# A Mathematical Study on a Fractional-Order SEIR Mpox Model: Analysis and Vaccination Influence
## 4 Stability Analysis
### Local Sensitivity and Elasticity Analysis of \(R_{0}^{*}\)  
In the field of epidemiological models, the parameters' values have a vital role in computing \(R_{0}^{*}\). Under the assumption that all parameters are retained at specific values (estimated) including the parameter \(\varrho\), we call this measure \"local sensitivity\" [17]. In such a measure, the formula used to evaluate the effect of changing one parameter \(\varrho\) on \(R_{0}^{*}\) is written as  
\[S_{\rm e}^{R_{0}^{*}}=\frac{\partial R_{0}^{*}}{\partial\varrho}. \tag{33}\]  
However, although this method does not consider a simultaneous changing of all parameters, it is influenced highly by the values of \(\varrho\) and \(R_{0}^{*}\). Hence, a better indicator will be the elasticity which is defined by the change in the quantity of \(R_{0}^{*}\), as a percentage, due to the value change of the parameter \(\varrho\), as a percentage. The formula of elasticity is written as  
\[\Upsilon_{\varrho}^{R_{0}^{*}}=\left(\frac{\partial R_{0}^{*}}{\partial \varrho}\right)\left(\frac{\varrho}{R_{0}^{*}}\right)\approx\mp\frac{\% \Delta R_{0}^{*}}{\%\Delta Q}, \tag{34}\]  
where a positive elasticity indicates an increment in \(R_{0}^{*}\) in terms of the parameter \(\varrho\), while a negative elasticity indicates a decrement in \(R_{0}^{*}\), in terms of the parameter \(\varrho\). From Equation (32) of \(R_{0}^{*}\) and using the elasticity Equation (34), we investigate the elasticity for all parameters of the model. From this point of view, we can calculate the local elasticity of \(R_{0}^{*}\) in relation to the vaccination rate (\(\
u\)) as  
\[\Upsilon_{\
u}^{R_{0}^{*}}=\left(\frac{\partial R_{0}^{*}}{\partial\
u} \right)\left(\frac{\
u}{R_{0}^{*}}\right)=\frac{\beta\gamma\Lambda}{(\gamma+ \mu)(\mu+\
u)^{2}(\alpha+\mu+\sigma)}\left(\frac{\
u}{R_{0}^{*}}\right)=- \frac{\
u}{\mu+\
u}<0. \tag{35}\]  
Since the value of \(\
u\) indicates the vaccination rate, then the increment of this rate leads to the decrement of \(R_{0}^{*}\) and, thus, negatively affects the spreading of the disease. To see this, we plot Figure 2 based on the data reported in Table 2. This figure clearly shows the influence of the vaccination rate \(\upsilon\) on \(R_{0}^{*}\) and, hence, confirms our aforesaid claim. On the other hand, the local elasticity of \(R_{0}^{*}\), in relation to the contact transmission rate (\(\beta\)), can be computed as \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(\Lambda\) & 163/10,000 \\ \(\beta\) & 0.5 \\ \(\gamma\) & 0.05 \\ \(\alpha\) & 1/17 \\ \(\mu\) & 91/10,000 \\ \(\
u\) & Changes within \([0,1]\) \\ \(\sigma\) & 0.03 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The parameters of the fractional-order Mpox model. Figure 2: Influence of the vaccination rate \(\upsilon\) on \(R_{0}^{*}\). \[\mathrm{Y}_{\beta}^{R_{0}^{*}}=\left(\frac{\partial R_{0}^{*}}{\partial\beta} \right)\left(\frac{\beta}{R_{0}^{*}}\right)=\frac{\gamma\Lambda}{(\gamma+\mu)( \mu+\
u)(\alpha+\mu+\sigma)}\left(\frac{\beta}{R_{0}^{*}}\right)=1>0. \tag{36}\]  
Thus,the increment of the transmission rate \(\beta\) by \(1\%\) leads to a rise in \(R_{0}^{*}\). This is the case of spreading infectious diseases due to the increment in personal contact and communications between individuals and due to the pathogen specifications. However, we plot Figure 3 on the basis of the same values reported in Table 2, but here we take \(\
u=0.15\) and allow \(\beta\) to be varied from \(0.045\) to \(0.18\). Such a figure also shows the influence of the transmission rate \(\beta\) on \(R_{0}^{*}\), which completely coincides with our algebraic calculations mentioned previously.