44}a_{55}a_{66}+a_{12}a_{21}a_{33}a_{44}(a_{55}+a_{66})\\&\quad +a_{12}a_{21}a_{55}a_{66}(a_{33}+a_{44})-a_{14}a_{21}a_{33}a_{42}(a_{55}\\&\quad +a_{66})-a_{14}a_{21}a_{42}a_{55}a_{66}+a_{15}a_{21}a_{32}a_{53}(a_{44}+a_{66})\\&\quad +a_{15}a_{21}a_{42}a_{54}(a_{33}+a_{66})+a_{16}a_{21}a_{32}a_{63}(a_{44}\\&\quad +a_{55})-a_{16}a_{21}a_{65}(a_{32}a_{53}+a_{42}a_{54})\\&\quad +a_{16}a_{22}a_{33}a_{61}(a_{44}+a_{55})+a_{16}a_{22}a_{44}a_{55}a_{61}\\&\quad -a_{16}a_{24}a_{42}a_{61}(a_{33}+a_{55})+a_{16}a_{25}a_{61}(a_{32}a_{53}+a_{42}a_{54})\\&\quad +a_{16}a_{33}a_{44}a_{55}a_{61}-a_{22}a_{33}a_{44}a_{55}a_{66}\\&\quad +a_{24}a_{33}a_{42}a_{55}a_{66}-a_{25}a_{32}a_{44}a_{53}a_{66}-a_{25}a_{33}a_{42}a_{54}a_{66}, \\ B_{6}&=(\beta _{1}A^{*}+\beta _{2}I^{*}+\beta _{3}H^{*})[(d+\psi )\alpha _{0}\alpha _{1}\alpha _{2}\alpha _{3}\\&\quad +p\psi \alpha _{2}(\alpha \phi +\xi \alpha _{3})+\kappa \omega \phi \eta \alpha _{1}]\\&>0,\ \text {for}\ I^{*}>0. \end{aligned}$$
Let us consider
$$\begin{aligned} \begin{aligned} \Delta _{1}&=B_{1},\ \Delta _{2}=\left| \begin{array}{cc} B_{1} &{} 1 \\ B_{3} &{} B_{2} \end{array} \right| ,\\ \Delta _{3}&=\left| \begin{array}{ccc} B_{1} &{} 1 &{} 0\\ B_{3} &{} B_{2} &{} B_{1} \\ B_{5} &{} B_{4} &{} B_{3} \end{array} \right| ,\\ \Delta _{4}&= \left| \begin{array}{cccc} B_{1} &{} 1 &{} 0 &{} 0\\ B_{3} &{} B_{2} &{} B_{1} &{} 1 \\ B_{5} &{} B_{4} &{} B_{3} &{} B_{2} \\ 0 &{} B_{6} &{} B_{5} &{} B_{4} \end{array} \right| , \\ \Delta _{5}&=\left| \begin{array}{ccccc} B_{1} &{} 1 &{} 0 &{} 0 &{} 0\\ B_{3} &{} B_{2} &{} B_{1} &{} 1 &{} 0 \\ B_{5} &{} B_{4} &{} B_{3} &{} B_{2} &{} B_{1} \\ 0 &{} B_{6} &{} B_{5} &{} B_{4} &{} B_{3} \\ 0 &{} 0 &{} 0 &{} B_{6} &{} B_{5} \end{array} \right| , \ \Delta _{6}=Det(\overline{J}\big |_{E^{*}})=B_{6}. \end{aligned} \end{aligned}$$
By Routh-Hurwitz criterion, \(E^{*}\) is locally asymptomatically stable (LAS) if and only if \(\Delta _{i}>0\) for \(i=1,2,3,4,5,6\), i.e., equivalently
-
(i)
\(B_{i}>0\) for \(i=1,6;\)
-
(ii)
\(\Delta _i >0\) for \(i=2,3,4,5\).
