AppendixMathematical analysis

The present section studies the proposed model’s positivity and boundedness, basic \(_)\) reproduction number, the threshold of points of equilibrium (disease-free and endemic), stability of disease-free \(_)\) and endemic (\(_)\) equilibrium point, effective \(_)\) reproduction number, the relation between \(_\) and crucial harmonies of vaccination, the first- and second-order Lyapunov functions (LF), existence, and uniqueness theorem for validating the model’s stability.

Models positivity and boundedness [63,64,65]

Let the assumed state vector be represented as \(} = \left( }_ , }_ , }_ , }_ , }_ , }_ , }_ , }_ , }_ } \right) = \left( , I_ , R_ } \right)\). Then, Eq. (1) can be signed as some initial set of values.

$$\user2} = f\left( }, t} \right), }\left( 0 \right) = }_ .$$

(12.1)

It is evident that the above-defined function f is locally Lipschitz for the first argument and continuous for the second in \(}^ \times }\), which implies that the solution \(}\left( t \right)\) holds in \(t \in \left( \right)\) for some \(T,\) according to the system of nonlinear ODEs existence and uniqueness theorem [63, 66]. For the invariance of sets under a flow, we used the Bony–Brezis theorem [67]. 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 \(T < \infty\), \(\mathop \limits_ }} \parallel }\left( t \right)\parallel \to \infty .\)

We have to show that under the flow \(}\left( t \right)\) for \(t \in \left( \right),\) if

$$}_ \in = \left\} \in }^ :}_ \ge 0\;}\;i = 1,2,3, \cdots ,9, \mathop \sum \limits_^ }_ \le N\left( 0 \right)} \right\},$$

the set \(\) is positively invariant, which concludes that \(T = \infty\), i.e., for initial conditions \(}_ \in \), the solution \(}\left( t \right) \in \) holds globally with time [63, 65, 66].

Theorem 1

The closed set.

$$: = \left\} = \left( , I_ , R_ } \right) \in }^ :}_ \ge 0\;}\;i = 1,2,3, \ldots ,9, \mathop \sum \limits_^ }_ \le N\left( 0 \right)} \right\},$$

is entirely uniform under the flow generated by Eqs. 1.1–1.9. As a result, solution \(}\left( } \right) \in \) occurs globally in the time given initial conditions \(}_ \in \).

Proof

Let the boundary segment is \(_ , i = 1,2, \cdots , 10\).

$$\begin _ & = \left\} \in : }_}} = 0} \right\}, i = 1,2, \ldots ,9 \\ _ & = \left\} \in : \mathop \sum \limits_^ }_ = 0} \right\}. \\ \end$$

It is evident that \(\partial = \cup_^ _ .\)

To complete the proof of the above invariance of the set \(\), it is enough to prove \(\forall\) inward normal, \(}.\user2}\left( t \right) \ge 0\) on \(\partial .\) The inward normal on \(_ \) for \(i = 1,2, \cdots , 9\) is unambiguously provided by \(}_ = }_ = [0\), \(\cdots\), 1, \(\cdots\), 0], where only the \(i\)-component is nonzero, but an inward normal on \(_\) is given by \(}_ = \left( - 1, \cdots , - 1} \right)\).

Now, on \(_ \) for \(i = 1,2, \cdots ,9\)

$$\begin }_ .\user2} = l_ }_ \ge 0, \forall } \in _ , \hfill \\ }_ .\user2} = l}_ \ge 0, \forall } \in _ , \hfill \\ }_ .\user2} = \beta }_ \left( }_ + }_ } \right) + \left( \right)\beta }_ \left( }_ + }_ } \right) \ge 0, \forall } \in _ , \hfill \\ }_ .\user2} = \alpha }_ \ge 0,\forall } \in _ , \hfill \\ }_ .\user2} = \gamma }_ \ge 0, \forall } \in _ , \hfill \\ }_ .\user2} = \delta }_ \ge 0,\forall } \in _ , \hfill \\ }_ .\user2} = \left( \right)\beta }_ \left( }_ + }_ } \right) \ge 0,\forall } \in _ , \hfill \\ }_ .\user2} = \alpha }_ \ge 0,\forall } \in _ , \hfill \\ }_ .\user2} = \gamma }_ \ge 0,\forall } \in _ , \hfill \\ \end$$

while \(} = \mathop \sum \limits_^ }_\) is readily seen to satisfy \(\frac} = 0\), which follows that on \(_\) such that

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

Derivation of the basic reproduction number \(_)\)

Before starting the formal analysis, we will describe how the so-called basic reproduction number \(_)\) is computed in detail, which is vital in epidemiological modeling because it has been found to help comprehend stability conditions. It has been demonstrated that stability conditions can be expected if this number is less than one, but instability conditions occur if it is more significant than one. Nevertheless, it has been emphasized that the value can be derived using various approaches; among them, the next-generation matrix techniques [68] are well-established. Thus, the detailed process \(FV^ \left( V_^ ,F_ V_^ } \right)\) of finding the value of the basic reproduction number \((R_ )\) is

$$F_ = \left[ c} 0 & \right)\beta } \\ 0 & 0 \\ \end } \right], V_ = \left[ c} \alpha & 0 \\ & \gamma \\ \end } \right].$$

$$\therefore F_ V_^ = \frac\left[ c} & \right)\beta } \right\}} \\ 0 & 0 \\ \end } \right].$$

The eigenvalues of \(F_ V_^\) are \(\lambda_ = \frac and \lambda_ = \frac \right)\beta }}.\)

Correspondingly, for the vaccination part,

$$F_ = \left[ c} 0 & \right)} \\ 0 & 0 \\ \end } \right], V_ = \left[ c} \alpha & 0 \\ & \gamma \\ \end } \right].$$

$$\therefore F_ V_^ = \frac\left[ c} & \right)\beta } \\ 0 & 0 \\ \end } \right].$$

