Let the assumed state vector be represented as $x=\left(x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9\right)=\left(S,L,E,I,R,V,E_V,I_V,R_V\right)$. Then Equations (1.1)-(1.9) are possible to signify in the form of some initial set of values

$x^{\prime }=f\left(x,t\right),x\left(0\right)=x_0$ (2.3)

It is evident that the above-defined function f is locally Lipschitz for the first argument and continuous for the second in ${ℝ}^9\times ℝ$, which implies that the solution $x\left(t\right)$ holds in $t\in \left(0,T\right)$ for some T, according to the system of nonlinear ODEs existence and uniqueness theorem [38] [41] . For the invariance of sets under a flow, we used the Bony-Brezis theorem [42] . The theorem applies to smooth manifolds. There are some corner points at the boundary of the invariant set. Shortly, by making an argument that if we have an initial condition at the corner points, it cannot go outside; for instance, the direction of the flow is towards the interior at those points. Thus, T might be re-defined as the supremum

overall mentioned periods. Finally, for $T<\infty$, ${\lim }_{t\to T^{-}}\Vert x\left(t\right)\Vert \to \infty$.

We have to show that under the flow $x\left(t\right)$ for $t\in \left(0,T\right)$, if

$x_0\in \Lambda =\left\{x\in {ℝ}^9:x_i\ge 0\text{for}i=1,2,3,\cdots ,9,{\displaystyle \underset{i=1}{\overset{9}{\sum }}{x}_i}\le N\left(0\right)\right\},$

the set Λ is positively invariant, which concludes that $T=\infty$, i.e., for initial conditions $x_0\in \text{Λ}$, the solution $x\left(t\right)\in \text{Λ}$ holds globally with time [38] [40] [41] .

Theorem 1: The closed set

$\Lambda :=\left\{x=\left(S,L,E,I,R,V,E_V,I_V,R_V\right)\in {ℝ}^9:x_i\ge 0\text{for}i=1,2,3,\cdots ,9,{\displaystyle \underset{i=1}{\overset{9}{\sum }}{x}_i}\le N\left(0\right)\right\},$

is entirely uniform under the flow generated by the Equations (1.1)-(1.9). As a result, the solution $x\left(t\right)\in \text{Λ}$ occurs globally in the time given initial conditions $x_0\in \text{Λ}$.

Proof: Let the boundary segment is ${\Pi }_i,i=1,2,\cdots ,10$

${\Pi }_i=\left\{x\in \Lambda :x_i=0\right\},i=1,2,\cdots ,9$

${\Pi }_{10}=\left\{x\in \Lambda :{\displaystyle \underset{i=1}{\overset{10}{\sum }}{x}_i}=0\right\}.$

It is evident that $\partial \Lambda ={\displaystyle {\cup }_{i=1}^{10}{\Pi }_i}$.

To complete the proof of the above invariance of the set Λ, it is enough to prove $\forall$ inward normal, $n\cdot x^{\prime }\left(t\right)\ge 0$ on $\partial \Lambda$. The inward normal on ${\Pi }_i$ for $i=1,2,\cdots ,9$ is unambiguously provided by $n_i=v_i=\left[0,\cdots ,1,\cdots ,0\right]$, where only the i-component is nonzero, but an inward normal on ${\Pi }_{10}$ is given by $n_{10}=\left(-1,-1,\cdots ,-1\right)$.

Now, on ${\Pi }_i$ for $i=1,2,\cdots ,9$

$v_1\cdot x^{\prime }=l_d{x}_2\ge 0,\forall x\in {\Pi }_1,$

$v_2\cdot x^{\prime }=l{x}_1\ge 0,\forall x\in {\Pi }_2,$

$v_3\cdot x^{\prime }=\beta x_1\left(x_4+x_8\right)+\left(1-q\right)\beta x_2\left(x_4+x_8\right)\ge 0,\forall x\in {\Pi }_3,$

$v_4\cdot x^{\prime }=\alpha x_3\ge 0,\forall x\in {\Pi }_4,$

$v_5\cdot x^{\prime }=\gamma x_4\ge 0,\forall x\in {\Pi }_5,$

$v_6\cdot x^{\prime }=\delta x_1\ge 0,\forall x\in {\Pi }_6,$

$v_7\cdot x^{\prime }=\left(1-\eta \right)\beta x_6\left(x_4+x_8\right)\ge 0,\forall x\in {\Pi }_7,$

