Positivity and boundedness of the solutions
From Eq. (1), we obtain
$$\begin{aligned}&\frac{\mathrm{d}S}{\mathrm{d}t}\bigg |_{S=0}=\Lambda>0, \ \frac{\mathrm{d}E}{\mathrm{d}t}\bigg |_{E=0}=\beta _1(1-\sigma _mp_m)SI_s+\beta _2(1-\sigma _mp_m)SI_a>0,\\&\frac{\mathrm{d}I_a}{\mathrm{d}t}\bigg |_{I_s=0}\!=\!a\sigma E\!\ge \! 0, \ \frac{\mathrm{d}I_s}{\mathrm{d}t}\bigg |_{I_a=0}=(1-a)\sigma E\!+\!\alpha I_a\!+\!\gamma _a Q_i\ge 0, \ \frac{\mathrm{d}Q_i}{\mathrm{d}t}\bigg |_{Q_i=0}=\alpha _aI_a\ge 0,\\&\frac{\mathrm{d}H}{\mathrm{d}t}\bigg |_{H=0}=T(I_s)\ge 0, \ \frac{\mathrm{d}R}{\mathrm{d}t}\bigg |_{R=0}=\eta _aQ_i+\gamma _hH\ge 0. \end{aligned}$$
All the above rates are nonnegative on the bounding planes. Hence, if we begin in the interior of the nonnegative bounded cone \({R}^{7}_+\), we will end up remaining in this cone. This is in view of the fact that the vector field direction is inward on all the bounding planes. Thus, the solutions of (1) will be nonnegative is guaranteed. Furthermore, from the model (1), we conclude that the total population N satisfies,
$$\begin{aligned} \frac{\mathrm{d}N}{\mathrm{d}t}=\Lambda -\mu N-\mu _sI_s-\mu _hH. \end{aligned}$$
This gives,
$$\begin{aligned} \limsup \limits _{t\rightarrow \infty }N \le \frac{\Lambda }{\mu }. \end{aligned}$$
Therefore, every solution S(t), E(t), \(I_s(t)\), \(I_a(t)\), H(t), \(Q_i(t)\) and R(t) is bounded by \(\frac{\Lambda }{\mu }\). This gives us the biologically feasible region of the system (1) by the below positively invariant set:
$$\begin{aligned} \Omega _1=\{(S, E, I_s, I_a, H, Q_i, R)\in {R}^7_+ : 0 \le S, E, I_s, I_a, H, Q_i, R \le \Lambda /\mu \}. \end{aligned}$$
Existence of Equilibrium Points and the Basic Reproduction Number (\({\mathcal {R}}_0\))
Disease-free equilibrium \(E_0\)
We consider the system (1) and find the disease-free equilibrium. For our model we have disease-free equilibrium as
$$\begin{aligned} E_0={({S}^{0}, {E}^{0}, {I_a}^{0}, {I_s}^{0}, {Q_i}^{0}, {H}^{0}, R^{0})}= \left( \dfrac{\Lambda }{\mu }, 0, 0, 0, 0, 0, 0\right) . \end{aligned}$$
The basic reproduction number \({\mathcal {R}}_0\)
We find the basic reproduction number \({\mathcal {R}}_0\) by following the next generation matrix method [38] we find the vector \({\mathcal {F}}\) and \({\mathcal {V}}\) as follows:
$$\begin{aligned} {\mathcal {F}}= & {} \left( \begin{array}{c} (1-\sigma _mp_m)\beta _1 S I_s+(1-\sigma _mp_m)\beta _2 S I_a\\ 0 \\ 0 \end{array} \right) ,\\ {\mathcal {V}}= & {} \left( \begin{array}{c} (\sigma +\mu )E\\ -a\sigma E+(\alpha +\alpha _a+\mu ) I_a \\ -(1-a)\sigma E-\alpha I_a-\gamma _a Q_i+(\mu _s+\alpha _s+\mu ) I_a \end{array} \right) , \end{aligned}$$
F= Jacobian of \({\mathcal {F}}\) at \(E_0=\left( \begin{array}{ccc} 0 &{} (1-\sigma _mp_m)\beta _1S^0 &{} (1-\sigma _mp_m)\beta _2 S^0\\ 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 \end{array} \right) \) and V = Jacobian of \({\mathcal {V}}\) at \(E_0 = \left( \begin{array}{ccc} (\sigma +\mu ) &{} 0 &{} 0 \\ -a\sigma &{} (\alpha +\alpha _a+\mu ) &{} 0 \\ -(1-a) \sigma &{} -\alpha &{} (\mu _s+\alpha _s+\mu ) \end{array} \right) \)
and it follows that
$$\begin{aligned} {FV^{-1}}=\left( \begin{array}{ccc} a_{11} &{} a_{12} &{} a_{13} \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \end{array} \right) , \end{aligned}$$
where
$$\begin{aligned} a_{11}= & {} \dfrac{(1-\sigma _mp_m)\beta _1 a \sigma S^0}{(\mu +\sigma )(\alpha _a+\alpha +\mu )}+\dfrac{\beta _2 a \sigma S^0 [\alpha +(1-a)(\alpha _a+\mu )]}{(\mu +\sigma )(\alpha _a+\alpha +\mu )(\mu _s+\alpha _s+\mu ) },\\ ~~a_{12}= & {} \dfrac{(1-\sigma _mp_m)\beta _1S^0}{(\alpha _a+\alpha +\mu )}+\dfrac{(1-\sigma _mp_m)\beta _2 \alpha S^0}{(\alpha _a+\alpha +\mu )(\mu _s+\alpha _s+\mu )}, ~~~a_{13}=\dfrac{\beta _2S^0}{(\mu _s+\alpha _s+\mu )}, \end{aligned}$$
The largest eigenvalue of \(FV^{-1}\) is the basic reproduction number \(({\mathcal {R}}_0)\) and clearly it is \(a_{11}\) and is given as below:
$$\begin{aligned} R_0=\dfrac{(1-\sigma _mp_m)\beta _1 a \sigma S^0}{(\mu +\sigma )(\alpha _a+\alpha +\mu )}+\dfrac{(1-\sigma _mp_m)\beta _2 a \sigma S^0 [\alpha +(1-a)(\alpha _a+\mu )]}{(\mu +\sigma )(\alpha _a+\alpha +\mu )(\mu _s+\alpha _s+\mu ) }, \end{aligned}$$
The quantity \({\mathcal {R}}_0\) is in fact the average number of secondary cases produced in completely susceptible population, by an infected individual in his/her whole infectious period.
Existence of Endemic Equilibrium Point
The following algebraic equations are satisfied by an endemic equilibrium of model (1):
$$\begin{aligned} \dfrac{\mathrm{d}S}{\mathrm{d}t}=0,~~\dfrac{\mathrm{d}E}{\mathrm{d}t}=0,~~\dfrac{\mathrm{d}I_a}{\mathrm{d}t}=0,~~\dfrac{\mathrm{d}I_s}{\mathrm{d}t}=0,~~\dfrac{\mathrm{d}Q_i}{\mathrm{d}t}=0,~~\dfrac{\mathrm{d}H}{\mathrm{d}t}=0~~\dfrac{\mathrm{d}R}{\mathrm{d}t}=0. \end{aligned}$$
(3)
When \(0 <I_s \le I_0,\) then system (1) becomes;
