# A Mathematical Study on a Fractional-Order SEIR Mpox Model: Analysis and Vaccination Influence
## 5 Numerical Findings  
In the following content, we intend to employ two numerical methods (the FEM and MFEM) to obtain approximate solutions of the fractional-order Mpox system (19). In fact, these methods represent two fractional versions of the conventional Euler method. One can refer to [8] for the purpose of obtaining a comprehensive view about such methods. In this context, Theorem 1 can lay the foundation to recall the FEM and MFEM, which are considered for the following fractional problem:  
\[{}^{C}D_{*}^{\rho}z(x)=g(t,z(t)),\;z(0)=z_{0},\;t>0,\;0<\rho\leq 1. \tag{38}\]  
To address problem (38), we assume \([0,a]\) as the interval over which we wish to obtain the desired numerical solution. In general, we are not capable to find \(z(t)\), which represents the analytical solution to the problem at hand. As an alternative, we can establish a set of points \((t_{j},z(t_{i}))\), and then employ them in finding the approximate solution [8; 18]. For simplification purposes, we divide \([0,a]\) into \(k\) sub-intervals \([t_{j},t_{i+1}]\) with \(h=a/k\) through \(t_{i}=ih\), for \(i=0,1,2,\ldots,k\). Now, we assume that \(z(t)\), \({}^{C}D_{*}^{\rho}z(t)\), and \({}^{C}D_{*}^{2\rho}z(t)\) are continuous on \((0,a]\). Therefore, by means of Theorem 1, one might expand \(z(t)\) about \(t=t_{i}\) as  
\[z(t)=z(t_{i})+\frac{(t-t_{i})^{\rho}}{\Gamma(\rho+1)}\;{}^{C}D_{*}^{\rho}z(t_ {i})+\frac{(t-t_{i})^{2\rho}}{\Gamma(2\rho+1)}\;{}^{C}D_{*}^{2\rho}z(\xi), \tag{39}\]  
for some \(\xi\in(t_{i},t_{i+1})\). Thus, if we substitute \(t_{i+1}\) instead of \(t\) in (39), we obtain  
\[z(t_{i+1})=z(t_{i})+\frac{h^{\rho}}{\Gamma(\rho+1)}\;{}^{C}D_{*}^{\rho}z(t_{i })+\frac{(t_{i+1}-t_{i})^{2\rho}}{\Gamma(2\rho+1)}\;{}^{C}D_{*}^{2\rho}z(\xi), \tag{40}\]  
for some \(\xi\in(t_{i},t_{i+1})\). Now, if one chooses \(h=t_{i+1}-t_{i}\) too small, then the last term of (40) can be eliminated to obtain  
\[z(t_{i+1})=z(t_{i})+\frac{h^{\rho}}{\Gamma(\rho+1)}\;{}^{C}D_{*}^{\rho}z(t_{i }). \tag{41}\]  
Actually, Equation (41) represents the primary formula of the FEM. In the same regard, if one substitutes  
\[{}^{C}D_{*}^{\rho}z(t_{i+1})=g\bigg{(}t_{i}+\frac{h^{\rho}}{\Gamma(\rho+1)},z (t_{i})+\frac{h^{\rho}}{\Gamma(\rho+1)}g(t_{i},z(t_{i}))\bigg{)}\]  
instead of (41), the result of \(z(t_{i+1})\) will be consequently yielded. In other words, we obtain  
\[z(t_{i+1})=z(t_{i})+\frac{h^{\rho}}{\Gamma(\rho+1)}\times g\bigg{(}t_{i}+\frac {h^{\rho}}{\Gamma(\rho+1)},z(t_{i})+\frac{h^{\rho}}{\Gamma(\rho+1)}g(t_{i},z (t_{i}))\bigg{)}, \tag{42}\]  
for \(i=0,1,2,\ldots,k-1\). Equation (42) represents the main formula of the MFEM that would be compared next with the FEM's formula. In what follows, we attempt to implement only the MFEM on the fractional-order Mpox system (19), as the FEM is similar to the MFEM. To do so, we reconsider such a system again as  
\[{}^{C}D_{*}^{\rho}S(t) =g_{1}(t,S(t),E(t),I(t),R(t)) \tag{43}\] \[{}^{C}D_{*}^{\rho}E(t) =g_{2}(t,S(t),E(t),I(t),R(t))\] \[{}^{C}D_{*}^{\rho}I(t) =g_{3}(t,S(t),E(t),I(t),R(t))\] \[{}^{C}D_{*}^{\rho}R(t) =g_{4}(t,S(t),E(t),I(t),R(t)),\]where  
\[ \begin{split} g_{1}(t,S(t),E(t),I(t),R(t))&=\Lambda- \beta S(t)I(t)-(\
u+\mu)S(t)\\ g_{2}(t,S(t),E(t),I(t),R(t))&=\beta S(t)I(t)-(\mu+ \gamma)E(t)\\ g_{3}(t,S(t),E(t),I(t),R(t))&=\gamma E(t)-(\mu+ \sigma+\alpha)I(t)\\ g_{4}(t,S(t),E(t),I(t),R(t))&=\
u S(t)+\alpha I(t)- \mu R(t).\end{split}  \tag{44}\]  
More precisely, with the aim of generating \((t_{k},S(t_{k}))\) in relation to the compartment \(S\), one should suppose that \(S(t),{}^{\mathbb{C}}D_{*}^{\rho}S(t)\), and \({}^{\mathbb{C}}D_{*}^{2\rho}S(t)\) are all continuous on \((0,T]\).