We can express Eq. (7) as
$$\begin{aligned} \begin{array}{l} _0^{PCAB}D_t^{\alpha } S(t) = \left\{ \begin{array}{l}D_t^{\alpha } S\left( t \right) = {G_1^ \cdot }\left( {S,E,I,R,D,H,B,C,t} \right) ,0
(10)
We will now determine whether there is a solution to the hypothetical piecewise derivable function and what the specifics of that solution are. To do this, we can utilize the system (10) from lemma 2.1. Additionally, we can write the following for more clarification.
$$\begin{aligned} _0^{PCAB}D_t^{{\alpha }}\chi (t) = {G^ \cdot }(t,\chi (t)), \end{aligned}$$
and
$$\begin{aligned} \chi (t) = \left\{ {\begin{array}{l}{{\chi _0} + \int \limits _0^t {{G^ \cdot }\left( {\varsigma ,\chi (\varsigma )} \right) d\varsigma ,0
(11)
where
$$\begin{aligned} \chi = \left\{ {\begin{array}{l}{S(t),}\\ {E(t),}\\ {I(t),}\\ {R(t),}\\ {D(t),}\\ {H(t),}\\ {B(t),}\\ {C(t).}\end{array}} \right. ,\,\,{\chi _0} = \left\{ {\begin{array}{l}{S(0),}\\ {E(0),}\\ {I(0),}\\ {R(0),}\\ {D(0),}\\ {H(0),}\\ {B(0),}\\ {C(0).}\end{array}} \right. ,\,\,{\chi _{{t_1}}} = \left\{ {\begin{array}{l}{S({t_1}),}\\ {E({t_1}),}\\ {I({t_1}),}\\ {R({t_1}),}\\ {D({t_1}),}\\ {H({t_1}),}\\ {B({t_1}),}\\ {C({t_1}).}\end{array}} \right. \end{aligned}$$
$$\begin{aligned} {G^ \cdot }(t,\chi (t)) = \left\{ {\begin{array}{l}{{G_1^ \cdot } = \left\{ {\begin{array}{l}{\frac{d}{{dt}}{\Im _1}(S,E,I,R,D,H,B,C),}\\ {^{AB}{\Im _1}(S,E,I,R,D,H,B,C).}\end{array}} \right. }\\ {{G_2^ \cdot } = \left\{ {\begin{array}{l}{\frac{d}{{dt}}{\Im _2}(S,E,I,R,D,H,B,C),}\\ {^{AB}{\Im _2}(S,E,I,R,D,H,B,C).}\end{array}} \right. }\\ {{G_3^ \cdot } = \left\{ {\begin{array}{l}{\frac{d}{{dt}}{\Im _3}(S,E,I,R,D,H,B,C),}\\ {^{AB}{\Im _3}(S,E,I,R,D,H,B,C).}\end{array}} \right. }\\ {{G_4^ \cdot } = \left\{ {\begin{array}{l}{\frac{d}{{dt}}{\Im _4}(S,E,I,R,D,H,B,C),}\\ {^{AB}{\Im _4}(S,E,I,R,D,H,B,C).}\end{array}} \right. }\\ {{G_5^ \cdot } = \left\{ {\begin{array}{l}{\frac{d}{{dt}}{\Im _5}(S,E,I,R,D,H,B,C),}\\ {^{AB}{\Im _5}(S,E,I,R,D,H,B,C).}\end{array}} \right. }\\ {{G_6^ \cdot } = \left\{ {\begin{array}{l}{\frac{d}{{dt}}{\Im _6}(S,E,I,R,D,H,B,C),}\\ {^{AB}{\Im _6}(S,E,I,R,D,H,B,C).}\end{array}} \right. }\\ {{G_7^ \cdot } = \left\{ {\begin{array}{l}{\frac{d}{{dt}}{\Im _7}(S,E,I,R,D,H,B,C),}\\ {^{AB}{\Im _7}(S,E,I,R,D,H,B,C).}\end{array}} \right. }\\ {{G_8^ \cdot } = \left\{ {\begin{array}{l}{\frac{d}{{dt}}{\Im _8}(S,E,I,R,D,H,B,C),}\\ {^{AB}{\Im _8}(S,E,I,R,D,H,B,C).}\end{array}} \right. }\end{array}} \right. \end{aligned}$$