$v_8\cdot x^{\prime }=\alpha x_7\ge 0,\forall x\in {\Pi }_8,$

$v_9\cdot x^{\prime }=\gamma x_8\ge 0,\forall x\in {\Pi }_9,$

while $N={\displaystyle {\sum }_{i=1}^9{x}_i}$ is readily seen to satisfy $\frac{\text{d}N}{\text{d}t}=0$, which follows that on ${\Pi }_{10}$ such that

$n_{10}\cdot x^{\prime }=0.$

Thus, for given initial conditions $x_0\in \Lambda$, solution $x\left(t\right)\in \Lambda$ holds globally over time on the positively invariant domain Λ.

Before commencing the formal study, we will provide a comprehensive explanation of the methodology used to calculate the basic reproduction number (R₀), which is crucial in epidemiological modeling due to its significant contribution to understanding stability conditions. Research has shown that if this numerical value is less than one, it indicates stable circumstances, whereas values more than one indicate unstable situations. However, it has been stressed that the value may be determined using other methods, with the next-generation matrix techniques [43] being widely recognized. Thus, the detailed process

$F{V}^{-1}\left(\equiv F_1{V}_1^{-1},F_2{V}_2^{-1}\right)$ of finding the value of the basic reproduction number (R₀) is

$F_1=\left[\begin{array}{cc}0& \beta +\left(1-q\right)\beta \\ 0& 0\end{array}\right],V_1=\left[\begin{array}{cc}\alpha & 0\\ -\alpha & \gamma \end{array}\right].$

$\therefore F_1{V}_1^{-1}=\frac{1}{\alpha \gamma }\left[\begin{array}{cc}\alpha \beta & \alpha \left\{\beta +\left(1-q\right)\beta \right\}\\ 0& 0\end{array}\right].$

The eigenvalues of $F_1{V}_1^{-1}$ are ${\lambda }_1=\frac{\beta }{\gamma }$ and ${\lambda }_2=\frac{\beta +\left(1-q\right)\beta }{\gamma }$.

Correspondingly, for the vaccination part,

$F_2=\left[\begin{array}{cc}0& \beta \left(1-\eta \right)\\ 0& 0\end{array}\right],V_2=\left[\begin{array}{cc}\alpha & 0\\ -\alpha & \gamma \end{array}\right].$

$\therefore F_2{V}_2^{-1}=\frac{1}{\alpha \gamma }\left[\begin{array}{cc}\alpha \beta & \alpha \left(1-\eta \right)\beta \\ 0& 0\end{array}\right].$

The eigenvalues of $F_2{V}_2^{-1}$ are ${\lambda }_3=\frac{\beta }{\gamma }$ and ${\lambda }_4=\frac{\left(1-\eta \right)\beta }{\gamma }$.

Thus, the basic reproduction number (R₀) is

$R_0=\frac{\beta }{\gamma }+\frac{\left(1-q\right)\beta }{\gamma }+\frac{\left(1-\eta \right)\beta }{\gamma }.$ (2.4)

For this model, the disease-free equilibrium point threshold is the solution of the nonlinear system of Equations (1.1)-(1.9). Thus,

$\frac{\text{d}S}{\text{d}t}=\frac{\text{d}L}{\text{d}t}=\frac{\text{d}E}{\text{d}t}=\frac{\text{d}I}{\text{d}t}=\frac{\text{d}R}{\text{d}t}=\frac{\text{d}V}{\text{d}t}=\frac{E_V}{\text{d}t}=\frac{I_V}{\text{d}t}=\frac{R_V}{\text{d}t}=0,$

gives $E_0=\left(N\left(=1\right),0,0,0,0,0,0,0,0\right)$, the disease-free equilibrium point threshold of the current model.