The eigenvalues of \(F_ V_^\) are \(\lambda_ = \frac\;}\;\lambda_ = \frac \right)\beta }} .\)

Thus, the basic reproduction number \((R_ )\) is

$$R_ = \frac + \frac \right)\beta }} + \frac \right)\beta }}.$$

(12.2)

Analysis of disease-free and endemic equilibrium point threshold

For this model, the disease-free equilibrium point threshold is the solution of the nonlinear system of Eq. (1). Thus,

$$\frac}S}}}t}} = \frac}L}}}t}} = \frac}E}}}t}} = \frac}I}}}t}} = \frac}R}}}t}} = \frac}V}}}t}} = \frac }}}t}} = \frac }}}t}} = \frac }}}t}} = 0,$$

gives \(E_ = \left( }^ ,0,0,0,0,}^ ,0,0,0} \right),\) the disease-free equilibrium point threshold of the current model.

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

$$\begin 0 & = - \beta S\left( t \right)\left( \left( t \right)} \right) - \delta S\left( t \right) - lS\left( t \right) + l_ L\left( t \right), \\ 0 & = lS\left( t \right) - \left( \right)\beta L\left( t \right)\left( \left( t \right)} \right) - l_ L\left( t \right), \\ 0 & = \beta S\left( t \right)\left( \left( t \right)} \right) + \left( \right)\beta L\left( t \right)\left( \left( t \right)} \right) - \alpha E\left( t \right), \\ 0 & = \alpha E\left( t \right) - \gamma I\left( t \right), \\ 0 & = \gamma I\left( t \right), \\ 0 & = \delta S\left( t \right) - \left( \right)\beta V\left( t \right)\left( \left( t \right)} \right), \\ 0 & = \left( \right)\beta V\left( t \right)\left( \left( t \right)} \right) - \alpha E_ \left( t \right), \\ 0 & = \alpha E_ \left( t \right) - \gamma I_ \left( t \right), \\ 0 & = \gamma I_ \left( t \right). \\ \end$$

Solving the above system of equations setting \(I \ne 0,I_ \ne 0\), the endemic equilibrium point is.

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

where

$$S^ = \frac }} + \delta + l}},L^ = \frac \right)\beta }} - \frac }} \right)\left( + \delta + l} \right)}},E^ = \frac L^ }},I^ = \frac L^ }},R^ = l_ L^$$

$$V^ = \frac \right)\beta }},E_ }} = \frac }},I_ }} = \frac }},R_ }} = \delta S^ .$$

Stability of disease-free \((E_ )\) equilibrium point

In this section, we will show that for \(_<1\) and \(_>1\) the disease-free \(_)\) equilibrium point is asymptotically stable 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 \(_<1\), then the unique disease-free equilibrium \(_\) is locally asymptotically stable. If \(_>1\), the unique disease-free equilibrium is unstable.

Proof

To validate the local stability, the Jacobian matrix of the proposed system (1) is.

$$J = \left[ c} ) - \delta - l} & } & 0 & & 0 & 0 & 0 & & 0 \\ l & ) - l_ } & 0 & & 0 & 0 & 0 & & 0 \\ )} & )} & & & 0 & 0 & 0 & & 0 \\ 0 & 0 & \alpha & & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \gamma & 0 & 0 & 0 & 0 & 0 \\ \delta & 0 & 0 & & 0 & )} & 0 & & 0 \\ 0 & 0 & 0 & & 0 & )} & & & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \alpha & & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \gamma & 0 \\ \end } \right].$$

At the DFE point \(_,\) we have

$$J(E_ ) = \left[ c} & } & & & 0 & 0 & 0 & 0 & 0 \\ l & } & & & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & & & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & & & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & & & 0 & 0 & 0 & 0 & 0 \\ \delta & 0 & & & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & & & 0 & 0 & & 0 & 0 \\ 0 & 0 & & & 0 & 0 & & & 0 \\ 0 & 0 & & & 0 & 0 & & & 0 \\ \end } \right].$$

The characteristic equation \(\left| } \right) - \lambda I} \right| = 0\) has nine roots, which are \(\lambda_ = \lambda_ = \lambda_ = 0, \lambda_ = \lambda_ = - \alpha ,\) \(\lambda_ = \lambda_ = - \gamma ,\) i.e., the first seven eigenvalues are less or equal to zero. Thus, the stability of the DFE point depends on the remaining two roots (\(\lambda_ , \lambda_ ),\) the solution of the following quadratic characteristic equation:

$$\lambda^ + a_ \lambda + a_ = 0,$$

where

$$\begin a_ & = \delta + l + l_ , \\ a_ & = \delta l_ . \\ \end$$

It is clear that the coefficients represented above are positive, i.e., \(a_ > 0\) and \(a_ > 0\). As a result, it implies that \(a_ > 0\) for \(i = 1,2\) are positive and follow the Routh–Hurwitz criterion [7]. On top of that, we conclude that all eigenvalues \(\lambda_ \left( \right)\) calculated from the Jacobian matrix at the DFE point have negative real parts. Therefore, the model is locally asymptotically stable at the unique DFE point whenever \(R_ < 1\) and unstable whenever \(R_ > 1\).

