In this section, we use the generalized Adams–Bashforth–Moulton technique given in [35] to prove the uniqueness of the solution for system (3) as follows:

$$\begin{aligned}& \Psi ^{\alpha - 1}{}^{C}D_{t}^{\alpha } S(t) = Q_{1}\bigl(t,S(t)\bigr), \\& \Psi ^{\alpha - 1}{}^{C}D_{t}^{\alpha } E(t) = Q_{2}\bigl(t,E(t)\bigr), \\& \Psi ^{\alpha - 1}{}^{C}D_{t}^{\alpha } E_{1}(t) = Q_{3}\bigl(t,E_{1}(t)\bigr), \\& \Psi ^{\alpha - 1}{}^{C}D_{t}^{\alpha } E_{2}(t) = Q_{4}\bigl(t,E_{2}(t)\bigr), \\& \Psi ^{\alpha - 1}{}^{C}D_{t}^{\alpha } I(t) = Q_{5}\bigl(t,I(t)\bigr), \\& \Psi ^{\alpha - 1}{}^{C}D_{t}^{\alpha } I_{1}(t) = Q_{6}\bigl(t,I_{1}(t)\bigr), \\& \Psi ^{\alpha - 1}{}^{C}D_{t}^{\alpha } I_{2}(t) = Q_{7}\bigl(t,I_{2}(t)\bigr), \\& \Psi ^{\alpha - 1}{}^{C}D_{t}^{\alpha } T(t) = Q_{8}\bigl(t,T(t)\bigr) , \\& \Psi ^{\alpha - 1}{}^{C}D_{t}^{\alpha } R(t) = Q_{9}\bigl(t,R(t)\bigr). \end{aligned}$$

By using Lemma 3.1, the system is given as

$$\begin{aligned}& S(t) - S(0) = \frac{\Psi ^{1 - \alpha }}{\Gamma \alpha } \int _{0}^{t} Q_{1}(\tau ,S) (t - \tau )^{\alpha - 1}\,d\tau, \\& E(t) - E(0) = \frac{\Psi ^{1 - \alpha }}{\Gamma \alpha } \int _{0}^{t} Q_{2}(\tau ,E) (t - \tau )^{\alpha - 1}\,d\tau , \\& E_{1}(t) - E_{1}(0) = \frac{\Psi ^{1 - \alpha }}{\Gamma \alpha } \int _{0}^{t} Q_{3}(\tau ,E_{1}) (t - \tau )^{\alpha - 1}\,d\tau , \\& E_{2}(t) - E_{2}(0) = \frac{\Psi ^{1 - \alpha }}{\Gamma \alpha } \int _{0}^{t} Q_{4}(\tau ,E_{2}) (t - \tau )^{\alpha - 1}\,d\tau , \\& I(t) - I(0) = \frac{\Psi ^{1 - \alpha }}{\Gamma \alpha } \int _{0}^{t} Q_{5}(\tau ,I) (t - \tau )^{\alpha - 1}\,d\tau , \\& I_{1}(t) - I_{1}(0) = \frac{\Psi ^{1 - \alpha }}{\Gamma \alpha } \int _{0}^{t} Q_{6}(\tau ,I_{1}) (t - \tau )^{\alpha - 1}\,d\tau , \\& I_{2}(t) - I_{2}(0) = \frac{\Psi ^{1 - \alpha }}{\Gamma \alpha } \int _{0}^{t} Q_{7}(\tau ,I_{2}) (t - \tau )^{\alpha - 1}\,d\tau , \\& T(t) - T(0) = \frac{\Psi ^{1 - \alpha }}{\Gamma \alpha } \int _{0}^{t} Q_{8}(\tau ,I) (t - \tau )^{\alpha - 1}\,d\tau, \\& R(t) - R(0) = \frac{\Psi ^{1 - \alpha }}{\Gamma \alpha } \int _{0}^{t} Q_{9}(\tau ,R) (t - \tau )^{\alpha - 1}\,d\tau . \end{aligned}$$

(5)

In the ensuing theorem the kernels \(Q_{i}\), \(i = 1,2,3,4,5,6,7,8,9\), satisfy the Lipschitz condition and contraction.

### Theorem 7.1

*The kernel* \(Q_{1}\) *satisfies the Lipschitz condition and contraction if the inequality given below holds* \(0 \le \lambda + \mu < 1\).