(12)
With \(\infty> {t_2} \ge t> {t_1} > 0\) and the norm-containing Banach space employed Let \(\mathbb {B} = \mathbb {C}[0,\mathfrak {T}].A\)
$$\begin{aligned} \left\| \chi \right\| = {\max }_{t \in [0,\mathfrak {T}]} \left| \chi \right| . \end{aligned}$$
(13)
Theorem 4.1
(Leray-Schauder fixed point theorem52). Assume that X is a Banach space and that \(\mathcal {D} \subset \mathcal {G}\) is an open set with \(0 \in \mathcal {D}\). Let \(\mathcal {G} \subset X\) also be a bounded, closed, convex set. Next, with the compact and continuous mapping \(\mathfrak {T} : \mathcal {D}^*\mathcal {G},\) one can either:
\((\mathcal {J_1})\exists j \epsilon \mathcal {D}^*\) such that \(z=\mathfrak {T}(j),\) or
\((\mathcal {J_1})\exists \partial \mathcal {D}, \nu \in (0,1)\) such that \(j=\nu \mathfrak {T}(j).\)
Theorem 4.2
Let \({G^ \cdot } \in \mathbb {C}\mathrm{ }\left( {\mathfrak {T} \times \mathfrak {F},\mathfrak {F}} \right).\) If \({\mathfrak {Z}_1}\exists \mathfrak {P} \in {\mathcal {K}^{ – 1}}\left( {\mathfrak {T},{\mathbb {R}^ + }} \right)\) and \(\exists A \in C ([0,\infty ),(0, \infty ))\) A non decreasing such that \(\chi \in X,\) and \(\forall t \in \mathfrak {T}\)
$$\begin{aligned} \left| {G^\cdot \left( {t,\chi \left( t \right) } \right) } \right| \le \psi \left( t \right) A\left( {\left| {\chi \left( t \right) } \right| } \right) , \end{aligned}$$
$$\begin{aligned} {\mathfrak {Z}_2}\exists {\mathfrak {P}_2} \in {\psi _1} > 1 \end{aligned}$$
such that
$$\begin{aligned} \left\{ \begin{array}{l}\frac{{{\psi _1}}}{{{\chi _0}}}> 1,\\ \frac{{{\psi _1}}}{{{\chi _0}\left[ {\frac{{\left( {1 – {\alpha }} \right) }}{{\textbf{AB}\left( {{\alpha }} \right) }} + \frac{{{\alpha }}}{{\textbf{AB}\left( {{\alpha }} \right) \Gamma \left( {{\alpha }} \right) }}} \right] \mathfrak {P}_0^*\mathfrak {U}\left( {{\psi _1}} \right) }}\end{array}>0 \right. \end{aligned}$$
(14)
with \(\mathfrak {P}_0^* = {\sup _{t \in \mathfrak {T}}}\left| {\mathfrak {P}\left( t \right) } \right|,\) then there exists a solution of system (7).