$$\begin{aligned} \begin{aligned} \dfrac{\mathrm{d}{S}}{\mathrm{d}t}&=\Lambda -(1-\sigma _mp_m)\beta _1 S I_a-(1-\sigma _mp_m)\beta _2S I_s-\mu S \\ \dfrac{\mathrm{d}{E}}{\mathrm{d}t}&= (1-\sigma _mp_m)\beta _1 S I_a+(1-\sigma _mp_m)\beta _2S I_s-\sigma E-\mu E \\ \dfrac{\mathrm{d}{I_a}}{\mathrm{d}t}&=a\sigma E-\alpha I_a-\alpha _a I_a-\mu I_a.\\ \dfrac{\mathrm{d}{I_{s}}}{\mathrm{d}t}&=(1-a)\sigma E+\alpha I_a+\gamma _a Q_i-(\mu _s+\mu ) I_s- \alpha _s I_s \\ \dfrac{\mathrm{d}{Q_i}}{\mathrm{d}t}&=\alpha _a I_a-\gamma _a Q_i-\eta _a Q_i-\mu Q_i. \\ \dfrac{\mathrm{d}{H}}{\mathrm{d}t}&=\alpha _s I_s -(\gamma _h+\mu _h+\mu )H. \\ \dfrac{\mathrm{d}{R}}{\mathrm{d}t}&=\eta _a Q_i+\gamma _h H-\mu R. \end{aligned} \end{aligned}$$
(4)
When \(I_s > I_0,\) then system (1) becomes;
$$\begin{aligned} \begin{aligned} \dfrac{\mathrm{d}{S}}{\mathrm{d}t}&=\Lambda -(1-\sigma _mp_m)\beta _1 S I_a-(1-\sigma _mp_m)\beta _2S I_s-\mu S \\ \dfrac{\mathrm{d}{E}}{\mathrm{d}t}&= (1-\sigma _mp_m)\beta _1 S I_a+(1-\sigma _mp_m)\beta _2S I_s-\sigma E-\mu E\\ \dfrac{\mathrm{d}{I_a}}{\mathrm{d}t}&=a\sigma E-\alpha I_a-\alpha _a I_a-\mu I_a.\\ \dfrac{\mathrm{d}{I_{s}}}{\mathrm{d}t}&=(1-a)\sigma E+\alpha I_a+\gamma _a Q_i-(\mu _s+\mu ) I_s- k \\ \dfrac{\mathrm{d}{Q_i}}{\mathrm{d}t}&=\alpha _a I_a-\gamma _a Q_i-\eta _a Q_i-\mu Q_i. \\ \dfrac{\mathrm{d}{H}}{\mathrm{d}t}&=k -(\gamma _h+\mu _h+\mu )H. \\ \dfrac{\mathrm{d}{R}}{\mathrm{d}t}&=\eta _a Q_i+\gamma _h H-\mu R. \end{aligned} \end{aligned}$$
(5)
The system (4) admits a unique positive solution \(E_1= {(S^*, E^*, I_a^*, I_s^*, Q_i^*, H^*,R^*)}\) where
$$\begin{aligned} R^*= & {} A_1 H^*+A_6 Q_i^*,~~ H^*= A_2 I_s^*,~~ Q_i^*=A_3 I_a^*,~~ I_a^*=A_4 E^*,~~ I_s^*=A_5 E^*,\\ S^*= & {} \dfrac{\Lambda }{(1-\sigma _mp_m)\beta _1 I_a^*+(1-\sigma _mp_m)\beta _2 I_s^*+\mu },~E^*\!=\!\dfrac{({\mathcal {R}}_0-1)}{(1-\sigma _mp_m)\beta _1 A_4\!+\!(1-\sigma _mp_m)\beta _2 A_5}, \end{aligned}$$
and
$$\begin{aligned} A_1= & {} \dfrac{\gamma _h}{\mu },~~A_2=\dfrac{\alpha _s}{\gamma _h+\mu _h+\mu },~~A_3=\dfrac{\alpha _a}{\gamma _a+\eta _a+\mu },~~A_4=\dfrac{a \sigma }{\alpha +\alpha _a+\mu },\\ A_5= & {} \dfrac{(1-a)\sigma +\alpha A_4+\gamma _a A_3 A_4}{\mu +\mu _s+\alpha _s}, A_6=\dfrac{\eta _a}{\mu }. \end{aligned}$$
So from above relations, we can see that all the variables are in terms \(E^*\) and \(E^*\) is positive constant when \({\mathcal {R}}_0 >1\). Hence, the endemic equilibrium \(E^*_1\) exists when \({\mathcal {R}}_0 >1\) for the system (4).