The proposed model endemic equilibrium point $E_{*}=\left(S^{*},L^{*},E^{*},I^{*},R^{*},V^{*},E_{v^{*}},I_{v^{*}},R_{v^{*}}\right)$ is a solution of the following system:

$0=-\beta S\left(t\right)\left(I\left(t\right)+I_V\left(t\right)\right)-\delta S\left(t\right)-lS\left(t\right)+l_{d}L\left(t\right),$

$0=lS\left(t\right)-\left(1-q\right)\beta L\left(t\right)\left(I\left(t\right)+I_V\left(t\right)\right)-l_{d}L\left(t\right),$

$0=\beta S\left(t\right)\left(I\left(t\right)+I_V\left(t\right)\right)+\left(1-q\right)\beta L\left(t\right)\left(I\left(t\right)+I_V\left(t\right)\right)-\alpha E\left(t\right),$

$0=\alpha E\left(t\right)-\gamma I\left(t\right),$

$0=\gamma I\left(t\right),$

$0=\delta S\left(t\right)-\left(1-\eta \right)\beta V\left(t\right)\left(I\left(t\right)+I_V\left(t\right)\right),$

$0=\left(1-\eta \right)\beta V\left(t\right)\left(I\left(t\right)+I_V\left(t\right)\right)-\alpha E_V\left(t\right),$

$0=\alpha E_V\left(t\right)-\gamma I_V\left(t\right),$

$0=\gamma I_V\left(t\right).$

Solving the above system of equations setting $I\ne 0,I_V\ne 0$, the endemic equilibrium point is

$E_{*}=\left(S^{*},L^{*},E^{*},I^{*},R^{*},V^{*},E_{v^{*}},I_{v^{*}},R_{v^{*}}\right)$,

where

$\begin{array}{l}{S}^{*}=\frac{\gamma I^{*}}{\beta I^{*}+\delta +l},L^{*}=\frac{\gamma }{\left(1-q\right)\beta }-\frac{\gamma I^{*}}{\left(1-q\right)\left(\beta I^{*}+\delta +l\right)},E^{*}=\frac{l_d{L}^{*}}{\alpha },I^{*}=\frac{l_d{L}^{*}}{\gamma },\\ R^{*}=l_d{L}^{*},V^{*}=\frac{\gamma }{\left(1-\eta \right)\beta },E_{v^{*}}=\frac{\delta S^{*}}{\alpha },I_{v^{*}}=\frac{\delta S^{*}}{\gamma },R_{v^{*}}=\delta S^{*}.\end{array}$

In this section, we will show that for $R_0<1$ the disease-free $(E_0)$ equilibrium point isasymptoticallystable locally. The disease will be eradicated and continue biologically in society when the primary reproduction number is less than and more significant than unity.

Theorem 2: If $R_0<1$, then the unique disease-free equilibrium E₀ is locally asymptotically stable. If $R_0>1$, the unique disease-free equilibrium is unstable.

Proof: To validate the local stability, the Jacobian matrix of the proposed system (1.1)-(1.9) is

$J=\left[\begin{array}{ccccccccc}-\beta \left(I+I_V\right)-\delta -l& l_d& 0& -\beta S& 0& 0& 0& -\beta S& 0\\ l& -\beta \left(1-q\right)\left(I+I_V\right)-l_d& 0& -\beta \left(1-q\right)L& 0& 0& 0& -\beta \left(1-q\right)L& 0\\ \beta \left(I+I_V\right)& \beta \left(1-q\right)\left(I+I_V\right)& -\alpha & \beta S+\beta \left(1-q\right)L& 0& 0& 0& \beta S+\beta \left(1-q\right)L& 0\\ 0& 0& \alpha & -\gamma & 0& 0& 0& 0& 0\\ 0& 0& 0& \gamma & 0& 0& 0& 0& 0\\ \delta & 0& 0& -\beta \left(1-\eta \right)V& 0& -\beta \left(1-\eta \right)\left(I+I_V\right)& 0& -\beta \left(1-\eta \right)V& 0\\ 0& 0& 0& \beta \left(1-\eta \right)V& 0& \beta \left(1-\eta \right)\left(I+I_V\right)& -\alpha & \beta \left(1-\eta \right)V& 0\\ 0& 0& 0& 0& 0& 0& \alpha & -\gamma & 0\\ 0& 0& 0& 0& 0& 0& 0& \gamma & 0\end{array}\right].$

At the DFE point $E_0$, we have

$J\left(E_0\right)=\left[\begin{array}{ccccccccc}-\delta -l& l_d& 0& -\beta & 0& 0& 0& -\beta & 0\\ l& -l_d& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& -\alpha & \beta & 0& 0& 0& \beta & 0\\ 0& 0& \alpha & -\gamma & 0& 0& 0& 0& 0\\ 0& 0& 0& \gamma & 0& 0& 0& 0& 0\\ \delta & 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& -\alpha & 0& 0\\ 0& 0& 0& 0& 0& 0& \alpha & -\gamma & 0\\ 0& 0& 0& 0& 0& 0& 0& \gamma & 0\end{array}\right]$