Proof
Initially, let us take into consideration \(\mathfrak {T}:\mathfrak {F} \rightarrow \mathfrak {F},\) as stated in (11). We suppose
$$\begin{aligned} {\mathbb {N}_w} = \left\{ {\chi \in \mathfrak {F}:{{\left\| \chi \right\| }_\mathfrak {F}} \le w} \right\} , \end{aligned}$$
for a certain \(w > 0.\) It follows that since \(G^\cdot\) is continuous, so is \(\mathfrak {T}\). We obtain from (\(\mathfrak {Z}_1\))
$$\begin{aligned} \mathfrak {T}(\chi ) = \left\{ {\begin{array}{l}{{\chi _0} + \int \limits _0^{t_1} {{G^ \cdot }(t,\chi (\varsigma ))d\varsigma ,0
(15)
$$\begin{aligned} \left| {\mathfrak {T}(\chi )} \right| \le \left\{ {\begin{array}{l}{\left| {{\chi _0}} \right| + \int \limits _0^{{t_1}} {\left| {{G^ \cdot }(t,\chi (\varsigma ))} \right| d\varsigma ,} }\\ {\left| {\chi ({t_1})} \right| + \frac{{1 – \alpha }}{{{\textbf {AB}}\left( \alpha \right) }}\left| {\chi (t)} \right| + \frac{\alpha }{{{\textbf {AB}}\left( \eta \right) \Gamma (\alpha )}}\int \limits _{{t_1}}^{{t_2}} {{{(t – \varsigma )}^{\alpha – 1}}\left| {{G^ \cdot }(t,\chi (\varsigma ))} \right| } d\varsigma .}\end{array}} \right. \end{aligned}$$
$$\begin{aligned} \le \left\{ {\begin{array}{l}{\left| {{\chi _0}} \right| + \int \limits _0^{{t_1}} {\mathfrak {P}\left( \varsigma \right) \mathfrak {U}\left| {(\chi (\varsigma ))} \right| d\varsigma ,} }\\ {\left| {\chi ({t_1})} \right| + \frac{{1 – {\alpha }}}{{\mathbf{{AB}}\left( {{\alpha }} \right) }}\left| {\chi (t)} \right| + \frac{{{\alpha }}}{{\mathbf{{AB}}\left( \eta \right) \Gamma ({\alpha })}}\int \limits _{{t_1}}^{{t_2}} {{{(t – \varsigma )}^{{\alpha } – 1}}\mathfrak {P}\left( \varsigma \right) U\left| {(\chi (\varsigma ))} \right| } d\varsigma ,}\end{array}} \right. \end{aligned}$$
$$\begin{aligned} \le \left\{ {\begin{array}{l}{ {{\chi _0}} + \mathfrak {P}_0^*\mathfrak {U}\left( w \right) }\\ { {\chi ({t_1})} + \frac{{1 – {\alpha }}}{{\mathbf{{AB}}\left( {{\alpha }} \right) }}\mathfrak {P}_0^*\mathfrak {U}\left( w \right) + \frac{{{\alpha }}}{{\mathbf{{AB}}\left( \eta \right) \Gamma ({\alpha })}}\mathfrak {P}_0^*\mathfrak {U}\left( w \right) ,}\end{array}} \right. \end{aligned}$$
for \(\chi (0) \in \mathbb {N}_{w}.\) Hence,
$$\begin{aligned} {\left\| {\mathfrak {T}\chi } \right\| _X} \le \left\{ {\begin{array}{l}{{\chi _0} + \mathfrak {P}_0^*\mathfrak {U}\left( w \right)
(16)
Thus, \(\mathfrak {T}\) is uniformly bounded on X. When \({t_j} \rightarrow {t_i},\) then
$$\begin{aligned} \left| {\mathfrak {T}(\chi )({t_i}) – \mathfrak {T}(\chi )({t_j})} \right| \rightarrow 0. \end{aligned}$$
(17)
Thus, \([0,{t_1}]\) provides evidence of the operator \(\mathfrak {T}\) equicontinuity. Consider the following: \({t_i},{t_j} \in [{t_1},\mathfrak {T}]\) in the ABC context as
$$\begin{aligned} & \left| {\mathfrak {T}(\chi )({t_i}) – \mathfrak {T}(\chi )({t_j})} \right| , \\ & = \left| \begin{array}{l}\frac{{1 – \alpha }}{{{\textbf {AB}}\left( \alpha \right) }} + \frac{\alpha }{{{\textbf {AB}}\left( \alpha \right) \Gamma (\alpha )}}\int \limits _0^{{t_i}} {{{(t – \varsigma )}^{\alpha – 1}}{G^ \cdot }\left( {\varsigma ,\chi (\varsigma )} \right) d\varsigma } \\ – \frac{{1 – \alpha }}{{{\textbf {AB}}\left( \alpha \right) }} + \frac{\alpha }{{{\textbf {AB}}\left( \alpha \right) \Gamma (\alpha )}}\int \limits _0^{{t_i}} {{{(t – \varsigma )}^{\alpha – 1}}{G^ \cdot }\left( {\varsigma ,\chi (\varsigma )} \right) d\varsigma } \end{array} \right| , \\ & \begin{array}{l} \le \frac{\alpha }{{{\textbf {AB}}\left( \alpha \right) \Gamma (\alpha )}}\int \limits _0^{{t_j}} {\left[ {{{({t_j} – \varsigma )}^{\alpha – 1}} – {{({t_i} – \varsigma )}^{\alpha – 1}}} \right] } \left| {{G^ \cdot }(\varsigma ,\chi (\varsigma ))} \right| d\varsigma \\ + \frac{\alpha }{{{\textbf {AB}}\left( \alpha \right) \Gamma (\alpha )}}\int \limits _{{t_j}}^{{t_i}} {{{({t_i} – \varsigma )}^{\alpha – 1}}} \left| {{G^ \cdot }(\varsigma ,\chi (\varsigma ))} \right| d\varsigma ,\end{array} \\ & \le \frac{\alpha }{{{\textbf {AB}}\left( \alpha \right) \Gamma (\alpha )}}\left[ {\int \limits _0^{{t_j}} {\left\{ {{{({t_j} – \varsigma )}^{\alpha – 1}} – {{({t_i} – \varsigma )}^{\alpha – 1}}} \right\} d\varsigma + } \frac{1}{{\Gamma (\alpha )}}\int \limits _{{t_j}}^{{t_i}} {{{({t_i} – \varsigma )}^{\alpha – 1}}} d\varsigma } \right] \left( {{C_{G^ \cdot }}\left| \chi \right| + {\mathfrak {N}_{G^ \cdot }}} \right) , \\ & \le \frac{{\alpha \left( {{C_{G^ \cdot }}\left| \chi \right| + {\mathfrak {N}_{G^ \cdot }}} \right) }}{{{\textbf {AB}}\left( \alpha \right) \Gamma (\alpha + 1)}}\left[ {t_i^{\alpha } + t_j^{\alpha } + 2{{\left( {{t_i} – {t_j}} \right) }^{\alpha } }} \right] . \end{aligned}$$
If \({t_j} \rightarrow {t_i},\) then
$$\begin{aligned} \left| {\mathfrak {T}(\chi )({t_i}) – \mathfrak {T}(\chi )({t_j})} \right| \rightarrow 0.\, \end{aligned}$$
\(t_i\longrightarrow t_j\) at this time. The ArzelÃ!‘-Ascoli theorem uses this to determine the equicontinuity of \(\mathfrak {T}\) and, in turn, the compactness of \(\mathfrak {T}\) on \(\mathbb {N}.\) We have one of \((\mathcal {J}_1)\) or (\(\mathcal {J}_2\)), as Theorem 4.1 is satisfied on \(\mathfrak {T}\). We set from (\(\mathfrak {Z_2}\)).
$$\begin{aligned} \Psi = \left\{ {\chi \in X:{{\left\| \chi \right\| }_X}
for some \(\psi _1 > 0\), such that
$$\begin{aligned} \left\{ {\begin{array}{l}{{\chi _0} + \mathfrak {P}_0^*\mathfrak {U}\left( w \right)
From (\(\mathfrak {Z}_1\)) and (16), we have
$$\begin{aligned} {\left\| {\mathfrak {T}\chi } \right\| _X} \le \left\{ {\begin{array}{l}{{\chi _0} + \mathfrak {P}_0^*\mathfrak {U}\left( {{{\left\| \chi \right\| }_X}} \right) ,}\\ {\chi ({t_1}) + \left[ {\frac{{1 – {\alpha }}}{{\mathbf{{AB}}\left( {{\alpha }} \right) }} + \frac{{{\alpha }}}{{\mathbf{{AB}}\left( \eta \right) \Gamma ({\alpha })}}} \right] \mathfrak {P}_0^*\mathfrak {U}\left( {{{\left\| \chi \right\| }_X}} \right) .}\end{array}} \right. \end{aligned}$$
(18)
Suppose that there are \(\chi \in \partial \Psi\) and \(0 such that \(\chi = \nu \mathfrak {T}(\chi ).\) Then, by (37), we write
$$\begin{aligned} \psi _{1} & = \left\| \chi \right\|_{X} = \nu \left\| {{\mathfrak{T}}\chi } \right\|_{X}
which is incorrect. Therefore, under Theorem 4.1, \(\mathfrak {T}\) admits a fixed point in \(\Psi\) and (\(\mathcal {J}_2\)) is not satisfied. This demonstrates that a solution exists (7). \(\square\)
Uniqueness
To demonstrate the uniqueness of our solution to the problem (7), we first look at a Lipschitz property for the model (7).