\(\square \)
Proof of Theorem 6.4
Followig Theorem (6.3), we can write system (1) as:
$$\begin{aligned} \frac{dX}{dt}&=F(X,Y),\\ \frac{dY}{dt}&=G(X,Y),\ G(X,O) = O,\\ \text {where}\ F(X,Y)&=\left( \begin{array}{ccc} \Lambda -(\beta _{1}A+\beta _{2}I+\beta _{3}H)S+\psi R-(d+\kappa )S\\ \kappa S+\xi Q+\phi H-(d+\psi )R \end{array} \right) \\ \text {and}\ G(X,Y)&=\left( \begin{array}{ccc} (\beta _{1}A+\beta _{2}I+\beta _{3}H)S-\alpha _{0}A \\ pA-\alpha _{1}Q \\ \eta A-\alpha _{2}I \\ \alpha Q+\omega I-\alpha _{3}H \end{array} \right) \end{aligned}$$
Here \(G(X,O)=O\) taking \(X=(S,R)^{T}\) and \(Y=(A,Q,I,H)^{T}\). The DFE of system (1) is \(U_{0}=E_{0}=(X_{0},O)^{T}\) with \(\displaystyle X_{0}\equiv (S_{0},R_{0})=\left( \frac{\Lambda (d+\psi )}{d(d+\psi +\kappa )},\frac{\Lambda \kappa }{d(d+\psi +\kappa )}\right) \). Clearly \(X_{0}\) is globally asymptotically stable for \(\displaystyle \frac{dX}{dt}=F(X,O)\) as \(X\rightarrow (S_{0},R_{0})^{T}\) whenever \(t\rightarrow \infty \). Further we have
$$\begin{aligned} \begin{aligned} G(X,Y)&=D_{Y}G(X_{0},O)Y-\overline{G}(X,Y) \\&=\left( \begin{array}{cccc} \beta _{1}S_{0}-\alpha _{0} &{} 0 &{} \beta _{2}S_{0} &{} \beta _{3}S_{0} \\ p &{} -\alpha _{1} &{} 0 &{} 0 \\ \eta &{} 0 &{} -\alpha _{2} &{} 0 \\ 0 &{} \alpha &{} \omega &{} -\alpha _{3} \end{array} \right) .\left( \begin{array}{c} A\\ Q\\ I\\ H \end{array} \right) \\&-\left( \begin{array}{c} (S_{0}-S)(\beta _{1}A+\beta _{2}I+\beta _{3}H)\\ 0\\ 0\\ 0 \end{array} \right) \end{aligned} \end{aligned}$$
Here \(B(X,Y)=D_{Y}G(X_{0},O)\) is a stable matrix when local stability criterion of \(E_{0}\) holds along with \(R_{c}<1\) and the off-diagonal elements are non-negative also implying B(X, Y) is a M-matrix. Moreover, \(\overline{G}(X,Y)\ge O\) when \(S\le S_{0}\le \frac{\Lambda }{d}\). Hence the assumptions (H1) and (H2) are satisfied and by Theorem (6.3), DFE \(E_{0}\) of system (1) is globally asymptotically stable when \(S\le S_{0}\) for \(R_{c}<1\). \(\square \)
Proof of Theorem 6.5
Consider a Lyapunov function V(t) as:
$$\begin{aligned} \begin{aligned} V(t)&=\left( S-S^{*}-S^{*}ln\frac{S}{S^{*}}\right) +m_{1}\left( A-A^{*}-A^{*} ln\frac{A}{A^{*}}\right) \\&\quad +m_{2}\left( Q-Q^{*}-Q^{*}ln\frac{Q}{Q^{*}}\right) \\&\quad +m_{3}\left( I-I^{*}-I^{*}ln\frac{I}{I^{*}}\right) +m_{4}\left( H-H^{*}-H^{*}ln\frac{H}{H^{*}}\right) \\&\quad +m_{5}\left( R-R^{*} -R^{*}ln\frac{R}{R^{*}}\right) \end{aligned} \end{aligned}$$
Time derivative of V along the solutions of system (1) is given by