The characteristic equation $\left|J\left(E_0\right)-\lambda I\right|=0$, has nine roots, which are

${\lambda }_1={\lambda }_2={\lambda }_3=0,{\lambda }_4=-\alpha ,{\lambda }_5=-\gamma ,$

${\lambda }_6=\frac{1}{2}\left(-\alpha -\gamma -\sqrt{{\alpha }^2+4\alpha \beta -2\alpha \gamma +{\gamma }^2}\right),$

${\lambda }_7=\frac{1}{2}\left(-\alpha -\gamma +\sqrt{{\alpha }^2+4\alpha \beta -2\alpha \gamma +{\gamma }^2}\right),$

${\lambda }_8=\frac{1}{2}\left(-l_d-l-\delta -\sqrt{{\left(l_d+l+\delta \right)}^2+4l_d\delta }\right)\text{and}$

${\lambda }_9=\frac{1}{2}\left(-l_d-l-\delta +\sqrt{{\left(l_d+l+\delta \right)}^2+4l_d\delta }\right).$

i.e., all nine eigenvalues are less than or equal to zero. Therefore, according to the Routh-Hurwitz criterion, the model is locally asymptotically stable at the unique DFE point whenever $R_0<1$ and unstable whenever $R_0>1$.

In general, a population in the real world can infrequently be entirely susceptible to infection. Some contacts will be immune because of previous infections, which have provided long-term immunity or are a consequence of prior vaccinations. Consequently, all communications will not infect, and the mean per secondary case of disease is less than the number of basic reproduction numbers. Thus, the effective reproductive number (R_e) is the mean per infectious case in a population of susceptible and non-susceptible people. When $R_e>1$, the number of cases will increase and start an epidemic. When $R_e=1$, the disease is endemic, and when $R_e<1$, there will be a decline in the number of cases. The effective reproduction number is sensitive to the multiple of the basic reproduction number and the fraction of the host population. Therefore,

$R_e=\frac{\beta }{\gamma }S\left(t\right)+\frac{\left(1-q\right)\beta }{\gamma }L\left(t\right)+\frac{\left(1-\eta \right)\beta }{\gamma }V\left(t\right).$ (2.5)

In the recommended model, $\frac{\beta S\left(t\right)\left(I\left(t\right)+I_V\left(t\right)\right)}{N}$ is used to compute the sum of secondary infections of susceptible and vaccinated infected people per unit of time. On top of that, it is equal to $\frac{\beta S\left(t\right)}{N}$ for the single infected individual. Furthermore, $\frac{1}{\gamma }$ and $\frac{\beta }{\gamma }$ indicates the lifespan of an infectious individual and the total secondary infections of susceptible individuals that may produce one infected individual in a disease-free society, respectively, whereas, for vaccinated individuals, both of the values are $\frac{1}{\gamma }$ and $\frac{\left(1-\eta \right)\beta }{\gamma }$. Analogously, secondary infections of vaccinated individuals equal $\frac{\beta S\left(t\right)I_V\left(t\right)}{N}$ and per unit of time, the sum is $\frac{\beta S\left(t\right)}{N}$. Realistically, due to the mass vaccination rate $\frac{1}{\delta }$, the basic reproduction number must be a decreasing function. Opposite scenarios for less vaccine efficacy rate $\left(1-\eta \right)$.

Besides the reproduction number commonly employed in epidemiology to determine whether or not a disease will spread, we introduce the adequate strength number to measure wave tendency in the epidemiology model. We use the well-established next-generation matrix technique to calculate the adequate strength number by assuming the total population is constant (N). Then,