Theorem 3

The endemic equilibrium point \(E^\) is locally asymptotically stable and unstable whenever \(R_ > 1.\)

Proof

The endemic equilibrium point \(E_ ,\) the desired Jacobian matrix is

$$J = \left[ c} + I_^ ) - \delta - l} & } & 0 & } & 0 & 0 & 0 & } & 0 \\ l & + I_^ ) - l_ } & 0 & } & 0 & 0 & 0 & } & 0 \\ + I_^ )} & + I_^ )} & & } & 0 & 0 & 0 & } & 0 \\ 0 & 0 & \alpha & & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \gamma & 0 & 0 & 0 & 0 & 0 \\ \delta & 0 & 0 & } & 0 & + I_^ )} & 0 & } & 0 \\ 0 & 0 & 0 & } & 0 & + I_^ )} & & } & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \alpha & & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \gamma & 0 \\ \end } \right].$$

The roots of the characteristic equations \(|J(E_ - \lambda I_ |\) satisfy the following equation:

$$\lambda^ + a_ \lambda^ + a_ \lambda^ + a_ \lambda^ + a_ \lambda + a_ = 0,$$

(12.3)

where,

$$a_ = - A_ + A_ - A_ + \delta + \omega ,$$

\(a_ = - A_ A_ + A_ A_ - A_ A_ + A_ A_ - A_ f - a\delta + A_ \delta - f\delta - a\omega + A_ \omega - A_ \omega + \delta \omega ,\)

$$\begin a_ & = - A_ A_ A_ - A_ A_ A_ + A_ A_ A_ - A_ A_ x - A_ A_ \delta + A_ A_ \delta - A_ A_ \delta \\ & \quad + A_ A_ \delta - A_ A_ \delta - A_ A_ \omega + A_ A_ \omega - A_ A_ \omega + A_ A_ \omega \\ & \quad - A_ A_ \omega - A_ \gamma \omega - A_ \delta \omega + A_ \delta \omega - A_ \delta \omega , \\ \end$$

$$\begin a_ & = - A_ A_ A_ \delta - A_ A_ A_ \delta + A_ A_ A_ \delta - A_ A_ x\delta - A_ A_ e\omega + A_ A_ A_ \omega \\ & \quad - A_ A_ x\omega - A_ A_ \gamma \omega - A_ x\gamma \omega - A_ A_ \delta \omega + A_ A_ \delta \omega - A_ A_ \delta \omega \\ & \quad + A_ A_ \delta \omega - A_ A_ \delta \omega - A_ \gamma \delta \omega - A_ \delta \tau \omega - A_ A_ A_ \omega , \\ \end$$

\(a_ = - A_ \left( A_ \delta \omega + A_ A_ \delta \omega - A_ A_ \delta \omega + A_ x\delta \omega + A_ \gamma \delta \omega + x\gamma \delta \omega + A_ \delta \tau \omega + x\delta \tau \omega } \right),\)

$$A_ = \left( \right)\beta I^ ,$$

$$A_ = \left( \right)\beta V^ ,$$

$$A_ = \beta S^ + \left( \right)\beta V^ - \gamma - \tau .$$

It is simple to demonstrate that all of the roots of Eq. (12.3) will have a negative real portion if \(_>1\) and that the coefficients of (12.3) will fulfill the Routh–Hurwitz condition [7]. The endemic equilibrium point will thus be locally asymptotically stable for \(_>1\).

Derivation of the effective reproduction number \(_)\)

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 the 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 (\(_\)) is the mean per infectious case in a population of susceptible and non-susceptible people. When \(_>1,\) the number of cases will increase and start an epidemic. When \(_=1\), the disease is endemic, and when \(_<1\), 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,

$$R_ = \fracS\left( t \right) + \frac \right)\beta }}L\left( t \right) + \frac \right)\beta }}V\left( t \right).$$

(12.4)

Relation between \(R_\) and crucial harmonies of vaccination

In the recommended model, \(\frac \left( t \right)} \right) }}\) 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 equals \(\frac\) for the single infected individual. Furthermore, \(\frac\) and \(\frac\) 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 \(\frac\) and \(\frac \right)\beta }}\). Analogously, secondary infections of vaccinated individuals equal \(\frac_(t)}\) and per unit of time, the sum is \(\frac\). Realistically, due to the mass vaccination rate \(\frac,\) the basic reproduction number must be a decreasing function. Opposite scenarios for less vaccine efficacy rate\(\left(1-\eta \right)\).

The first derivative of the Lyapunov function (LF)

Let us assume models independent variables for the endemic LF, \(\left\_,_,_\right\}, _<0\) is the detrimental equilibrium point \(_\).

Theorem 4

For the value of the basic reproductive number \(_-1>0\), the proposed \(SLEIRV___\) models endemic equilibrium point \(_\) is globally asymptotically stable.

Proof

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

$$\begin & L_ \left( ,I_ ,R_ } \right) = \left( - S^ \log \frac }}} \right) + \left( - L^ \log \frac }}} \right) \\ & \quad + \left( - E^ \log \frac }}} \right) + \left( - I^ \log \frac }}} \right) + \left( - R^ \log \frac }}} \right) \\ & \left( - V\log \frac }}} \right) + \left( - E_^ - E_^ \log \frac^ }} }}} \right) + \left( - I_^ - I_^ \log \frac^ }} }}} \right) \\ & \quad + \left( - R_^ - R_^ \log \frac^ }} }}} \right). \\ \end$$

(12.5)

Differentiating both sides in terms of t, we get,

$$\begin \frac}L_ }}}t}} & = \dot_ = \left( }}} \right)\dot + \left( }}} \right)\dot + \left( }}} \right)\dot + \left( }}} \right)\dot + \left( }}} \right)\dot \\ & \quad + \left( }}} \right)\dot + \left( - E_^ }} }}} \right)\dot_ + \left( - I_^ }} }}} \right)\dot_ + \left( - R_^ }} }}} \right)\dot_ . \\ \end$$

(12.6)