### Proof

For *S* and \(S_{*}\) we have

$$ \bigl\Vert Q_{1}(t,S) - Q_{1}(t,S_{*}) \bigr\Vert \le (\lambda + \mu ) \Vert S - S_{*} \Vert . $$

Suppose that \(d_{1} = \lambda + \mu \), where \(\|S\| \le M_{1}\), \(\|E\| \le M_{2}\), \(\|E_{1}\| \le M_{3}\), \(\|E_{2}\| \le M_{4}\), \(\|I\| \le M_{5}\), \(\|I_{1}\| \le M_{6}\), \(\|I_{2}\| \le M_{7} \|T\| \le M_{8}\) and \(\|R\| \le M_{9}\) is a bounded function. So

$$ \bigl\Vert Q_{1}(t,S) - Q_{1}(t,S_{*}) \bigr\Vert \le d_{1} \bigl\Vert S(t) - S_{*}(t) \bigr\Vert . $$

(6)

Thus, for \(Q_{1}\) the Lipchitz condition is obtained, and if \(0 \le \lambda + \mu < 1\) then \(Q_{1}\) is a contraction.

Similarly, the Lipschitz condition for \(Q_{i}\), \(i = 2,3,4,5,6,7,8,9\), is given as follows:

$$\begin{aligned}& \bigl\Vert Q_{2}(t,E) - Q_{2}(t,E_{*}) \bigr\Vert \le d_{2} \bigl\Vert E(t) - E_{*}(t) \bigr\Vert , \\& \bigl\Vert Q_{3}(t,E_{1}) - Q_{3}(t,E_{1*}) \bigr\Vert \le d_{3} \bigl\Vert E_{1}(t) - E_{1*}(t) \bigr\Vert , \\& \bigl\Vert Q_{4}(t,E_{2}) - Q_{4}(t,E_{2*}) \bigr\Vert \le d_{4} \bigl\Vert E_{2}(t) - E_{2*}(t) \bigr\Vert , \\& \bigl\Vert Q_{5}(t,I) - Q_{5}(t,I_{*}) \bigr\Vert \le d_{5} \bigl\Vert I(t) - I_{*}(t) \bigr\Vert , \\& \bigl\Vert Q_{6}(t,I_{1}) - Q_{6}(t,I_{1*}) \bigr\Vert \le d_{6} \bigl\Vert I_{1}(t) - I_{1*}(t) \bigr\Vert , \\& \bigl\Vert Q_{7}(t,I_{2}) - Q_{7}(t,I_{2*}) \bigr\Vert \le d_{7} \bigl\Vert I_{2}(t) - I_{2*}(t) \bigr\Vert , \\& \bigl\Vert Q_{8}(t,T) - Q_{8}(t,T_{*}) \bigr\Vert \le d_{8} \bigl\Vert T(t) - T_{*}(t) \bigr\Vert , \\& \bigl\Vert Q_{9}(t,R) - Q_{9}(t,R_{*}) \bigr\Vert \le d_{9} \bigl\Vert R(t) - R_{*}(t) \bigr\Vert , \end{aligned}$$

where \(d_{2} = \sigma + \mu \), \(d_{3} = a_{7}\lambda + r_{0} + \mu \), \(d_{4} = a_{4}\lambda + k_{1} + a_{3} + \mu \), \(d_{5} = k_{2} + \mu \), \(d_{6} = r_{1} + \mu \), \(d_{7} = r_{2} + \mu \), \(d_{8} = a_{1}a_{5}\lambda + r_{3} + \mu \),and \(d_{9} = a_{8} + \mu \) are bounded functions, if \(0 \le d_{i} < 1\), \(i = 2,3,4,5,6,7,8,9\), then \(Q_{i}\), \(i = 2,3,4,5,6,7,8,9\), are contractions. According to system (5), consider the following recursive forms:

$$\begin{aligned}& P_{1n}(t) = S_{n}(t) - S_{n - 1}(0) = \frac{\Psi ^{1 - \alpha }}{\Gamma \alpha } \int _{0}^{t} \bigl[Q_{1}(\tau ,S_{n - 1}) - Q_{1}(\tau ,S_{n - 2})\bigr](t - \tau )^{\alpha - 1}\,d\tau, \\& P_{2n}(t) = E_{n}(t) - E_{n - 1}(0) = \frac{\Psi ^{1 - \alpha }}{\Gamma \alpha } \int _{0}^{t} \bigl[Q_{2}(\tau ,E_{n - 1}) - Q_{2}(\tau ,E_{n - 2})\bigr](t - \tau )^{\alpha - 1}\,d\tau, \\& P_{3n}(t) = (E_{1})_{n}(t) - (E_{1})_{n - 1}(0) \\& \hphantom{P_{3n}(t)}= \frac{\Psi ^{1 - \alpha }}{\Gamma \alpha } \int _{0}^{t} \bigl[Q_{3}\bigl(\tau ,(E_{1})_{n - 1}\bigr) - Q_{3}\bigl(\tau ,(E_{1})_{n - 2}\bigr)\bigr](t - \tau )^{\alpha - 1}\,d\tau, \\& P_{4n}(t) = (E_{2})_{n}(t) - (E_{2})_{n - 1}(0) \\& \hphantom{P_{4n}(t)} = \frac{\Psi ^{1 - \alpha }}{\Gamma \alpha } \int _{0}^{t} \bigl[Q_{4}\bigl(\tau ,(E_{2})_{n - 1}\bigr) - Q_{4}\bigl(\tau ,(E_{2})_{n - 2}\bigr)\bigr](t - \tau )^{\alpha - 1}\,d\tau, \\& P_{5n}(t) = I_{n}(t) - I_{n - 1}(0) = \frac{\Psi ^{1 - \alpha }}{\Gamma \alpha } \int _{0}^{t} \bigl[Q_{5}(\tau ,I_{n - 1}) - Q_{5}(\tau ,I_{n - 2})\bigr](t - \tau )^{\alpha - 1}\,d\tau, \\& P_{6n}(t) = (I_{1})_{n}(t) - (I_{1})_{n - 1}(0) = \frac{\Psi ^{1 - \alpha }}{\Gamma \alpha } \int _{0}^{t} \bigl[Q_{6}\bigl(\tau ,(I_{1})_{n - 1}\bigr) - Q_{6}\bigl(\tau ,(I_{1})_{n - 2}\bigr)\bigr](t - \tau )^{\alpha - 1}\,d\tau, \\& P_{7n}(t) = (I_{2})_{n}(t) - (I_{2})_{n - 1}(0) = \frac{\Psi ^{1 - \alpha }}{\Gamma \alpha } \int _{0}^{t} \bigl[Q_{7}\bigl(\tau ,(I_{2})_{n - 1}\bigr) - Q_{7}\bigl(\tau ,(I_{2})_{n - 2}\bigr)\bigr](t - \tau )^{\alpha - 1}\,d\tau, \\& P_{8n}(t) = T_{n}(t) - T_{n - 1}(0) = \frac{\Psi ^{1 - \alpha }}{\Gamma \alpha } \int _{0}^{t} \bigl[Q_{8}(\tau ,T_{n - 1}) - Q_{8}(\tau ,T_{n - 2})\bigr](t - \tau )^{\alpha - 1}\,d\tau, \\& P_{9n}(t) = R_{n}(t) - R_{n - 1}(0) = \frac{\Psi ^{1 - \alpha }}{\Gamma \alpha } \int _{0}^{t} \bigl[Q_{9}(\tau ,R_{n - 1}) - Q_{9}(\tau ,R_{n - 2})\bigr](t - \tau )^{\alpha - 1}\,d\tau \end{aligned}$$

with the initial conditions

$$\begin{aligned}& S_{0}(t) = S(0),\qquad E_{0}(t) = E(0),\qquad (E_{1})_{0}(t) = (E_{1}) (0), \\& (E_{2})_{0}(t) = (E_{2}) (0),\qquad I_{0}(t) = I(0), \\& (I_{1})_{0}(t) = (I_{1}) (0),\qquad (I_{2})_{0}(t) = (I_{2}) (0),\qquad T_{0}(t) = T(0)\quad \text{and}\quad R_{0}(t) = R(0). \end{aligned}$$

We take the norm of the first equation in the above system, then

$$\begin{aligned} \bigl\Vert P_{1n}(t) \bigr\Vert =& \bigl\Vert S_{n}(t) - S_{n - 1}(0) \bigr\Vert \\ =& \biggl\Vert \frac{\Psi ^{1 - \alpha }}{\Gamma \alpha } \int _{0}^{t} \bigl[Q_{1}(\tau ,S_{n - 1}) - Q_{1}(\tau ,S_{n - 2})\bigr](t - \tau )^{\alpha - 1}\,d\tau \biggr\Vert \\ \le& \frac{\Psi ^{1 - \alpha }}{\Gamma \alpha } \int _{0}^{t} \bigl\Vert \bigl[Q_{1}( \tau ,S_{n - 1}) - Q_{1}(\tau ,S_{n - 2})\bigr](t - \tau )^{\alpha - 1} \bigr\Vert \,d\tau . \end{aligned}$$

With Lipchitz condition (6), we have

$$ \bigl\Vert P_{1n}(t) \bigr\Vert \le \frac{\Psi ^{1 - \alpha }}{\Gamma \alpha } d_{1} \int _{0}^{t} \bigl\Vert P_{1(n - 1)}(\tau ) \bigr\Vert \,d\tau . $$