Note that from the set of Eq. (5), we obtain the following equilibrium points:
$$\begin{aligned} R^*=A_1 H^*+A_7Q_i^*,~~ H^*= A_2,~~ Q_i^*=A_3 I_a^*,~~ I_a^*=A_4 E^*,~~ I_s^*=A_5 E^*-A_6 \end{aligned}$$
where
$$\begin{aligned} A_1= & {} \dfrac{\gamma _h}{\mu },~~A_2=\dfrac{k}{\gamma _h+\mu _h+\mu },~~A_3=\dfrac{\alpha _a}{\gamma _a+\eta _a+\mu },~~A_4=\dfrac{a \sigma }{\alpha +\alpha _a+\mu },\\ A_5= & {} \dfrac{(1-a)\sigma +\alpha A_4+\gamma _a A_3 A_4}{\mu +\mu _s},~~ A_6=\dfrac{k}{\mu +\mu _s}, A_7=\dfrac{\eta _a}{\mu }. \end{aligned}$$
and
$$\begin{aligned} S^*=\dfrac{\Lambda }{(1-\sigma _mp_m)\beta _1 I_a^*+(1-\sigma _mp_m)\beta _2 I_s^*+\mu }, \end{aligned}$$
We see that \(E^*\) is the positive root of the following quadratic equation
$$\begin{aligned} C_1 {E^*}^2-C_2 E^*+C_3=0 \end{aligned}$$
where
$$\begin{aligned} C_1= & {} (\sigma +\mu )(1-\sigma _mp_m)(\beta _1 A_4+\beta _2 A_5) \\ C_2= & {} [(\sigma +\mu )((1-\sigma _mp_m)(\beta _2A_6-\mu )+\Lambda (1-\sigma _mp_m)(\beta _1 A_4+\beta _2 A_5)] \\ C_3= & {} (1-\sigma _mp_m) \Lambda \beta _2 A_6. \end{aligned}$$
Here, it can be noted that \(C_3>0\) and \(C_1>0, C_2>0\) provided \(\beta _2A_6>\mu \). The roots of the last quadratic equation are \(\dfrac{C_2+\sqrt{C_2^2-4C_1C_3}}{2C_1}\) and \(\dfrac{C_2-\sqrt{C_2^2-4C_1C_3}}{2C_1}.\)
So assuming \(\sqrt{C_2^2-4C_1C_3}>0\), then we have the following cases:
Case 1: One positive root:
So from the quadratic equation, we will get one positive if \(C_2<\sqrt{C_2^2-4C_1C_3}\). Hence, unique positive equilibrium exists for the system (5) where \(I_s \ge I_0.\)
Case 2: Two positive roots:
We get two positive roots if \(C_2>\sqrt{C_2^2-4C_1C_3}\). Here, two positive equilibria exist for the system (5) where \(I_s \ge I_0.\) The existence of two values of \(E^*\) suggests that there is a probability of backward bifurcation and this is exhibited in Fig. 3a. The diagram on bifurcation is obtained by taking into consideration \(\beta _2\), which is the transmission coefficient as the critical parameter and plotting it against the corresponding values of \({\mathcal {R}}_0.\) The obtained plot suffices that mere reduction of the reproduction number \({\mathcal {R}}_0\) below one is not enough for elimination of the disease, since the existence of backward bifurcation demands reduction in value of \({\mathcal {R}}_0\) quite lesser than one to obtain stability of the unique infection-free equilibrium. In this case, we have two endemic equilibrium is named as \(E_2\) and \(E_3\). In Fig. 3b, the diagram on bifurcation is obtained by considering the threshold value of the infective population \(I_0\) as the critical parameter. In this case, it is observed that this parameter has involvement in the treatment. This is due to the assumption that treatment is proportional to the number of infective until the infective population reaches a threshold value \(I_0\) , after which the treatment function becomes a constant. As a result, in this figure it is noted that the equilibrium level of the infective population decreases with the rise in \(I_0\) until it comes to a saturation point. The situation when the reproduction number \({\mathcal {R}}_0<1\) and either the infection free equilibrium \(E_0\) is stable or the equilibrium \(E_3\) is stable is described in Fig. 3b. In this situation when we increase \(I_0\), the equilibrium level of the exposed population (corresponding to the equilibrium \(E_2\)) decreases up until we arrive at \(I_0=43.457\) where increasing it further has no effect and in this situation only the infection free equilibrium \(E_0\) is stable.
Theorem 3.1
When \(R_0 < 1\), the disease-free equilibrium \(E_0= {(S^0, 0, 0, 0, 0, 0, 0)}\) is locally asymptotically stable under some restriction on the parameters otherwise it is unstable.
Proof
See Appendix. \(\square \)
Theorem 3.2
When \(R_0 > 1\), the endemic Equilibrium \(E_1= {(S^*, E^*, I_a^*, I_s^*, Q_i^*, H^*, R^*)}\) is locally asymptotically stable under some restriction on the parameters otherwise it is unstable.
Proof
See Appendix. \(\square \)