$$\begin{aligned} \frac{dV}{dt}&= \left( 1-\frac{S^{*}}{S}\right) [\Lambda -(\beta _{1}A+\beta _{2}I+\beta _{3}H)S\\&\quad +\psi R-(d+\kappa )S]\\&+m_{1}\left( 1-\frac{A^{*}}{A}\right) [(\beta _{1}A+\beta _{2}I+\beta _{3}H)S-\alpha _{0}A]\\&+m_{2}\left( 1-\frac{Q^{*}}{Q}\right) [pA-\alpha _{1}Q]\\&+m_{3}\left( 1-\frac{I^{*}}{I}\right) [\eta A-\alpha _{2}I]\\&\quad +m_{4}\left( 1-\frac{H^{*}}{H}\right) [\alpha Q+\omega I-\alpha _{3}H]\\&+m_{5}\left( 1-\frac{R^{*}}{R}\right) [\kappa S+\xi Q+\phi H-(d+\psi )R] \\&=\left( 1-\frac{S^{*}}{S}\right) [\beta _{1}(A^{*}S^{*}-SA)+\beta _{2}(I^{*}S^{*}-SI)\\&+\beta _{3}(H^{*}S^{*}-SH)+\psi (R-R^{*})\\&-(d+\kappa )(S-S^{*})]+m_{1}\left( 1-\frac{A^{*}}{A}\right) \\&\quad [-\beta _{1}(A^{*}S^{*}-SA)\\&-\beta _{2}(I^{*}S^{*}-SI)-\beta _{3}(H^{*}S^{*}-SH)\\&-\alpha _{0}(A-A^{*})]+m_{2}\left( 1-\frac{Q^{*}}{Q}\right) \\&\quad [p(A-A^{*})-\alpha _{1}(Q-Q^{*})]\\&+m_{3}\left( 1-\frac{I^{*}}{I}\right) [\eta (A-A^{*})\\&-\alpha _{2}(I-I^{*})]+m_{4}\left( 1-\frac{H^{*}}{H}\right) [\alpha (Q-Q^{*})\\&+\omega (I-I^{*})-\alpha _{3}(H-H^{*})]\\&+m_{5}\left( 1-\frac{R^{*}}{R}\right) [\kappa (S-S^{*})\\&+\xi (Q-Q^{*})+\phi (H-H^{*})-(d+\psi )(R-R^{*})] \end{aligned}$$
Let, \(\frac{S}{S^{*}}=x,\ \frac{A}{A^{*}}=y,\ \frac{Q}{Q^{*}}=z,\ \frac{I}{I^{*}}=u,\ \frac{H}{H^{*}}=v\) and \(\frac{R}{R^{*}}=l.\) So, we have
$$\begin{aligned} \frac{dV}{dt}&= \left( 1-\frac{1}{x}\right) [-S^{*}(\beta _{1}A^{*}+\beta _{2}I^{*} +\beta _{3}H^{*}+d+\kappa )(x-1)\\&\quad -\beta _{1}SA^{*}(y-1)-\beta _{2}SI^{*}(u-1)\\&\quad -\beta _{3}SH^{*}(v-1)+\psi R^{*}(l-1)]+m_{1}\left( 1-\frac{1}{y}\right) \\&\quad [S^{*}(\beta _{1}A^{*}+\beta _{2}I^{*}+\beta _{3}H^{*})(x-1)+\beta _{1}SA^{*}(y-1)\\&\quad +\beta _{2}SI^{*}(u-1)+\beta _{3}SH^{*}(v-1)-\alpha _{0}A^{*}(y-1)]\\&\quad +m_{2}\left( 1-\frac{1}{z}\right) [pA^{*}(y-1)-\alpha _{1}Q^{*}(z-1)]\\&\quad +m_{3}\left( 1-\frac{1}{u}\right) [\eta A^{*}(y-1)-\alpha _{2}I{*}(u-1)]\\&\quad +m_{4}\left( 1-\frac{1}{v}\right) [\alpha Q^{*}(z-1)+\omega I^{*}(u-1)\\&\quad -\alpha _{3}H^{*}(v-1)]+m_{5}\left( 1-\frac{1}{l}\right) [\kappa S^{*}(x-1)\\&\quad +\xi Q^{*}(z-1)+\phi H^{*}(v-1)-(d+\psi )R^{*}(l-1)] \end{aligned}$$
Now steady state of system (1) at \(E^{*}\) gives \(\displaystyle \Lambda -(\beta _{1}A^{*}+\beta _{2}I^{*}+\beta _{3}H^{*})S^{*}+\psi R^{*}=(d+\kappa )S^{*}, \ \ (\beta _{1}A^{*}+\beta _{2}I^{*}+\beta _{3}H^{*})S = \alpha _{0}A^{*}, pA^{*}=\alpha _{1}Q^{*}, \ \ \eta A^{*}=\alpha _{2}I^{*}, \ \ \alpha Q^{*}+\omega I^{*}=\alpha _{3}H^{*}, \ \ \kappa S^{*}+\xi Q^{*}+\phi H^{*}=(d+\psi )R^{*}\). Also, we take \(m_{1},\ m_{5}=1\). So, we have
