# A Mathematical Study on a Fractional-Order SEIR Mpox Model: Analysis and Vaccination Influence
## 3 Mpox Model  
Mathematical models are extensively engaged in simulating the dynamics and the transmission of infectious diseases. In accordance with an effective vaccine, we study the progression and dynamics of the Mpox disease in compliance with various scenarios of the vaccination rate through a compartmental model of the type SEIR. In this paper, we consider the intervention of vaccination playing a significant role as a parameter, impacting on the model's dynamics. The equilibrium of the disease-free state will be determined, and then its stability with computing the BRN \(R_{0}^{*}\) will be examined in Section 4. The diagram of the compartment model, including the vaccination rate for the aimed Mpox model, is shown in Figure 1, in which the total of the susceptible population is denoted by \(S\), the total of the exposed population is denoted by \(E\), the total of the infected population is denoted by \(I\), and, finally, the total of the recovery population by \(R\). In the same context, Figure 1 exhibits the parameters \((\Lambda,\sigma,\
u,\mu,\alpha,\gamma,\beta)\) that are responsible for the dynamics of the considered model. These parameters are outlined in Table 1.  
It is noteworthy that the rate of transmission \(\beta\) indicates the transmission's contact from \(S\) to \(E\) as a result of interacting with \(I\) cases. Additionally, \(\gamma^{-1}\) indicates the period of incubation, whereas \(\alpha^{-1}\) indicates the period of recovery. In light of the aforementioned considerations, the conventional Mpox model might be described in the following manner [10]:  
\[ \begin{split}\frac{dS}{dt}&=\Lambda-\beta S(t)I(t)-( \
u+\mu)S(t)\\ \frac{dE}{dt}&=-(\mu+\gamma)E(t)+\beta S(t)I(t)\\ \frac{dI}{dt}&=\gamma E(t)-(\mu+\sigma+\alpha)I(t) \\ \frac{dR}{dt}&=\
u S(t)+\alpha I(t)-\mu R(t). \end{split}  \tag{10}\]  
with initial conditions  
\[S(0)=S_{0},\ E(0)=E_{0},\ I(0)=I_{0},R(0)=R_{0}. \tag{11}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Denotation** & **Value** \\ \hline \(\Lambda\) & Rate of birth & Depending on country \\ \(\gamma\) & Rate of transmission (\(E\) to \(I\)) & \(\left[\frac{1}{2},\frac{1}{2}\right]\) \\ \(\beta\) & Rate of transmission (\(S\) to \(E\)) & \(\left[0,045,0.18\right]\) \\ \(\alpha\) & Rate of transmission (\(I\) to \(R\)) & \(\left[\frac{1}{32},\frac{1}{15}\right]\) \\ \(\sigma\) & Rate of fatality & \(\left[0.03,0.06\right]\) \\ \(\
u\) & Rate of vaccination & Variable \([0,1]\) \\ \(\mu\) & Rate of natural death & Dependent on country \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Model parameters. Figure 1: The compartment model, including vaccination rate. For more clarification, model (10) is established in accordance with particular conditions, which might be stated as follows:  
* The contact transmission parameter \(\beta\) does not take into account some factors, like climate, age, marital status, or gender. * All of the model's parameters have no negative values. * Inter-individual relations within society are distinguished by uniformity and homogeneity. This is because of the inherited suppositions for all of the model's states in which all persons have the same parameter value of the contact transmission regardless of their situations, such as their health circumstances or age. * The size of population \(N\) is considered constant at any time, and it meets the following equality: \[S(t)+E(t)+I(t)+R(t)=N(t).\] (12)
* In the proposed model, the demography is taken into account, in which the natural death \(\mu\) and natural birth \(\Lambda\) are incorporated. * System (10) satisfies the following property: \[\frac{dN}{dt}\leq h(N),\] (13) where \(h\) is a function depending only on \(N\)[11].