(7)

As a similar way, we obtained

$$ \bigl\Vert P_{in}(t) \bigr\Vert \le \frac{\Psi ^{1 - \alpha }}{\Gamma \alpha } d_{i} \int _{0}^{t} \bigl\Vert P_{i(n - 1)}(\tau ) \bigr\Vert \,d\tau ,\quad i = 2,3,\ldots,9. $$

(8)

Thus, we can write that

$$\begin{aligned}& S_{n}(t) = \sum_{j = 1}^{n} P_{1j}(t),\qquad E_{n}(t) = \sum _{j = 1}^{n} P_{2j}(t),\qquad (E_{1})_{n}(t) = \sum_{j = 1}^{n} P_{3j}(t), \\& (E_{2})_{n}(t) = \sum_{j = 1}^{n} P_{4j}(t), \\& I_{n}(t) = \sum_{j = 1}^{n} P_{5j}(t),\qquad (I_{1})_{n}(t) = \sum _{j = 1}^{n} P_{6j}(t),\qquad (I_{2})_{n}(t) = \sum_{j = 1}^{n} P_{7j}(t), \\& T_{n}(t) = \sum_{j = 1}^{n} P_{8j}(t), \qquad R_{n}(t) = \sum _{j = 1}^{n} P_{9j}(t). \end{aligned}$$

□

The existence of a solution is given in the next theorem.

### Theorem 7.2

*A system of solutions given by the fractional SEITR model* (1) *exists if there exists* \(t_{1}\) *such that*

$$ \frac{\Psi ^{1 - \alpha }}{\Gamma \alpha } t_{1}d_{i} < 1. $$

### Proof

From the recursive technique and Eq. (7) and Eq. (8) we conclude that

$$\begin{aligned}& \bigl\Vert P_{1n}(t) \bigr\Vert \le \bigl\Vert S_{n}(0) \bigr\Vert \biggl[ \frac{\Psi ^{1 - \alpha }}{\Gamma \alpha } d_{1}t \biggr]^{n}, \\& \bigl\Vert P_{2n}(t) \bigr\Vert \le \bigl\Vert E_{n}(0) \bigr\Vert \biggl[ \frac{\Psi ^{1 - \alpha }}{\Gamma \alpha } d_{2}t \biggr]^{n}, \\& \bigl\Vert P_{3n}(t) \bigr\Vert \le \bigl\Vert (E_{1})_{n}(0) \bigr\Vert \biggl[ \frac{\Psi ^{1 - \alpha }}{\Gamma \alpha } d_{3}t \biggr]^{n}, \\& \bigl\Vert P_{4n}(t) \bigr\Vert \le \bigl\Vert (E_{2})_{n}(0) \bigr\Vert \biggl[ \frac{\Psi ^{1 - \alpha }}{\Gamma \alpha } d_{4}t \biggr]^{n},\qquad \bigl\Vert P_{5n}(t) \bigr\Vert \le \bigl\Vert I_{n}(0) \bigr\Vert \biggl[ \frac{\Psi ^{1 - \alpha }}{\Gamma \alpha } d_{5}t \biggr]^{n}, \\& \bigl\Vert P_{6n}(t) \bigr\Vert \le \bigl\Vert (I_{1})_{n}(0) \bigr\Vert \biggl[ \frac{\Psi ^{1 - \alpha }}{\Gamma \alpha } d_{6}t \biggr]^{n}, \\& \bigl\Vert P_{7n}(t) \bigr\Vert \le \bigl\Vert (I_{2})_{n}(0) \bigr\Vert \biggl[ \frac{\Psi ^{1 - \alpha }}{\Gamma \alpha } d_{7}t \biggr]^{n},\qquad \bigl\Vert P_{8n}(t) \bigr\Vert \le \bigl\Vert T_{n}(0) \bigr\Vert \biggl[ \frac{\Psi ^{1 - \alpha }}{\Gamma \alpha } d_{8}t \biggr]^{n}, \\& \bigl\Vert P_{9n}(t) \bigr\Vert \le \bigl\Vert R_{n}(0) \bigr\Vert \biggl[ \frac{\chi ^{1 - \alpha }}{\Gamma \alpha } d_{9}t \biggr]^{n}. \end{aligned}$$