From Eq. (1), putting the value of \(\dot, \dot, \dot,\dot,\dot,\dot_ ,\dot_ ,\dot_\) in Eq. (12.6), we have,

$$\begin \frac}L_ }}}t}} & = \left( }}} \right)\left\ } \right) - \delta S - lS + l_ L} \right\} \\ & \quad + \left( }}} \right)\left\ \right)\beta L\left( } \right) - l_ L} \right\} \\ & \quad + \left( }}} \right)\left\ } \right) + \left( \right)\beta L\left( } \right) - \alpha E} \right\} \\ & \quad + \left( }}} \right)\left( \right) + \left( }}} \right)\gamma I \\ & \quad + \left( }}} \right)\left\ \right)\beta V\left( } \right)} \right\} \\ & \quad + \left( - E_^ }} }}} \right)\left\ \right)\beta V\left( } \right) - \alpha E_ } \right\} \\ & \quad + \left( - I_^ }} }}} \right)\left( - \gamma I_ } \right) \\ & \quad + \left( - R_^ }} }}} \right)\gamma I_ . \\ \end$$

(12.7)

In Eq. (12.7), substitute \(S - S^ ,L - L^ ,E - E^ ,I - I^ ,V - V^ ,E_ - E_^ ,I_ - I_^ ,R_ - R_^\) instead of \(S, L, E,I,R, V,E_ ,I_ ,R_ .\) Then,

$$\begin \frac}L_ }}}t}} & = \left( }}} \right)\left[ } \right)\left\ } \right) + \left( - I_^ } \right)} \right\} - \delta \left( } \right) - l\left( } \right) + l_ \left( } \right)} \right] \\ & \quad + \left( }}} \right)\left[ } \right) - \left( \right)\beta \left( } \right)\left\ } \right) + \left( - I_^ } \right)} \right\} - l_ \left( } \right)} \right] \\ & \quad + \left( }}} \right)[\beta \left( } \right)\left\ } \right) + \left( - I_^ } \right)} \right\} + \left( \right)\beta \left( } \right)\left\ } \right) + \left( - I_^ } \right)} \right\} \\ & \quad - \alpha \left( } \right)] + \left( }}} \right)\left\ } \right) - \gamma \left( } \right)} \right\} + \left( }}} \right)\gamma \left( } \right) \\ & \quad + \left( }}} \right)\left[ } \right) - \left( \right)\beta \left( } \right)\left\ } \right) + \left( - I_^ } \right)} \right\}} \right] \\ & \quad + \left( - E_^ }} }}} \right)\left[ \right)\beta \left( } \right)\left\ } \right) + \left( - I_^ } \right)} \right\} - \alpha \left( - E_^ } \right)} \right] \\ & \quad + \left( - I_^ }} }}} \right)\left\ - E_^ } \right) - \gamma \left( - I_^ } \right)} \right\} \\ & \quad + \left( - R_^ }} }}} \right)\gamma \left( - I_^ } \right). \\ \end$$

(12.8)

After simplifying, we may write,

$$\frac}L_ }}}t}} = _ - _ ,$$

(12.9)

where

$$\begin _ & = \beta \frac } \right)^ }}I^ + \beta \frac } \right)^ }}I_^ + l_ L + l_ \frac }}L^ + lS + l\frac }}S^ + \beta \left( \right)LI^ + \beta \left( \right)LI_^ \\ & \quad + \beta \left( \right)L^ I + \beta \left( \right)L^ I_ + \beta \left( \right)\frac }}LI + \beta \left( \right)\frac }}LI_ + \beta \left( \right)\frac }}L^ I^ \\ & \quad + \beta \left( \right)\frac }}L^ I_^ + \beta SI + \beta SI_ + \beta S^ I^ + \beta S^ I_^ + \beta \frac }}SI^ + \beta \frac }}SI_^ + \beta \frac }}S^ I + \beta \frac }}S^ I_ \\ & \quad + \beta \left( \right)LI + \beta \left( \right)LI_ + \beta \left( \right)L^ I^ + \beta \left( \right)L^ I_^ + \beta \left( \right)\frac }}LI^ + \beta \left( \right)\frac }}LI_^ \\ & \quad + \beta \left( \right)\frac }}L^ I + \beta \left( \right)\frac }}L^ I_ + \alpha E + \alpha \frac }}E^ + \gamma I + \gamma \frac }}I^ + \delta S + \delta \frac }}S^ \\ & \quad + \beta \left( \right)\frac } \right)^ }}I^ + \beta \left( \right)\frac } \right)^ }}I_^ + \beta \left( \right)\frac^ }} }}VI^ + \beta \left( \right)\frac^ }} }}VI_^ \\ & \quad + \beta \left( \right)\frac^ }} }}V^ I + \beta \left( \right)\frac^ }} }}V^ I_ + \gamma I_ + \gamma \frac^ }} }}I_^ + \alpha E_ + \alpha \frac^ }} }}E_^ , \\ \end$$

and

$$\begin _ & = \beta \frac } \right)^ }}I + \beta \frac } \right)^ }}I_ + \left( \right)\frac } \right)^ }} + l_ L^ + l_ \frac }}L + lS^ + l\frac }}S \\ & \quad + \beta \left( \right)LI + \beta \left( \right)LI_ + \beta \left( \right)L^ I^ + \beta \left( \right)L^ I_^ + \beta \left( \right)\frac }}LI^ \\ & \quad + \beta \left( \right)\frac }}LI_^ + \beta \left( \right)\frac }}L^ I + \beta \left( \right)\frac }}L^ I_ + l_ \frac } \right)^ }} + \beta SI^ \\ & \quad + \beta SI_^ + \beta S^ I + \beta S^ I_ + \beta \frac }}SI + \beta \frac }}SI_ + \beta \frac }}S^ I^ + \beta \frac }}S^ I_^ + \beta \left( \right)LI^ \\ & \quad + \beta \left( \right)LI_^ + \beta \left( \right)L^ I + \beta \left( \right)L^ I_ + \beta \left( \right)\frac }}LI + \beta \left( \right)\frac }}LI_ \\ & \quad + \beta \left( \right)\frac }}L^ I^ + \beta \left( \right)\frac }}L^ I_^ + \alpha \frac } \right)^ }} + \alpha E^ + \alpha \frac }}E \\ & \quad + \gamma \frac } \right)^ }} + \gamma I^ + \gamma \frac }}I + \delta S^ + \delta \frac }}S + \beta \left( \right)\frac } \right)^ }}I + \beta \left( \right)\frac } \right)^ }}I_ \\ & \quad + \beta \left( \right)\frac^ }} }}VI + \beta \left( \right)\frac^ }} }}VI_ + \beta \left( \right)\frac^ }} }}V^ I^ \\ & \quad + \beta \left( \right)\frac^ }} }}V^ I_^ + \gamma I_^ + \gamma \frac^ }} }}I_ + \alpha E_^ + \alpha \frac^ }} }}E_ + \gamma \frac - I_^ } \right)^ }} }} + \alpha \frac - E_^ } \right)^ }} }}. \\ \end$$