$\beta SI\equiv \frac{\beta SI}{N},E\equiv \frac{E}{N}.$

Now,

$\begin{array}{l}\frac{\partial }{\partial I}\left\{\beta S\frac{I}{N}+\left(1-q\right)\beta L\frac{I}{N}-\alpha \frac{E}{N}\right\}\\ =\left\{\beta S+\left(1-q\right)\beta L\right\}\left(\frac{N-\stackrel{\dot{}}{N}I}{N^2}\right)-\alpha \left(\frac{N\stackrel{\dot{}}{E}-\stackrel{\dot{}}{N}E}{N^2}\right)\\ =\left\{\beta S+\left(1-q\right)\beta L\right\}\left(\frac{N-\stackrel{\dot{}}{N}I}{N^2}\right)-\alpha \frac{\left\{\beta S+\left(1-q\right)\beta L\right\}I-\alpha E-\stackrel{\dot{}}{N}E}{N^2}\\ =\left\{\beta S+\left(1-q\right)\beta L\right\}\left(\frac{N-\stackrel{\dot{}}{N}I}{N^2}\right)-\alpha \left\{\beta S+\left(1-q\right)\beta L\right\}\frac{I}{N^2}+\frac{{\alpha }^{2}E}{N^2}+\alpha \frac{\stackrel{\dot{}}{N}E}{N^2}.\end{array}$ (2.6)

Similarly,

$\begin{array}{l}\frac{\partial }{\partial I_V}\left\{\beta S\frac{I_V}{N}+\left(1-q\right)\beta L\frac{I_V}{N}-\alpha \frac{E}{N}\right\}\\ =\left\{\beta S+\left(1-q\right)\beta L\right\}\left(\frac{N-\stackrel{\dot{}}{N}{I}_V}{N^2}\right)-\alpha \left\{\beta S+\left(1-q\right)\beta L\right\}\frac{I_V}{N^2}+\frac{{\alpha }^{2}E}{N^2}+\alpha \frac{\stackrel{\dot{}}{N}E}{N^2}.\end{array}$ (2.7)

$\begin{array}{l}\frac{\partial }{\partial I}\left\{\left(1-\eta \right)\beta V\frac{I}{N}-\alpha \frac{E_V}{N}\right\}\\ =\left(1-\eta \right)\beta V\left(\frac{N-\stackrel{\dot{}}{N}I}{N^2}\right)-\left(1-\eta \right)\alpha \beta V\frac{I}{N^2}+\frac{{\alpha }^2{E}_V}{N^2}+\alpha \frac{\stackrel{\dot{}}{N}{E}_V}{N^2}.\end{array}$ (2.8)

$\begin{array}{l}\frac{\partial }{\partial I_V}\left\{\left(1-\eta \right)\beta V\frac{I_V}{N}-\alpha \frac{E_V}{N}\right\}\\ =\left(1-\eta \right)\beta V\left(\frac{N-\stackrel{\dot{}}{N}{I}_V}{N^2}\right)-\left(1-\eta \right)\alpha \beta V\frac{I_V}{N^2}+\frac{{\alpha }^2{E}_V}{N^2}+\alpha \frac{\stackrel{\dot{}}{N}{E}_V}{N^2}.\end{array}$ (2.9)

In this case,

$F_{SN\left(1\right)}=\left[\begin{array}{cc}0& -\frac{\beta +\left(1-q\right)\beta }{N^2}-\frac{\alpha \beta +\left(1-q\right)\alpha \beta }{N^2}\\ 0& 0\end{array}\right],$ (2.10)

$F_{SN\left(2\right)}=\left[\begin{array}{cc}0& -\frac{\left(1-\eta \right)\beta }{N^2}-\frac{\left(1-\eta \right)\alpha \beta }{N^2}\\ 0& 0\end{array}\right],$ (2.11)

and

$V_{SN\left(1\right)}=\left[\begin{array}{cc}-\frac{{\alpha }^2}{N^2}& 0\\ -\alpha & \gamma \end{array}\right],$ (2.12)

$V_{SN\left(2\right)}=\left[\begin{array}{cc}-\frac{{\alpha }^2}{N^2}& 0\\ -\alpha & \gamma \end{array}\right].$ (2.13)

$\therefore F_{SN\left(1\right)}{V}_{SN\left(1\right)}^{-1}=-\frac{N^2}{\gamma {\alpha }^2}\left[\begin{array}{cc}-a_{11}& a_{12}\\ 0& 0\end{array}\right],$ (2.14)

where

$a_{11}=\frac{\alpha \beta +\left(1-q\right)\alpha \beta }{N^2}+\frac{{\alpha }^2\beta +\left(1-q\right){\alpha }^2\beta }{N^2},$