Thus, the system has a continuous solution. To prove that the above functions construct a solution for model (2), we assume that

$$\begin{aligned}& S(t) - S(0) = S_{n}(t) - W_{1n}(t),\qquad E(t) - E(0) = E_{n}(t) - W_{2n}(t), \\& (E_{1}) (t) - (E_{1}) (0) = (E_{1})_{n}(t) - W_{3n}(t), \\& (E_{2}) (t) - (E_{2}) (0) = (E_{2})_{n}(t) - W_{4n}(t),\qquad I(t) - I(0) = I_{n}(t) - W_{5n}(t), \\& (I_{1}) (t) - (I_{1}) (0) = (I_{1})_{n}(t) - W_{6n}(t), \\& (I_{2}) (t) - (I_{2}) (0) = (I_{2})_{n}(t) - W_{7n}(t),\qquad T(t) - T(0) = T_{n}(t) - W_{8n}(t), \\& R(t) - R(0) = R_{n}(t) - W_{9n}(t). \end{aligned}$$

So

$$ \bigl\Vert W_{1n}(t) \bigr\Vert \le \frac{\Psi ^{1 - \alpha }}{\Gamma \alpha } \int _{0}^{t} \bigl\Vert Q_{1}(\tau ,S) - Q_{1}(\tau ,S_{n - 1}) \bigr\Vert \,d\tau \le \frac{\Psi ^{1 - \alpha }}{\Gamma \alpha } d_{1} \Vert S - S_{n - 1} \Vert t. $$

By repeating the method, we obtain

$$ \bigl\Vert W_{1n}(t) \bigr\Vert \le \biggl[ \frac{\Psi ^{1 - \alpha }}{\Gamma \alpha } t \biggr]^{n + 1}d_{1}^{n + 1}h. $$

At \(t_{1}\), we get

$$ \bigl\Vert W_{1n}(t) \bigr\Vert \le \biggl[ \frac{\Psi ^{1 - \alpha }}{\Gamma \alpha } t_{1} \biggr]^{n + 1}d_{1}^{n + 1}h. $$

As *n* approaches to ∞, this implies \(\Vert W_{1n}(t) \Vert \to 0\). Similarly, we can obtain \(\Vert W_{in}(t) \Vert \to 0\), \(i = 2,3,4,5,6,7,8,9\). Hence the theorem is proved.

To prove the uniqueness of the solution, consider that the system has another solution such as \(S_{\varpi } (t)\), \(E_{\varpi } (t)\), \(E_{1\varpi } (t)\), \(E_{2\varpi } (t)\), \(I_{\varpi } (t)\), \(I_{1\varpi } (t)\), \(I_{2\varpi } (t)\), \(T_{\varpi } (t)\), and \(R_{\varpi } (t)\), then we have

$$ S(t) - S_{\varpi } (t) = \frac{\Psi ^{1 - \alpha }}{\Gamma \alpha } \int _{0}^{t} \bigl(Q_{1}(\tau ,S) - Q_{1}(\tau ,S_{\varpi } ) \bigr)\,d\tau . $$

We take the norm of this equation

$$ \bigl\Vert S(t) - S_{\varpi } (t) \bigr\Vert \le \frac{\Psi ^{1 - \alpha }}{\Gamma \alpha } \int _{0}^{t} \bigl\Vert \bigl(Q_{1}( \tau ,S) - Q_{1}(\tau ,S_{\varpi } )\bigr) \bigr\Vert \,d\tau . $$

It follows from Lipschitz condition (3) that

$$ \bigl\Vert S(t) - S_{\varpi } (t) \bigr\Vert \le \frac{\Psi ^{1 - \alpha }}{\Gamma \alpha } d_{1}t \bigl\Vert S(t) - S_{\varpi } (t) \bigr\Vert . $$

Thus,

$$ \bigl\Vert S(t) - S_{\varpi } (t) \bigr\Vert \biggl( 1 - \frac{\Psi ^{1 - \alpha }}{ \Gamma \alpha } d_{1}t \biggr) \le 0. $$

(9)

□

### Theorem 7.3

*The solution of model* (3) *is unique if the following condition holds*:

$$ \biggl( 1 - \frac{\Psi ^{1 - \alpha }}{\Gamma \alpha } d_{1}t \biggr) > 0. $$

### Proof

Suppose that condition (9) holds

$$ \bigl\Vert S(t) - S_{\varpi } (t) \bigr\Vert \biggl( 1 - \frac{\Psi ^{1 - \alpha }}{\Gamma \alpha } d_{1}t \biggr) \le 0. $$

Then \(\Vert S(t) - S_{\varpi } (t) \Vert = 0\). Therefore, we get \(S(t) = S_{\varpi } (t)\). Likewise, the same equality can be shown for *E*, \(E_{1}\), \(E_{2}\), *I*, \(I_{1}\), \(I_{2}\), *T*, and *R*. □