Lemma 4.1
Consider \(S,E,I,R,D,H,B,C,S^*, E^*, I^*, R^*,D^*,H^*,B^*,C^*\in \mathbb {C}={\textbf {C}}(\mathfrak {J}, \mathbb {R})\) and let \(\mathcal {Q}\)\(\left\| S \right\| \le {\eta _1}.s\) \(\left\| E \right\| \le {\eta _2}\),\(\left\| I \right\| \le {\eta _3}\),\(\left\| R \right\| \le {\eta _4}\),\(\left\| D \right\| \le {\eta _5}\),\(\left\| H \right\| \le {\eta _6}\), \(\left\| B \right\| \le {\eta _7}\),\(\left\| C \right\| \le {\eta _8}\) for some constants \(\eta _1,\eta _2,…,\eta _8>0.\) Then, \(G_1^ \cdot , G_2^ \cdot , G_3^ \cdot ,…, G_8^ \cdot\) satisfy the Lipschitz property with constants \(\wp _1,\wp _2,…,\wp _8 > 0\) with regard to the pertinent components, as specified in (12), where
$$\begin{aligned} \begin{array}{l}{\wp _1} = \frac{{{\beta _i}}}{N}{w_1} + \frac{{{\beta _h}}}{N}{w_2} + \frac{{{\beta _d}}}{N}{w_3} + \frac{{{\beta _r}}}{N}{w_4} + \mu ,\\ {\wp _2} = – \left( {\sigma + \mu } \right) ,\,\,{\wp _3} = {\gamma _1} + \varepsilon + \tau + \mu ,\\ {\wp _4} = {\gamma _3} + \mu ,\,\,\,{\wp _5} = {\delta _1} + \xi ,\\ {\wp _6} = {\gamma _2} + {\delta _2} + \mu ,\,\,{\wp _7} = \xi ,\,{\wp _8} = \mu .\end{array} \end{aligned}$$
Proof: \(S,S^*\mathbb {C}={\textbf {C}}(\mathfrak {J}, \mathbb {R})\) provides the value we take for \(G_1^\cdot.\)arbitrary, and we have got
$$\begin{aligned} \left\| {G_{1}^{ \cdot } – G_{1}^{*} } \right\| & = \left\| {\begin{array}{*{20}l} {\left( {\mu {\mathbf{N}} – \frac{{\beta _{i} }}{{\mathbf{N}}}SI – \frac{{\beta _{h} }}{{\mathbf{N}}}SH – \frac{{\beta _{d} }}{{\mathbf{N}}}SD – \frac{{\beta _{r} }}{{\mathbf{N}}}SR – \mu S} \right)} \hfill \\ { – \left( {\mu {\mathbf{N}} – \frac{{\beta _{i} }}{{\mathbf{N}}}S^{*} I – \frac{{\beta _{h} }}{{\mathbf{N}}}S^{*} H – \frac{{\beta _{d} }}{{\mathbf{N}}}S^{*} D – \frac{{\beta _{r} }}{{\mathbf{N}}}S^{*} R – \mu S} \right)} \hfill \\ \end{array} } \right\| \\ & \le \left[ {\frac{{\beta _{i} }}{{\mathbf{N}}}w_{1} + \frac{{\beta _{h} }}{{\mathbf{N}}}w_{2} + \frac{{\beta _{d} }}{{\mathbf{N}}}w_{3} + \frac{{\beta _{r} }}{{\mathbf{N}}}w_{4} + \mu } \right]\left\| {S – S^{*} } \right\| \le \wp _{1} \left\| {S – S^{*} } \right\|. \\ \end{aligned}$$
(19)
We determine that, under the constant \(\wp _1 > 0\), \(G_1^\cdot\) is Lipschitz with regard to S from (38). In \(G_2^\cdot\), we estimate by selecting at arbitrary \(E,E^*\mathbb {C}={\textbf {C}}(\mathfrak {J}, \mathbb {R})\).
