# A Mathematical Study on a Fractional-Order SEIR Mpox Model: Analysis and Vaccination Influence
## 5 Numerical Findings
From this perspective, if we assume that  
\[g_{1}(t,S(t),E(t),I(t),R(t))=\Lambda-\beta S(t)I(t)-(\
u+\mu)S(t),\]  
so that  
\[{}^{\mathbb{C}}D_{*}^{\rho}S(t)=g_{1}(t,S(t),E(t),I(t),R(t)),\]  
then, with the use of (42), one might obtain  
\[S(t_{i+1})=S(t_{i})+\frac{h^{\rho}}{\Gamma(\rho+1)}\times g_{1}\bigg{(}t_{i} +\frac{h^{\rho}}{\Gamma(\rho+1)},S(t_{i})+\frac{h^{\rho}}{\Gamma(\rho+1)}g_{1} (t_{i},S(t_{i}))\bigg{)}, \tag{45}\]  
for \(i=0,1,2,\ldots,k-1\). In a similar manner, the aforesaid approach might be applied for the remaining classes to obtain their approximate solutions. In the long run, we can infer the following approximations of model (19):  
\[ \begin{split}& S(t_{i+1})=S(t_{i})+\frac{h^{\rho}}{\Gamma(\rho+1 )}\times g_{1}\bigg{(}t_{i}+\frac{h^{\rho}}{\Gamma(\rho+1)},S(t_{i})+\frac{h^ {\rho}}{\Gamma(\rho+1)}g_{1}(t_{i},S(t_{i}))\bigg{)},\\ & E(t_{i+1})=E(t_{i})+\frac{h^{\rho}}{\Gamma(\rho+1)}\times g_{2} \bigg{(}t_{i}+\frac{h^{\rho}}{\Gamma(\rho+1)},E(t_{i})+\frac{h^{\rho}}{\Gamma (\rho+1)}g_{2}(t_{i},E(t_{i}))\bigg{)},\\ & I(t_{i+1})=I(t_{i})+\frac{h^{\rho}}{\Gamma(\rho+1)}\times g_{3} \bigg{(}t_{i}+\frac{h^{\rho}}{\Gamma(\rho+1)},I(t_{i})+\frac{h^{\rho}}{\Gamma (\rho+1)}g_{3}(t_{i},I(t_{i}))\bigg{)},\\ & R(t_{i+1})=R(t_{i})+\frac{h^{\rho}}{\Gamma(\rho+1)}\times g_{4} \bigg{(}t_{i}+\frac{h^{\rho}}{\Gamma(\rho+1)},R(t_{i})+\frac{h^{\rho}}{\Gamma (\rho+1)}g_{4}(t_{i},R(t_{i}))\bigg{)},\end{split}  \tag{46}\]  
where \(g_{1},g_{2},g_{3},g_{4}\) are already outlined in (44), \(i=0,1,2,\ldots,k-1\). In the following content, we propose to depict certain numerical findings that demonstrate the dynamics of the fractional-order Mpox system (19). In this connection, we consider Table 3, which takes very close data to certain available data captured out of the Indian community, on the basis of reference [19; 20]. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(S(0)\) & 23 \\ \(E(0)\) & 6 \\ \(I(0)\) & 4 \\ \(R(0)\) & 0 \\ \(\Lambda\) & 0.01630 \\ \(\beta\) & 0.05000 \\ \(\gamma\) & 0.05882 \\ \(\alpha\) & 0.03125 \\ \(\mu\) & 0.00910 \\ \(v\) & \([0,1]\) \\ \(\sigma\) & 0.0300 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of system (19) [19]. For further description and in accordance with \(\rho=0.75\), we simulate, once again, the infected and exposed sizes of system (19), as shown in Figures 9 and 10, by using the MFEM with and without consideration of the vaccination rate \(\
u\). Accordingly, one might obviously notice that, if the vaccination rate is raised, then the amounts of the exposed and infected states are reduced, validating the impact of such a rate on the model at hand. For the same connection, it was demonstrated in [8] that the MFEM is a modified numerical scheme for the FEM. Thus, with the aim of validat  
Figure 8: Infected sizes \(I\times 10^{2}\) of system (19) for \(\rho=1,0.9,0.8,0.7,0.6\) through MFEM with \(\
u=0.5\). Figure 10: Infected sizes \(I\times 10^{2}\) of system (19) for \(\rho=0.75\) with and without considering \(\
u\). Figure 9: Exposed sizes \(E\times 10^{2}\) of system (19) for \(\rho=0.75\) with and without considering \(\
u\). In the same regard, for the purpose of showing the validity of the domain \(\psi\) reported in Section 4.1 and the validity of the boundedness of solution of system (19) discussed in Section 4.2, one can easily check, based on the data reported in Table 3, that the sum of all classes satisfies the inequality declared in (20). In other words, one can check  
\[S(t)+E(t)+I(t)+R(t)\leq 1.79121. \tag{47}\]  
This bound is further confirmed numerically by plotting the sum of \(S(t)+E(t)+I(t)+R(t)\) according to different fractional-order values, as shown in Figure 15.