It is evident that, \(\frac}L_ }}}t}} < 0\) if \(_ < _ .\) However, for \(S = S^ ,L = L^ ,E = E^ ,I = I^ ,R = R^ ,V = V^ ,E_ = E_^ ,I_ = I_^ ,R_ = R_^\) we may write,

$$\begin 0 & = _ - _ \\ & \Rightarrow \frac}L_ }}}t}} = 0. \\ \end$$

(12.10)

We conclude that for the recommended model, the leading compact invariant set in

$$\left\ , L^ , E^ ,I^ ,R^ ,V^ ,E_^ ,I_^ ,R_^ } \right) \in :\frac}L_ }}}t}} = 0} \right\}$$

(12.11)

is the endemic equilibrium point \(\left\ } \right\}\). Finally, it is evident that according to Lasalle’s invariance, if \(_ < _\), \(E_\) is globally asymptotically stable in \(\).

The second derivative of the Lyapunov function (LF)

Generally, the first derivative of LF assists researchers in checking the global stability of the models. However, it helps in knowing crucial information like disease sequence but not well enough to comprehend the variabilities. As a result, second derivative analysis is essential for further information, for example, curvature and sign. The second derivative, we believe, will give more details.

$$\begin \frac}\dot_ }}}t}} & = \frac}}}t}}\left\ }}} \right)\dot + \left( }}} \right)\dot + \left( }}} \right)\dot + \left( }}} \right)\dot + \left( }}} \right)\dot + \left( }}} \right)\dot} \right. \\ & \quad + \left( ^ }} }}} \right)\dot_ + \left( ^ }} }}} \right)\dot_ + \left( ^ }} }}} \right)\dot_ \\ & = \left( }}} \right)^ S^ + \left( }}} \right)^ L^ + \left( }}} \right)^ E^ + \left( }}} \right)^ I^ + \left( }}} \right)^ R^ + \left( }}} \right)^ V^ + \left( _ }} }}} \right)^ E_^ + \left( _ }} }}} \right)^ I_^ \\ & \quad + \left( _ }} }}} \right)^ R_^ + \left( }}} \right)\ddot + \left( }}} \right)\ddot + \left( }}} \right)\ddot + \left( }}} \right)\ddot + \left( }}} \right)\ddot \\ \end$$

(12.12)

Here, the second derivative of Eq. (1) is

$$\begin \ddot & = - \beta \dot\left( } \right) - \beta S\left( + \dot_ } \right) - \delta \dot - l\dot + l_ \dot \\ \ddot & = l\dot - \left( \right)\beta \dot\left( } \right) - \left( \right)\beta L\left( + \dot_ } \right) - l_ \dot \\ \ddot & = \beta \dot\left( } \right) + \beta S\left( + \dot_ } \right) + \left( \right)\beta \dot\left( } \right) + \left( \right)\beta L\left( + \dot_ } \right) - \alpha \dot \\ \ddot & = \alpha \dot - \gamma \dot \\ \ddot & = \gamma \dot \\ \ddot & = \delta \dot - \left( \right)\beta \dot\left( } \right) - \left( \right)\beta S\left( + \dot_ } \right) \\ \ddot_ & = \left( \right)\beta \dot\left( } \right) + \left( \right)\beta V\left( + \dot_ } \right) - \alpha \dot_ \\ \ddot_ & = \alpha \dot_ - \gamma \dot_ \\ \ddot_ & = \gamma \dot_ . \\ \end$$

Hence,

$$\begin & \frac}\dot_ }}}t}} = \left( }}} \right)^ S^ + \left( }}} \right)^ L^ + \left( }}} \right)^ E^ + \left( }}} \right)^ I^ + \left( }}} \right)^ R^ + \left( }}} \right)^ V^ + \left( _ }} }}} \right)^ E_^ + \left( _ }} }}} \right)^ I_^ \\ & \quad + \left( _ }} }}} \right)^ R_^ + \left( }}} \right)\left\\left( } \right) - \beta S\left( + \dot_ } \right) - \delta \dot - l\dot + l_ \dot} \right\} \\ & \quad + \left( }}} \right)\left\ - \left( \right)\beta \dot\left( } \right) - \left( \right)\beta L\left( + \dot_ } \right) - l_ \dot} \right\} \\ & \quad + \left( }}} \right)\left\\left( } \right) + \beta S\left( + \dot_ } \right) + \left( \right)\beta \dot\left( } \right) + \left( \right)\beta L\left( + \dot_ } \right) - \alpha \dot} \right\} \\ & \quad + \left( }}} \right)\left( - \gamma \dot} \right) + \left( }}} \right)\gamma \dot \\ & \quad + \left( }}} \right)\left\ - \left( \right)\beta \dot\left( } \right) - \left( \right)\beta V\left( + \dot_ } \right)} \right\} \\ & \quad + \left( ^ }} }}} \right)\left\ \right)\beta \dot\left( } \right) + \left( \right)\beta V\left( + \dot_ } \right) - \alpha \dot_ } \right\} \\ & \quad + \left( ^ }} }}} \right)\left( _ - \gamma \dot_ } \right) + \left( ^ }} }}} \right)\gamma \dot_ . \\ \end$$