$$\begin{aligned} \left\| {G_2^ \cdot – G_2^*} \right\| \end{aligned}$$
$$\begin{aligned} \begin{aligned}&= \left\| \begin{array}{l}\left( {\frac{{{\beta _i}}}{{\textbf {N}}}SI + \frac{{{\beta _h}}}{{\textbf {N}}}SH + \frac{{{\beta _d}}}{{\textbf {N}}}SD + \frac{{{\beta _r}}}{{\textbf {N}}}SR + \sigma E + \mu E} \right) \\ – \left( {\frac{{{\beta _i}}}{{\textbf {N}}}S I + \frac{{{\beta _h}}}{{\textbf {N}}}S H + \frac{{{\beta _d}}}{{\textbf {N}}}SD + \frac{{{\beta _r}}}{{\textbf {N}}}SR + \sigma E^*+ \mu E^*} \right) \end{array} \right\| \\ &\le – \left[ {\sigma + \mu } \right] \left\| {E – E^*} \right\| \\ &\le {\wp _2}\left\| {E – E^*} \right\| .\end{aligned} \end{aligned}$$
(20)
We determine that, under the constant \(\wp _2 > 0\), \(G_2^\cdot\) is Lipschitz with regard to S from (38). In \(G_3^\cdot\), we estimate by selecting at arbitrary \(I,I^*\mathbb {C}={\textbf {C}}(\mathfrak {J}, \mathbb {R})\).
$$\begin{aligned} \left\| {G_3^ \cdot – G_3^*} \right\| \end{aligned}$$
$$\begin{aligned} \begin{aligned}&= \left\| {\left( {\sigma E – \left( {{\gamma _1} + \varepsilon + \tau + \mu } \right) I} \right) – \left( {\sigma E – \left( {{\gamma _1} + \varepsilon + \tau + \mu } \right) I^*} \right) } \right\| \\ &\le \left[ {{\gamma _1} + \varepsilon + \tau + \mu } \right] \left\| {I – I^*} \right\| \\ &\le {\wp _3}\left\| {I – I^*} \right\| .\end{aligned} \end{aligned}$$
(21)
We determine that, under the constant \(\wp _3 > 0\), \(G_3^\cdot\) is Lipschitz with regard to I from (38). In \(G_4^\cdot\), we estimate by selecting at arbitrary \(R,R^*\mathbb {C}={\textbf {C}}(\mathfrak {J}, \mathbb {R})\).
$$\begin{aligned} \left\| {G_4^ \cdot – G_4^*} \right\| \end{aligned}$$
$$\begin{aligned} \begin{aligned}&= \left\| {\left( {{\gamma _1}I + {\gamma _2}H + \left( {{\gamma _3} + \mu } \right) R} \right) – \left( {{\gamma _1}I + {\gamma _2}H + \left( {{\gamma _3} + \mu } \right) R^*} \right) } \right\| \\ &\le \left[ {{\gamma _3} + \mu } \right] \left\| {R – R^*} \right\| \\ &\le {\wp _4}\left\| {R – R} \right\| .\end{aligned} \end{aligned}$$
(22)
We determine that, under the constant \(\wp _4 > 0\), \(G_4^\cdot\) is Lipschitz with regard to R from (38). In \(G_5^\cdot\), we estimate by selecting at arbitrary \(D,D^*\mathbb {C}={\textbf {C}}(\mathfrak {J}, \mathbb {R})\).