$$\begin{aligned} \frac{dV}{dt} =&S^{*}(\beta _{1}A^{*}+\beta _{2}I^{*}+\beta _{3}H^{*}+d+\kappa )\left( 2-x-\frac{1}{x}\right) \\&+\alpha _{0}A^{*}\left( 2-y-\frac{1}{y}\right) \\&+m_{2}\alpha _{1}Q^{*}\left( 2-z-\frac{1}{z}\right) +m_{3}\alpha _{2}I^{*}\left( 2-u-\frac{1}{u}\right) \\&+m_{4}\alpha _{3}H^{*}\left( 2-v-\frac{1}{v}\right) \\&+m_{5}(d+\psi )R^{*}\left( 2-l-\frac{1}{l}\right) \\&-\beta _{1}SA^{*}\left( 1-\frac{y}{x}+\frac{1}{x}-\frac{1}{y}\right) -\beta _{2}SI^{*}\left( \frac{u}{y}-\frac{u}{x}+\frac{1}{x}-\frac{1}{y}\right) \\&-\beta _{3}SH^{*}\left( \frac{v}{y}-\frac{v}{x}+\frac{1}{x}-\frac{1}{y}\right) +\psi R^{*}\left( l-1-\frac{l}{x}+\frac{1}{x}\right) \\&+m_{2}pA^{*}\left( y-1-\frac{y}{z}+\frac{1}{z}\right) \\&+(\beta _{1}A^{*}+\beta _{2}I^{*}+\beta _{3}H^{*})S^{*}\left( x-1-\frac{x}{y}+\frac{1}{y}\right) \\&+m_{3}\eta A^{*}\left( y-1-\frac{y}{u}+\frac{1}{u}\right) \\&+m_{4}\alpha Q^{*}\left( z-1-\frac{z}{v}+\frac{1}{v}\right) \\&+m_{4}\omega I^{*}\left( u-1-\frac{u}{v}+\frac{1}{v}\right) \\&+m_{5}\xi Q^{*}\left( z-1-\frac{z}{l}+\frac{1}{l}\right) \\&+m_{5}\kappa S^{*}\left( x-1-\frac{x}{l}+\frac{1}{l}\right) +m_{5}\phi H^{*}\left( v-1-\frac{v}{l}+\frac{1}{l}\right) \\&=\alpha _{0}A^{*}\left( 3-\frac{1}{x}-\frac{x}{y}-y\right) +(d+\kappa )\left( 2-x-\frac{1}{x}\right) \\&+m_{2}\alpha _{1}Q^{*}\left( 1+y-z-\frac{y}{z}\right) \\&+m_{3}\alpha _{2}I^{*}\left( 1+y-u-\frac{y}{u}\right) +m_{4}\alpha Q^{*}\left( 1+z-v-\frac{z}{v}\right) \\&+m_{4}\omega I^{*}\left( 1+u-v-\frac{u}{v}\right) \\&+dR^{*}\left( 2-l-\frac{1}{l}\right) +\beta _{1}SA^{*}\\&\left( \frac{1}{y}+\frac{y}{x}-1-\frac{1}{x}\right) \\&+\beta _{2}SI^{*}\left( \frac{1}{y}+\frac{u}{x}-\frac{1}{x}-\frac{u}{y}\right) \\&+\beta _{3}SH^{*}\left( \frac{1}{y}+\frac{v}{x}-\frac{1}{x}-\frac{v}{y}\right) +\psi R^{*}\left( 1+\frac{1}{x}-\frac{l}{x}-\frac{1}{l}\right) \\&+\xi Q^{*}\left( z-1-\frac{z}{l}+\frac{1}{l}\right) \\&+\kappa S^{*}\left( x-1-\frac{x}{l}+\frac{1}{l}\right) +\phi H^{*}\left( v-1-\frac{v}{l}+\frac{1}{l}\right) \end{aligned}$$
Let us consider \(m_4 = \frac{\phi + \beta _3 S^*}{\alpha 3},m_2 = \frac{\xi + m_4\alpha }{\alpha _1 }\) and \(m_3 = \frac{\beta _2 S^* + m_4\omega }{\alpha _2}.\) So, \(\displaystyle \frac{dV}{dt}\le 0\) in the region \(\Phi \) provided following conditions are satisfied,
\(\displaystyle (i)\ \psi S^{*}R^{*}<2\alpha _{0}S^{*}A^{*}\) and \(\displaystyle (ii)\ pm_{2}\alpha _{2}+\eta m_{3}\alpha _{2}+\beta _{1}\alpha _{2}S^{*}<\alpha _{0}\alpha _{2}\).