$a_{12}=\left\{\frac{\beta +\left(1-q\right)\beta }{N^2}+\frac{\alpha \beta +\left(1-q\right)\alpha \beta }{N^2}\right\}\frac{{\alpha }^2}{N^2}.$

Analogously,

$F_{SN\left(2\right)}{V}_{SN\left(2\right)}^{-1}=-\frac{N^2}{\gamma {\alpha }^2}\left[\begin{array}{cc}-\frac{\left(1-\eta \right)\alpha \beta }{N^2}-\frac{\left(1-\eta \right){\alpha }^2\beta }{N^2}& \left\{\frac{\left(1-\eta \right)\beta }{N^2}+\frac{\left(1-\eta \right)\alpha \beta }{N^2}\right\}\frac{{\alpha }^2}{N^2}\\ 0& 0\end{array}\right].$ (2.15)

Thus, the four Eigenvalues are

${\lambda }_{SN\left(1\right)}=\frac{\beta +\left(1-q\right)\beta }{\alpha \gamma }+\frac{\beta +\left(1-q\right)\beta }{\gamma },{\lambda }_{SN\left(2\right)}=-\frac{\beta +\left(1-q\right)\beta }{\gamma N^2}-\frac{\alpha \beta +\left(1-q\right)\alpha \beta }{\gamma N^2},$

${\lambda }_{SN\left(3\right)}=\frac{\left(1-\eta \right)\beta }{\alpha \gamma }+\frac{\left(1-\eta \right)\beta }{\gamma },{\lambda }_{SN\left(4\right)}=-\frac{\left(1-\eta \right)\beta }{\gamma N^2}-\frac{\left(1-\eta \right)\alpha \beta }{\gamma N^2}.$

At that juncture

$SN=\frac{\beta +\left(1-q\right)\beta }{\alpha \gamma }+\frac{\beta +\left(1-q\right)\beta }{\gamma }+\frac{\left(1-\eta \right)\beta }{\alpha \gamma }+\frac{\left(1-\eta \right)\beta }{\gamma }.$

Or,

$SN=\frac{\beta }{\alpha \gamma }\left(3-q-\eta \right)\left(1+\alpha \right).$ (2.16)

As stated above, the reproduction number (effective) must reflect the stability of the disease-free equilibrium state but failed to present wave tendency. According to equation (2.16), the strength number (SN) acts as a positive number (SN>0) for the parameters q, η, and α, all ranging from 0 to 1. Thus, SN > 0 determines that the spread has enough power to begin the revival stage, meaning more than one wave will occur.

Let us assume models independent variables for the endemic LF, $\left\{S,L,E,I,R,V,E_V,I_V,R_V\right\}$, $L_f<0$ is the detrimental equilibrium point $E_{*}$.

Theorem 3: For the value of the basic reproductive number $R_0-1>0$, the proposed $SLEIRV{E}_V{I}_V{R}_V$ model endemic equilibrium point $E_{*}$ is globally asymptotically stable.

Proof: The LF may be represented as follows to prove the theorem above:

$\begin{array}{l}{L}_f\left(S,L,E,I,R,V,E_V,I_V,R_V\right)\\ =\left(S-S^{*}-S^{*}\log \frac{S^{*}}{S}\right)+\left(L-L^{*}-L^{*}\log \frac{L^{*}}{L}\right)+\left(E-E^{*}-E^{*}\log \frac{E^{*}}{E}\right)\\ +\left(I-I^{*}-I^{*}\log \frac{I^{*}}{I}\right)+\left(R-R^{*}-R^{*}\log \frac{R^{*}}{R}\right)+\left(V-V^{*}-V\log \frac{V^{*}}{V}\right)\\ +\left(E_V-E_V^{*}-E_V^{*}\log \frac{E_V^{*}}{E_V}\right)+\left(I_V-I_V^{*}-I_V^{*}\log \frac{I_V^{*}}{I_V}\right)+\left(R_V-R_V^{*}-R_V^{*}\log \frac{R_V^{*}}{R_V}\right).\end{array}$(2.17)