$$\begin{aligned} \left\| {G_5^ \cdot – G_5^*} \right\| \end{aligned}$$
$$\begin{aligned} \begin{aligned}&= \left\| {\left( {\varepsilon I – {\delta _1}D – \xi D} \right) – \left( {\varepsilon I – {\delta _1}D^*- \xi D^*} \right) } \right\| \\ &\le \left[ {{\delta _1} + \xi } \right] \left\| {D – D^*} \right\| \\ &\le {\wp _5}\left\| {H – H^*} \right\| .\end{aligned} \end{aligned}$$
(23)
We determine that, under the constant \(\wp _5 > 0\), \(G_5^\cdot\) is Lipschitz with regard to D from (38). In \(G_6^\cdot\), we estimate by selecting at arbitrary \(H,H^*\mathbb {C}={\textbf {C}}(\mathfrak {J}, \mathbb {R})\).
$$\begin{aligned} \left\| {G_6^ \cdot – G_6^*} \right\| \end{aligned}$$
$$\begin{aligned} \begin{aligned}&= \left\| {\left( {\tau I – \left( {{\gamma _2} + {\delta _2} + \mu } \right) H} \right) – \left( {\tau I – \left( {{\gamma _2} + {\delta _2} + \mu } \right) H^*} \right) } \right\| \\ &\le \left[ {{\gamma _2} + {\delta _2} + \mu } \right] \left\| {H – H^*} \right\| \\ &\le {\wp _6}\left\| {H – H^*} \right\| .\end{aligned} \end{aligned}$$
(24)
We determine that, under the constant \(\wp _6 > 0\), \(G_6^\cdot\) is Lipschitz with regard to H from (38). In \(G_7^\cdot\), we estimate by selecting at arbitrary \(B,B^*\mathbb {C}={\textbf {C}}(\mathfrak {J}, \mathbb {R})\).
$$\begin{aligned} \left\| {G_7^ \cdot – G_7^*} \right\| \end{aligned}$$
$$\begin{aligned} \begin{aligned}&= \left\| {\left( {{\delta _1}D + {\delta _2}H – \xi B} \right) – \left( {{\delta _1}D + {\delta _2}H – \xi B^*} \right) } \right\| \\ &\le \left[ \xi \right] \left\| {B – B^*} \right\| \\ &\le {\wp _7}\left\| {B – B^*} \right\| .\end{aligned} \end{aligned}$$
(25)
We determine that, under the constant \(\wp _7 > 0\), \(G_7^\cdot\) is Lipschitz with regard to C from (38). In \(G_8^\cdot\), we estimate by selecting at arbitrary \(C,C^*\mathbb {C}={\textbf {C}}(\mathfrak {J}, \mathbb {R})\).
$$\begin{aligned} \left\| {G_8^ \cdot – G_8^*} \right\| \end{aligned}$$
$$\begin{aligned} \begin{aligned}&= \left\| {\left( {{\gamma _3}R – \mu C} \right) – \left( {{\gamma _3}R – \mu C^\star } \right) } \right\| \\ &\le \left[ \mu \right] \left\| {C – C^*} \right\| \\ &\le {\wp _8}\left\| {C – C^*} \right\| .\end{aligned} \end{aligned}$$
(26)
With \(\wp > 0\), this demonstrates that \(G_8^\cdot\) is Lipschitzian with respect to C. Thus, given constants \(\wp _1,\wp _2,…,\wp _8> 0,\) the kernel functions \(G_1^\cdot , G_2^\cdot ,…,G_8^\cdot\) are Lipschitz, respectively.
Theorem 4.3
Let \(\mathcal {Q}\) hold if
$$\begin{aligned} \left[ {\frac{{\left( {1 – {\alpha }} \right) }}{{AB\left( {{\alpha }} \right) }} + \frac{{{\alpha }{T^{{\alpha } – 1}}}}{{AB\left( {{\alpha }} \right) \Gamma \left( {{\alpha }} \right) }}} \right] {\delta _j}
(27)
for\(j\in {1,2,3,4,5,6,7,8}\) and where\(\wp _j>0\) are the Lipschitz constants, then there is only one solution for the system (7).