Moreover, \(\displaystyle \frac{dV}{dt}\bigg |_{E^{*}}=0\). So, by Lyapunov LaSalle’s theorem [41], \({E^{*}}\) is GAS in the interior of \(\Phi \) subject to the stated parametric conditions. \(\square \)
Existence of optimal control functions
Now we derive the conditions for existence of optimal control interventions which also minimize the cost function J in a finite time period.
Proof of Theorem 9.1
The optimal control variables, when exist, satisfy the following conditions:
-
(i)
Solutions of system (8) with control variables \(u_{1}\) and \(u_{2}\) in \(\Pi \ne \phi \).
-
(ii)
The mentioned set \(\Pi \) is closed, convex and the state system is represented with linear function of control variables where coefficients depend on time and also on state variables.
-
(iii)
Integrand of (7): L is convex on \(\Pi \) and \(L(S,A,Q,I,H,R,u_{1},u_{2})\ge h(u_{1},u_{2})\) where \(h(u_{1},u_{2})\) is continuous and \(||(u_{1},u_{2})||^{-1}h(u_{1},u_{2})\rightarrow \infty \) when \(||(u_{1},u_{2})||\rightarrow \infty \); ||.|| represents the \(L^{2}(0,T_{f})\) norm. From (8), the total population \(N=S+A+Q+I+H+R.\)
$$\begin{aligned} \begin{aligned} \text {So,}\ \frac{dN}{dt}&= \Lambda -dN-\delta _{1}A-\delta _{2}I-\delta _{3}H \le \Lambda -dN \\&\Rightarrow 0<N(t)\le N(0)e^{-dt} +\frac{\Lambda }{d}\left( 1-e^{-dt}\right) , \end{aligned} \end{aligned}$$
where \(N(0)=S(0)+A(0)+Q(0)+I(0)+H(0)+R(0)\). As \(\displaystyle t\rightarrow \infty ,\ 0<N(t)\le \frac{\Lambda }{d}.\) For each of the control variable in \(\Pi \), solution of (8) is bounded and right hand side functions are locally Lipschitzian too. \(Picard-Lindel\ddot{o}f\) theorem shows that condition (i) is satisfied [42]. The control set \(\Pi \) is closed and convex by definition. Again all the equations of system (8) are written as linear equations in \(u_{1}\) and \(u_{2}\) where state variables depend on coefficients and hence condition (ii) is satisfied also. Moreover, the quadratic nature of all control variables guarantee the convex property of integrand \(L(S,A,Q,I,H,R,u_{1},u_{2})\).