Differentiating both sides in terms of t, we get,

$\begin{array}{l}\frac{\text{d}{L}_f}{\text{d}t}={\stackrel{\dot{}}{L}}_f=\left(\frac{S-S^{*}}{S}\right)\stackrel{\dot{}}{S}+\left(\frac{L-L^{*}}{L}\right)\stackrel{\dot{}}{L}+\left(\frac{E-E^{*}}{E}\right)\stackrel{\dot{}}{E}+\left(\frac{I-I^{*}}{I}\right)\stackrel{\dot{}}{I}+\left(\frac{R-R^{*}}{R}\right)\stackrel{\dot{}}{R}\\ +\left(\frac{V-V^{*}}{V}\right)\stackrel{\dot{}}{V}+\left(\frac{E_V-E_V^{*}}{E_V}\right){\stackrel{\dot{}}{E}}_V+\left(\frac{I_V-I_V^{*}}{I_V}\right){\stackrel{\dot{}}{I}}_V+\left(\frac{R_V-R_V^{*}}{R_V}\right){\stackrel{\dot{}}{R}}_V.\end{array}$(2.18)

From Equations (1.1)-(1.9), putting the value of $\stackrel{\dot{}}{S},\stackrel{\dot{}}{L},\stackrel{\dot{}}{E},\stackrel{\dot{}}{I},\stackrel{\dot{}}{V},{\stackrel{\dot{}}{E}}_V,{\stackrel{\dot{}}{I}}_V,{\stackrel{\dot{}}{R}}_V$ in equation (2.18), we have,

$\begin{array}{c}\frac{\text{d}{L}_f}{\text{d}t}=\left(\frac{S-S^{*}}{S}\right)\left\{-\beta S\left(I+I_V\right)-\delta S-lS+l_{d}L\right\}\\ +\left(\frac{L-L^{*}}{L}\right)\left\{lS-\left(1-q\right)\beta L\left(I+I_V\right)-l_{d}L\right\}\\ +\left(\frac{E-E^{*}}{E}\right)\left\{\beta S\left(I+I_V\right)+\left(1-q\right)\beta L\left(I+I_V\right)-\alpha E\right\}\end{array}$

$\begin{array}{c}+\left(\frac{I-I^{*}}{I}\right)\left(\alpha E-\gamma I\right)+\left(\frac{R-R^{*}}{R}\right)\gamma I\\ +\left(\frac{V-V^{*}}{V}\right)\left\{\delta S-\left(1-\eta \right)\beta V\left(I+I_V\right)\right\}\\ +\left(\frac{E_V-E_V^{*}}{E_V}\right)\left\{\left(1-\eta \right)\beta V\left(I+I_V\right)-\alpha E_V\right\}\\ +\left(\frac{I_V-I_V^{*}}{I_V}\right)\left(\alpha E_V-\gamma I_V\right)+\left(\frac{R_V-R_V^{*}}{R_V}\right)\gamma I_V.\end{array}$ (2.19)

In Equation (2.19), substitute $S-S^{*}$, $L-L^{*}$, $E-E^{*}$, $I-I^{*}$, $V-V^{*}$, $E_V-E_V^{*}$, $I_V-I_V^{*}$, $R_V-R_V^{*}$ instead of $S,L,E,I,R,V,E_V,I_V,R_V$. Then,

$\begin{array}{c}\frac{\text{d}{L}_f}{\text{d}t}=\left(\frac{S-S^{*}}{S}\right)[-\beta \left(S-S^{*}\right)\left\{\left(I-I^{*}\right)+\left(I_V-I_V^{*}\right)\right\}-\delta \left(S-S^{*}\right)-l\left(S-S^{*}\right)\\ +l_d\left(L-L^{*}\right)]+\left(\frac{L-L^{*}}{L}\right)[l\left(S-S^{*}\right)-\left(1-q\right)\beta \left(L-L^{*}\right)\{\left(I-I^{*}\right)\\ +\left(I_V-I_V^{*}\right)\}-l_d\left(L-L^{*}\right)]+\left(\frac{E-E^{*}}{E}\right)[\beta \left(S-S^{*}\right)\left\{\left(I-I^{*}\right)+\left(I_V-I_V^{*}\right)\right\}\\ +\left(1-q\right)\beta \left(L-L^{*}\right)\left\{\left(I-I^{*}\right)+\left(I_V-I_V^{*}\right)\right\}-\alpha \left(E-E^{*}\right)]\end{array}$