Proof: By contradiction, we carry out the proof. Let us assume that there is an additional solution to the system (7), which is \((S^*, E^*, I^*, R^*,D^*,H^*,B^*,C^*)\), under initial conditions \(S^*=S(0),E^*=E(0),I^*=I(0),R^*=R(0), D^*=D(0),H^*=H(0),B^*=B(0),C^*=C(0).\) Now, we have
$$\begin{aligned} \begin{aligned}S^*&= S\left( 0 \right) + \frac{{\left( {1 – {\alpha }} \right) }}{{{\textbf {AB}}\left( {{\alpha }} \right) }}G_1^ \cdot \left( {t,S^*} \right) \\ &\quad + \frac{{{\alpha }}}{{{\textbf {AB}}\left( {{\alpha }} \right) \Gamma \left( {{\alpha }} \right) }}\int \limits _0^t {{{\left( {t – \varsigma } \right) }^{\omega – 1}}G_1^ \cdot \left( {\varsigma , S^*} \right) } d\varsigma ,\end{aligned} \end{aligned}$$
(28)
In this case, we estimate
$$\begin{aligned} \begin{aligned}\left| {S – {S^*}} \right|&\le \frac{{\left( {1 – {\alpha }} \right) }}{{{\textbf {AB}}\left( {{\alpha }} \right) }}\left| {G_1^ \cdot \left( {t,S} \right) } \right. \left. { – G_1^ \cdot \left( {t,{S^*}} \right) } \right| \\ &+ \frac{{{\alpha }}}{{{\textbf {AB}}\left( {{\alpha }} \right) \Gamma \left( {{\alpha }} \right) }}\int \limits _0^t {{{\left( {t – \varsigma } \right) }^{\omega – 1}}\left| {G_1^ \cdot \left( {\varsigma ,S} \right) } \right. } \left. { – G_1^ \cdot \left( {\varsigma ,{S^*}} \right) } \right| d\varsigma \\ &\le \frac{{\left( {1 – {\alpha }} \right) }}{{{\textbf {AB}}\left( {{\alpha }} \right) }}{\delta _1}\left\| {S – {S^*}} \right\| \\ &+ \frac{{{\alpha }}}{{{\textbf {AB}}\left( {{\alpha }} \right) \Gamma \left( {{\alpha }} \right) }}\int \limits _0^t {{{\left( {t – \varsigma } \right) }^{\omega – 1}}{\wp _1}} \left\| {S – {S^*}} \right\| d\varsigma \\ &\left[ {\frac{{\left( {1 – {\alpha }} \right) }}{{{\textbf {AB}}\left( {{\alpha }} \right) }} + \frac{{{\alpha }{T^{{\alpha } – 1}}}}{{{\textbf {AB}}\left( {{\alpha }} \right) \Gamma \left( {{\alpha }} \right) }}} \right] {\wp _1}\left\| {S – {S^*}} \right\| ,\end{aligned} \end{aligned}$$
(29)
and so
$$\begin{aligned} \left( {1 – \left[ {\frac{{\left( {1 – {\alpha }} \right) }}{{{\textbf {AB}}\left( {{\alpha }} \right) }} + \frac{{{\alpha }{T^{{\alpha } – 1}}}}{{{\textbf {AB}}\left( {{\alpha }} \right) \Gamma \left( {{\alpha }} \right) }}} \right] {\delta _1}} \right) \left\| {S – {S^*}} \right\| \le 0. \end{aligned}$$
From (27), we can assert that the above inequality holds if \(\left\| {S – {S^*}} \right\| =0\) or \(S= S^*.\) Similarly, for others.
As a consequence, \(S^*=S(0),E^*=E(0),I^*=I(0),R^*=R(0), D^*=D(0),H^*=H(0),B^*=B(0),C^*=C(0).\) which proves that the solution of system (7) is unique.