(12.13)

and

$$\begin \frac}^ L_ }}}t^ }} & = }\left( ,I_ ,R_ } \right) + \left( }}} \right)\left\\left( } \right) - \beta S\left( + \dot_ } \right) - \delta \dot - l\dot + l_ \dot} \right\} \\ & \quad + \left( }}} \right)\left\ - \left( \right)\beta \dot\left( } \right) - \left( \right)\beta L\left( + \dot_ } \right) - l_ \dot} \right\} \\ & \quad + \left( }}} \right)\left\\left( } \right) + \beta S\left( + \dot_ } \right) + \left( \right)\beta \dot\left( } \right) + \left( \right)\beta L\left( + \dot_ } \right) - \alpha \dot} \right\} \\ & \quad + \left( }}} \right)\left( - \gamma \dot} \right) + \left( }}} \right)\gamma \dot \\ & \quad + \left( }}} \right)\left\ - \left( \right)\beta \dot\left( } \right) - \left( \right)\beta V\left( + \dot_ } \right)} \right\} \\ & \quad + \left( ^ }} }}} \right)\left\ \right)\beta \dot\left( } \right) + \left( \right)\beta V\left( + \dot_ } \right) - \alpha \dot_ } \right\} \\ & \quad + \left( ^ }} }}} \right)\left( _ - \gamma \dot_ } \right) + \left( ^ }} }}} \right)\gamma \dot_ . \\ \end$$

(12.14)

Finally, replacing the value of \(\dot, \dot, \dot,\dot,\dot,\dot,\dot_ ,\dot_ ,\dot_\) in equation \(\left( \right)\), we have,

$$\frac}^ L_ }}}t^ }} = _ - _ ,$$

(12.15)

where \(_\) and \(_\) are the summation of all positive and negative terms.

Therefore,

\(\frac}^ L_ }}}t^ }} > 0\) if \(_ > _ ,\)

\(\frac}^ L_ }}}t^ }} < 0\) if \(_ < _ ,\)

\(\frac}^ L_ }}}t^ }} = 0\) if \(_ = _ .\)

Existence and uniqueness

The present sub-section examines the existence and uniqueness of the proposed model’s solution through the concept of classical calculus.

Theorem 5:

If \(\theta_\) and \(\overline_\) are the positive constants, then.

(i) \(\forall i \in \left\ \right\}\)

$$\left| \left( , t} \right) - f_ \left( , t} \right)} \right|^ \le \theta_ \left| - x_ } \right|^ .$$

(12.17)

(ii) \(\forall \left( \right) \in }^ \times \left( \right)\)

$$\left| \left( , t} \right)} \right|^ \le \overline_ \left( } \right|^ } \right) or \overline_ \left| } \right|^ .$$

(12.18)

We may represent the current model as follows

$$\begin \frac}S\left( t \right)}}}t}} & = - \beta S\left( t \right)\left( \left( t \right)} \right) - \delta S\left( t \right) - lS\left( t \right) + l_ L\left( t \right) = f_ \left( ,I_ ,R_ } \right), \\ \frac}L\left( t \right)}}}t}} & = lS\left( t \right) - \left( \right)\beta L\left( t \right)\left( \left( t \right)} \right) - l_ L\left( t \right) = f_ \left( ,I_ ,R_ } \right), \\ \frac}E\left( t \right)}}}t}} & = \beta S\left( t \right)\left( \left( t \right)} \right) + \left( \right)\beta L\left( t \right)\left( \left( t \right)} \right) - \alpha E\left( t \right) = f_ \left( ,I_ ,R_ } \right), \\ \frac}I\left( t \right)}}}t}} & = \alpha E\left( t \right) - \gamma I\left( t \right) = f_ \left( ,I_ ,R_ } \right), \\ \frac}R\left( t \right)}}}t}} & = \gamma I\left( t \right) = f_ \left( ,I_ ,R_ } \right), \\ \frac}V\left( t \right)}}}t}} & = \delta S\left( t \right) - \left( \right)\beta V\left( t \right)\left( \left( t \right)} \right) = f_ \left( ,I_ ,R_ } \right), \\ \frac}E_ \left( t \right)}}}t}} & = \left( \right)\beta V\left( t \right)\left( \left( t \right)} \right) - \alpha E_ \left( t \right) = f_ \left( ,I_ ,R_ } \right), \\ \frac}I_ \left( t \right)}}}t}} & = \alpha E_ \left( t \right) - \gamma I_ \left( t \right) = f_ \left( ,I_ ,R_ } \right), \\ \frac}R_ \left( t \right)}}}t}} & = \gamma I_ \left( t \right) = f_ \left( ,I_ ,R_ } \right), \\ \end$$

To begin, we will show that the given function \(f_ \left( ,I_ ,R_ } \right)\) satisfies

$$|f_ \left( , t} \right) - f_ \left( , t} \right)|^ \le \theta_ \left| - S_ } \right|^ .$$

(12.19)

Therefore,

$$\begin |f_ \left( , t} \right) - f_ \left( , t} \right)|^ & = \left| } \right)\left( - S_ } \right) - \delta \left( - S_ } \right) - l\left( - S_ } \right)} \right|^ \\ & = \left| } \right) - \delta - l} \right\}\left( - S_ } \right)} \right|^ \\ & \le \left\ \left( + \left| } \right|^ } \right) + 2\delta^ + 2l^ } \right\}\left| - S_ } \right|^ \\ & \le \left\ \left( }}\limits_ \left| I \right|^ + \mathop }}\limits_ \left| } \right|^ } \right) + 2\delta^ + 2l^ } \right\}\left| - S_ } \right|^ \\ & \le \left\ \left| \right|} \right|_^ + 2\beta^ \left| \left( t \right)} \right|} \right|_^ + 2\delta^ + 2l^ } \right\}\left| - S_ } \right|^ \\ & \le \theta_ \left| - S_ } \right|^ , \\ \end$$

where \(\theta_ = 2\beta^ \left| \right|} \right|_^ + 2\beta^ \left| \left( t \right)} \right|} \right|_^ + 2\delta^ + 2l^ .\)

In the same way, the other compartments may be shown to meet the inequality mentioned above.