$\begin{array}{c}+\left(\frac{I-I^{*}}{I}\right)\left\{\alpha \left(E-E^{*}\right)-\gamma \left(I-I^{*}\right)\right\}+\left(\frac{R-R^{*}}{R}\right)\gamma \left(I-I^{*}\right)\\ +\left(\frac{V-V^{*}}{V}\right)\left[\delta \left(S-S^{*}\right)-\left(1-\eta \right)\beta \left(V-V^{*}\right)\left\{\left(I-I^{*}\right)+\left(I_V-I_V^{*}\right)\right\}\right]\\ +\left(\frac{E_V-E_V^{*}}{E_V}\right)\left[\left(1-\eta \right)\beta \left(V-V^{*}\right)\left\{\left(I-I^{*}\right)+\left(I_V-I_V^{*}\right)\right\}-\alpha \left(E_V-E_V^{*}\right)\right]\\ +\left(\frac{I_V-I_V^{*}}{I_V}\right)\left\{\alpha \left(E_V-E_V^{*}\right)-\gamma \left(I_V-I_V^{*}\right)\right\}+\left(\frac{R_V-R_V^{*}}{R_V}\right)\gamma \left(I_V-I_V^{*}\right).\end{array}$(2.20)

After simplifying, we may write,

$\frac{\text{d}{L}_f}{\text{d}t}={\Pi }_1-{\Pi }_2,$ (2.21)

where

$\begin{array}{c}{\Pi }_1=\beta \frac{{\left(S-S^{*}\right)}^2}{S}{I}^{*}+\beta \frac{{\left(S-S^{*}\right)}^2}{S}{I}_V^{*}+l_{d}L+l_d\frac{S^{*}}{S}{L}^{*}+lS+l\frac{L^{*}}{L}{S}^{*}+\beta \left(1-q\right)L{I}^{*}\\ +\beta \left(1-q\right)L{I}_V^{*}+\beta \left(1-q\right)L^{*}I+\beta \left(1-q\right)L^{*}{I}_V+\beta \left(1-q\right)\frac{E^{*}}{E}LI\\ +\beta \left(1-q\right)\frac{E^{*}}{E}L{I}_V+\beta \left(1-q\right)\frac{E^{*}}{E}{L}^{*}{I}^{*}+\beta \left(1-q\right)\frac{E^{*}}{E}{L}^{*}{I}_V^{*}+\beta SI+\beta S{I}_V\\ +\beta S^{*}{I}^{*}+\beta S^{*}{I}_V^{*}+\beta \frac{E^{*}}{E}S{I}^{*}+\beta \frac{E^{*}}{E}S{I}_V^{*}+\beta \frac{E^{*}}{E}{S}^{*}I+\beta \frac{E^{*}}{E}{S}^{*}{I}_V\end{array}$

$\begin{array}{c}+\beta \left(1-q\right)LI+\beta \left(1-q\right)L{I}_V+\beta \left(1-q\right)L^{*}{I}^{*}+\beta \left(1-q\right)L^{*}{I}_V^{*}+\beta \left(1-q\right)\frac{E^{*}}{E}L{I}^{*}\\ +\beta \left(1-q\right)\frac{E^{*}}{E}L{I}_V^{*}+\beta \left(1-q\right)\frac{E^{*}}{E}{L}^{*}I+\beta \left(1-q\right)\frac{E^{*}}{E}{L}^{*}{I}_V+\alpha E+\alpha \frac{I^{*}}{I}{E}^{*}+\gamma I\\ +\gamma \frac{R^{*}}{R}{I}^{*}+\delta S+\delta \frac{V^{*}}{V}{S}^{*}+\beta \left(1-\eta \right)\frac{{\left(V-V^{*}\right)}^2}{V}{I}^{*}+\beta \left(1-\eta \right)\frac{{\left(V-V^{*}\right)}^2}{V}{I}_V^{*}\\ +\beta \left(1-\eta \right)\frac{E_V^{*}}{E_V}V{I}^{*}+\beta \left(1-\eta \right)\frac{E_V^{*}}{E_V}V{I}_V^{*}+\beta \left(1-\eta \right)\frac{E_V^{*}}{E_V}{V}^{*}I+\beta \left(1-\eta \right)\frac{E_V^{*}}{E_V}{V}^{*}{I}_V\\ +\gamma I_V+\gamma \frac{R_V^{*}}{R_V}{I}_V^{*}+\alpha E_V+\alpha \frac{I_V^{*}}{I_V}{E}_V^{*},\end{array}$