$$\begin{aligned} \begin{aligned}&\text {Also,}\ L(S,A,Q,I,H,R,u_{1},u_{2})\\&\quad = w_{1}A+w_{2}I+w_{3}u^{2}_{1}+w_{4}u^{2}_{2} \\&\quad \ge w_{3}u^{2}_{1}+w_{4}u^{2}_{2} \end{aligned} \end{aligned}$$
Let, \(\overline{w} = \min (w_{3},w_{4})>0\) and \(k(u_{1},u_{2})=\overline{w}(u^{2}_{1}+u^{2}_{2}).\) Then \(L(S,A,Q,I,H,R,u_{1},u_{2})\ge k(u_{1},u_{2}).\) Here k is continuous and \(||(u_{1},u_{2})||^{-1}k(u_{1},u_{2})\rightarrow \infty \) whenever \(||(u_{1},u_{2})||\rightarrow \infty .\) Hence, condition (iii) is also satisfied. So, it is concluded that there exist control variables \(u^{*}_{1}\) and \(u^{*}_{2}\) with the condition \(J[u^{*}_{1},u^{*}_{2}]=\min [J[u_{1},u_{2}]]\) [38, 43].
Characterization of optimal control functions
By Pontryagin’s Maximum Principle, we have derived here the necessary conditions for optimal control functions for system (7)-(8) [43,44,45,46,47]. Let us define the Hamiltonian function as:
$$\begin{aligned} \begin{aligned}&\overline{H}\left( S,A,Q,I,H,R,u_{1},u_{2},\lambda \right) \\&\quad =L(S,A,Q,I,H,R,u_{1},u_{2})+\lambda _{1}\frac{dS}{dt}+\lambda _{2}\frac{dA}{dt}+\lambda _{3}\frac{dQ}{dt}\\&\qquad +\lambda _{4}\frac{dI}{dt}+\lambda _{5}\frac{dH}{dt}+\lambda _{6}\frac{dR}{dt}\\&\text {So},\ \overline{H} \\&\quad =w_{1}A+w_{2}I+w_{3}u^{2}_{1}+w_{4}u^{2}_{2}+\lambda _{1}[\Lambda -(1-u_{1}(t))(\beta _{1}A\\&\qquad +\beta _{2}I)S-\beta _{3}HS-dS+\psi R \\&\qquad -u_{2}(t)S]+\lambda _{2}[(1-u_{1}(t))(\beta _{1}A+\beta _{2}I)S+\beta _{3}HS-\alpha _{0}A]\\&\qquad +\lambda _{3}[pA-\alpha _{1}Q]+\lambda _{4}[\eta A-\alpha _{2}I]\\&\qquad +\lambda _{5}[\alpha Q+\omega I-\alpha _{3}H]+\lambda _{6}[\xi Q+u_{2}(t)S\\&\qquad +\phi H-(d+\psi )R] \end{aligned} \end{aligned}$$
(11)
Here \(\lambda =\left( \lambda _{1},\lambda _{2},\lambda _{3},\lambda _{4},\lambda _{5},\lambda _{6}\right) \) are the adjoint variables. We get minimized Hamiltonian by Pontryagin’s Maximum Principle to minimize the cost functional. Pontryagin’s Maximum Principle mainly adjoin the cost functional with the state equations by introducing adjoint variables.
Proof of Theorem 9.2
Let \(u^{*}_{1}\) and \(u^{*}_{2}\) be optimal control variables and \(S^{*}, A^{*}, Q^{*}, I^{*}, H^{*}, R^{*}\) are corresponding optimal state variables of the control system (8) which minimize the cost functional (7). So, by Pontryagin’s Maximum Principle, there exist adjoint variables \(\lambda _{1},\ \lambda _{2},\ \lambda _{3},\ \lambda _{4},\ \lambda _{5},\ \lambda _{6}\) which satisfy the following canonical equations:
$$\begin{aligned} \begin{aligned}&\frac{d\lambda _{1}}{dt}=-\frac{\partial \overline{H}}{\partial S}, \quad \frac{d\lambda _{2}}{dt}=-\frac{\partial \overline{H}}{\partial A}, \quad \frac{d\lambda _{3}}{dt}=-\frac{\partial \overline{H}}{\partial Q},\\&\frac{d\lambda _{4}}{dt}=-\frac{\partial \overline{H}}{\partial I}, \quad \frac{d\lambda _{5}}{dt}=-\frac{\partial \overline{H}}{\partial H}, \\&\frac{d\lambda _{6}}{dt}= -\frac{\partial \overline{H}}{\partial R}. \end{aligned} \end{aligned}$$
So, we have
$$\begin{aligned} \frac{d\lambda _{1}}{dt}&=\lambda _{1}[(\beta _{1}A+\beta _{2}I)(1-u_{1})+\beta _{3}H+u_{2}+d] \nonumber \\&-\lambda _{2}[(\beta _{1}A+\beta _{2}I)(1-u_{1})+\beta _{3}H]-\lambda _{6}(u_{2})\nonumber \\ \frac{d\lambda _{2}}{dt}&=-w_{1}+\lambda _{1}[(1-u_{1})\beta _{1}S]\nonumber \\&\quad -\lambda _{2}[(1-u_{1})\beta _{1}S-\alpha _{0}]-\lambda _{3}(p)-\lambda _{4}(\eta ) \nonumber \\ \frac{d\lambda _{3}}{dt}&=\lambda _{3}(\alpha _{1})-\lambda _{5}(\alpha )-\lambda _{6}(\xi ) \nonumber \\ \frac{d\lambda _{4}}{dt}&=-w_{2}+\lambda _{1}[(1-u_{1})\beta _{2}S]-\lambda _{2}[(1-u_{1})\beta _{2}S]\nonumber \\&\quad +\lambda _{4}(\alpha _{2})-\lambda _{5}(\omega ) \nonumber \\ \frac{d\lambda _{5}}{dt}&=\lambda _{1}(\beta _{3}S)-\lambda _{2}(\beta _{3}S)+\lambda _{5}(\alpha _{3})-\lambda _{6}(\phi )\nonumber \\ \frac{d\lambda _{6}}{dt}&=-\lambda _{1}(\psi )+\lambda _{6}(d+\psi ) \end{aligned}$$
(12)
with the transversality conditions \(\lambda _{i}(T_{f})=0\), for \(i=1,2,3,4,5,6.\)
$$\begin{aligned} \text {From optimality conditions}: ~~~ \frac{\partial \overline{H}}{\partial u_{1}}\bigg |_{u_{1}=u^{*}_{1}}=0 \ \text {and} \ \frac{\partial \overline{H}}{\partial u_{2}}\bigg |_{u_{2}=u^{*}_{2}}=0. \end{aligned}$$
So, \(\displaystyle u^{*}_{1}= \frac{S^{*}(\beta _{1}A^{*}+\beta _{2}I^{*})}{2w_{3}}\left( \lambda _{2}-\lambda _{1}\right) ,\ u^{*}_{2}=\frac{S^{*}}{2w_{4}}\left( \lambda _{1}-\lambda _{6}\right) \).
Now from these findings along with the characteristics of control set \(\Pi ,\) we have
$$\begin{aligned} \begin{aligned} u^{*}_{1}&= {\left\{ \begin{array}{ll} 0, &{} \text {if}\ \frac{S^{*}(\beta _{1}A^{*}+\beta _{2}I^{*})}{2w_{3}}\left( \lambda _{2}-\lambda _{1}\right) < 0 \\ \\ \frac{S^{*}(\beta _{1}A^{*}+\beta _{2}I^{*})}{2w_{3}}\left( \lambda _{2}-\lambda _{1}\right) , &{} \text {if}\ 0\le \frac{S^{*}(\beta _{1}A^{*}+\beta _{2}I^{*})}{2w_{3}}\left( \lambda _{2}-\lambda _{1}\right) \le 1 \\ \\ 1, &{} \text {if}\ \frac{S^{*}(\beta _{1}A^{*}+\beta _{2}I^{*})}{2w_{3}}\left( \lambda _{2}-\lambda _{1}\right) > 1 \end{array}\right. } \end{aligned} \end{aligned}$$