Secondly, we shall demonstrate that

$$|f_ \left( \right)|^ \le \overline_ \left( } \right).$$

(12.20)

Then,

$$\begin |f_ \left( \right)|^ & = \left| } \right) - \delta S - lS + l_ L} \right|^ \\ & = \left| } \right) - \delta - l} \right\}S + l_ L} \right|^ \\ & \le \left\ \left( + \left| } \right|^ } \right) + 2\delta^ + 2l^ } \right\}\left| S \right|^ + 2l_^ \left| L \right|^ \\ & \le \left\ \left( }}\limits_ \left| I \right|^ + \mathop }}\limits_ \left| } \right|^ } \right) + 2\delta^ + 2l^ } \right\}\left| S \right|^ + 2l_^ \mathop }}\limits_ \left| L \right|^ \\ & \le \left\ \left( \right|} \right|_^ + \left| \left( t \right)} \right|} \right|_^ } \right) + 2\delta^ + 2l^ } \right\}\left| S \right|^ + 2l_^ \left| \right|} \right|_^ \\ & \le \overline_ \left( } \right), \\ \end$$

implies that

$$\frac \left( \right|} \right|_^ + \left| \left( t \right)} \right|} \right|_^ } \right) + 2\delta^ + 2l^ }}^ \left| \right|} \right|_^ }} < 1,$$

where \(\overline_ = 2l_^ \left| \right|} \right|_^ .\)

In the same way, as mentioned earlier, we can also show that inequality holds for the other compartments. To summarize, our system’s solution exists and is unique, as described in [49, 69].

Positivity and boundedness of solutions (network based model) Lemma 1

Let us assume that the solution.

\((_,_,_,_,_,V}_,_^,_^,_^,\dots ,_,_,_,_,_,_,_^,_^,_^)\) of the proposed system illustrated in Eq. (3.1–3.9) for the given initial condition in \(\Pi ,\) where \(\Theta \left(0\right)>0.\) Therefore, \(0<_\left(t\right)<1, 0<_\left(t\right)<1, 0<_\left(t\right)<1, 0<_\left(t\right)<1, 0<_\left(t\right)<\mathrm<_^\left(t\right)<\mathrm<_^\left(t\right)<\mathrm<_^\left(t\right)<1,and\Theta \left(}\right)>0, \forall \mathrm>0\), where \(k=\mathrm,3,\dots ,n.\)

Proof

Firstly, we assume that \(\left( } \right) > 0,\forall } > 0\). Then, we can write from Eq. (4)

$$\begin }\left( t \right) & = \frac\mathop \sum \limits_^ \left( \right)P\left( k \right)\left[ _ \left( t \right) + \dot_^ \left( t \right)} \right] \\ & = \frac\mathop \sum \limits_^ \left( \right)P\left( k \right)\left[ \left( t \right) - \gamma I_ \left( t \right) + \alpha E_^ \left( t \right) - \gamma I_^ \left( t \right)} \right] \\ & = \frac\mathop \sum \limits_^ \alpha \left( \right)P\left( k \right)[E_ \left( t \right) + E_^ \left( t \right)] - \gamma \left( t \right). \\ \end$$

Then, we can write,

$$\Theta \left( t \right) = - \gamma \mathop \smallint \limits_^ \Theta \left( u \right)}u + \mathop \smallint \limits_^ \frac \right)P\left( k \right)}}\left( ^ E_ \left( u \right) + \mathop \sum \limits_^ E_^ \left( u \right)} \right)}u.$$

As \(\left( 0 \right) > 0,\) and \(\left( } \right) > 0,\forall } > 0.\)

According to the initial condition, \(_\left(0\right)\ge 0,_^\left(0\right)\ge 0.\) Then from the continuity of exposed individuals \(_\left(t\right), \exists \gamma >0\), which implies that \(_\left(0\right)>0\) for \(t\in \left(0,\gamma \right).\) Therefore, we have to show that \(_\left(t\right)>0 \forall \mathrm.\) Otherwise, we can locate \(_\ge \gamma >0\) such that \(_\left(_\right)=0\) and \(_\left(t\right)>0\) for some \(t\in \left(0,_\right).\) Thus, from the third equation, we have,

$$\dot_ \left( t \right) \ge 0.$$

It means the fact that \(_\left(_\right)<0\) for some \(t\in \left(0,_\right)\) seems to be in a contradiction. As a result, \(_\left(t\right)>0 \forall \mathrm.\) The following relationship can be derived from equation seven using the positivity of \(_\left(t\right), _^\left(t\right),\) and \(\Theta \left(}\right)\):

$$\dot_^ \left( t \right) + \gamma E_^ \left( t \right) > 0\;}\;t > 0.$$

Then,

$$E_^ \left( t \right) > E_^ \left( 0 \right)\exp ( - \gamma t) \ge 0$$

and thus, \(E_^ \left( t \right) > 0 \forall t.\)

Analogously, we can write,

$$S_ \left( t \right),L_ \left( t \right),I_ \left( t \right),R_ \left( t \right),I_^ \left( t \right),R_^ \left( t \right) > 0 \forall t.$$

Here, the total population, \(N_ \left( t \right) = S_ \left( t \right) + L_ \left( t \right) + E_ \left( t \right) + I_ \left( t \right) + R_ \left( t \right) + V_ \left( t \right) + E_^ \left( t \right) + I_^ \left( t \right) + R_^ \left( t \right),\) where \(k = 1,2,3, \ldots ,n.\)

Then, \(\dot_ \left( t \right) = \dot_ \left( t \right) + \dot_ \left( t \right) + \dot_ \left( t \right) + \dot_ \left( t \right) + \dot_ \left( t \right) + \dot_ \left( t \right) + \dot_^ \left( t \right) + \dot_^ \left( t \right) + \dot_^ \left( t \right).\)

Substituting the value of \(\dot_ \left( t \right),\dot_ \left( t \right),\dot_ \left( t \right),\dot_ \left( t \right),\dot_ \left( t \right),\dot_ \left( t \right),\dot_^ \left( t \right),\dot_^ \left( t \right),\dot_^ \left( t \right)\) from Eq. (3) in the above equation, we have.

\(\dot_ \left( t \right) = 0 \forall } \ge 0,\) which assures that the total population \(N_ \left( t \right)\) is constant.

Now,

$$N_ \left( t \right) = S_ \left( t \right) + L_ \left( t \right) + E_ \left( t \right) + I_ \left( t \right) + R_ \left( t \right) + V_ \left( t \right) + E_^ \left( t \right) + I_^ \left( t \right) + R_^ \left( t \right) = 1 \forall } > 0.$$

As

$$S_ \left( t \right) > 0,L_ \left( t \right) > 0,E_ \left( t \right) > 0,I_ \left( t \right) > 0,R_ \left( t \right) > 0,V_ \left( t \right) > 0,E_^ \left( t \right) > 0,I_^ \left( t \right) > 0,R_^ \left( t \right) > 0;$$

as a result, we accomplish that \(0 < S_ \left( t \right) < 1, 0 < E_ \left( t \right) < 1, 0 < I_ \left( t \right) < 1, 0 < R_ \left( t \right) < 1, 0 < V_ \left( t \right) < 1,0 < E_^ \left( t \right) < 1,0 < I_^ \left( t \right) < 1,0\left\langle ^ \left( t \right)\left\langle \right\rangle 0, \forall t} \right\rangle 0\),

which concludes the proof.

Existence and uniqueness of the fractional-order derivative solutions Theorem 6

The current model’s kernels of Eq. (8) accomplish prominent Lipschitz continuity \(_\ge 0, i=1, 2,\dots ,9.\)

Proof

Suppose,

$$\begin _^} D_^ S\left( t \right) & = - \beta S\left( t \right)\left( \left( t \right)} \right) - \delta ~S\left( t \right) - lS\left( t \right) + l_ L\left( t \right), \\ _^} D_^ L\left( t \right) & = lS\left( t \right) - \left( \right)\beta L\left( t \right)\left( \left( t \right)} \right) - l_ L\left( t \right), \\ _^} D_^ E\left( t \right) & = \beta S\left( t \right)\left( \left( t \right)} \right) + \left( \right)\beta L\left( t \right)\left( \left( t \right)} \right)~ - \alpha E\left( t \right), \\ _^} D_^ I\left( t \right) & = \alpha E\left( t \right) - \gamma I\left( t \right), \\ _^} D_^ R\left( t \right) & = \gamma I\left( t \right), \\ _^} D_^ V\left( t \right) & = \delta ~S\left( t \right) - \left( \right)\beta V\left( t \right)\left( \left( t \right)} \right), \\ _^} D_^ E_ \left( t \right) & = \left( \right)\beta V\left( t \right)\left( \left( t \right)} \right) - \alpha E_ \left( t \right), \\ _^} D_^ I_ \left( t \right) & = \alpha E_ \left( t \right) - \gamma I_ \left( t \right), \\ _^} D_^ R_ \left( t \right) & = \gamma I_ \left( t \right). \\ \end$$

Now,

$$\begin & |f_ \left( , t} \right) - f_ \left( , t} \right)\left| = \right| - \beta \left( } \right)\left( - S_ } \right) - \delta \left( - S_ } \right) - l\left( - S_ } \right)| \\ & = \left| } \right) - \delta - l} \right\}} \right|\left( - S_ } \right) \\ & \le \left\}}\limits_ I + \mathop }}\limits_ I_ } \right)} \right| + \left| \delta \right| + \left| l \right|} \right\}\left| - S_ } \right|} \right| \\ & \le LC_ \left| - S_ } \right|} \right|, \\ \end$$

where \(LC_ = \left\}}\limits_ I + \mathop }}\limits_ I_ } \right)} \right| + \left| \delta \right| + \left| l \right|} \right\},\) which implies that

$$|f_ \left( , t} \right) - f_ \left( , t} \right)\left| } \right|\left| - S_ } \right||.$$

(12.21.1)

Similarly, we obtain the following inequality if we consider the natural death rates in every compartment.

$$|f_ \left( , t} \right) - f_ \left( , t} \right)\left| } \right|\left| - L_ } \right||,$$

(12.21.2)

$$|f_ \left( , t} \right) - f_ \left( , t} \right)\left| } \right|\left| - E_ } \right||,$$

(12.21.3)

$$|f_ \left( , t} \right) - f_ \left( , t} \right)\left| } \right|\left| - I_ } \right||,$$

(12.21.4)

$$|f_ \left( , t} \right) - f_ \left( , t} \right)\left| } \right|\left| - R_ } \right||,$$

(12.21.5)

$$|f_ \left( , t} \right) - f_ \left( , t} \right)\left| } \right|\left| - V_ } \right||,$$

(12.21.6)

$$|f_ \left( } ,~t} \right) - f_ \left( } ,~t} \right)\left| } \right|\left| } - E_} } \right||,$$

(12.21.7)

$$|f_ \left( } ,~t} \right) - f_ \left( } ,~t} \right)\left| } \right|\left| } - I_} } \right||,$$