$$\begin{aligned} \begin{aligned} u^{*}_{2}&= {\left\{ \begin{array}{ll} 0, &{} \text {if}\ \frac{S^{*}}{2w_{4}}\left( \lambda _{1}-\lambda _{6}\right) < 0 \\ \\ \frac{S^{*}}{2w_{4}}\left( \lambda _{1}-\lambda _{6}\right) , &{} \text {if}\ 0\le \frac{S^{*}}{2w_{4}}\left( \lambda _{1}-\lambda _{6}\right) \le 1 \\ \\ 1, &{} \text {if}\ \frac{S^{*}}{2w_{4}}\left( \lambda _{1}-\lambda _{6}\right) > 1 \end{array}\right. } \end{aligned} \end{aligned}$$
which is equivalent as (10). \(\square \)
Optimal system
We state the optimal system with optimal control variables \(u^{*}_{1}\) and \(u^{*}_{2}\) below. The optimal system with minimized Hamiltonian \(\overline{H}^{*}\) at \((S^{*},A^{*}, Q^{*}, I^{*}, H^{*}, R^{*}, \lambda _{1}, \lambda _{2}, \lambda _{3}, \lambda _{4}, \lambda _{5},\ \lambda _{6})\) is as follows:
$$\begin{aligned} \frac{dS^{*}}{dt}&= \Lambda -(1-u_{1}^{*})(\beta _{1}A^{*}+\beta _{2}I^{*})S^{*}\nonumber \\&\quad -\beta _{3}H^{*}S^{*}-dS^{*}+\psi R^{*}-u_{2}^{*}S^{*}, \nonumber \\ \frac{dA^{*}}{dt}&= (1-u_{1}^{*})(\beta _{1}A^{*}+\beta _{2}I^{*})S^{*}+\beta _{3}H^{*}S^{*}-\alpha _{0}A^{*}, \nonumber \\ \frac{dQ^{*}}{dt}&= pA^{*}-\alpha _{1}Q^{*},\nonumber \\ \frac{dI^{*}}{dt}&= \eta A^{*}-\alpha _{2}I^{*},\nonumber \\ \frac{dH^{*}}{dt}&=\alpha Q^{*}+\omega I^{*}-\alpha _{3}H^{*},\nonumber \\ \frac{dR^{*}}{dt}&= u_{2}^{*}S^{*}+\xi Q^{*}+\phi H^{*}-(d+\psi )R^{*}, \end{aligned}$$
(13)
with initial conditions: \(S^{*}(0)>0,\ A^{*}(0)\ge 0,\ Q^{*}(0)\ge 0,\ I^{*}(0)\ge 0, H^{*}\ge 0\) and \(R^{*}\ge 0\). The corresponding adjoint system is given as:
$$\begin{aligned} \begin{aligned} \frac{d\lambda _{1}}{dt}&=\lambda _{1}[(\beta _{1}A^{*}+\beta _{2}I^{*})(1-u_{1}^{*})+\beta _{3}H^{*}+u_{2}^{*}+d]\\&\quad -\lambda _{2}[(\beta _{1}A^{*}+\beta _{2}I^{*})(1-u_{1}^{*})+\beta _{3}H^{*}]-\lambda _{6}(u_{2}^{*})\\ \frac{d\lambda _{2}}{dt}&=-w_{1}+\lambda _{1}[(1-u_{1}^{*})\beta _{1}S^{*}]\\&\quad -\lambda _{2}[(1-u_{1}^{*})\beta _{1}S^{*}-\alpha _{0}]-\lambda _{3}(p)-\lambda _{4}(\eta ) \\ \frac{d\lambda _{3}}{dt}&=\lambda _{3}(\alpha _{1})-\lambda _{5}(\alpha )-\lambda _{6}(\xi ) \\ \frac{d\lambda _{4}}{dt}&=-w_{2}+\lambda _{1}[(1-u_{1}^{*})\beta _{2}S^{*}] -\lambda _{2}[(1-u_{1}^{*})\beta _{2}S^{*}]\\&\quad +\lambda _{4}(\alpha _{2})-\lambda _{5}(\omega ) \\ \frac{d\lambda _{5}}{dt}&=\lambda _{1}(\beta _{3}S^{*})-\lambda _{2}(\beta _{3}S^{*})+\lambda _{5}(\alpha _{3})-\lambda _{6}(\phi ) \\ \frac{d\lambda _{6}}{dt}&=-\lambda _{1}(\psi )+\lambda _{6}(d+\psi ), \end{aligned} \end{aligned}$$
(14)
with transversality conditions \(\lambda _{i}(T_{f})=0\), for \(i=1,2,\ldots ,6\) and \(u_{1}^{*}\) and \(u_{2}^{*}\) are same as in (10).