# A Mathematical Study on a Fractional-Order SEIR Mpox Model: Analysis and Vaccination Influence
## 1 Introduction  
Mathematical modeling has turned into an invaluable tool in revealing the behaviors of infectious diseases and formulating effective control strategies. Mpox, a viral disease with similarities to smallpox, has attracted considerable attention in the field of epidemiological models due to its potential impact on human health. As the Mpox disease is a viral zoonotic infection, it might transmit from animals to people. In addition, it might transmit from the environment to a person, as it can transmit from one person to another [1]. In 2022, the Mpox disease began to spread like a pandemic in numerous countries all over the world. It is spreading now in some African countries [2, 3]. To gain deeper insights into the transmission dynamics and develop accurate predictive models for Mpox, many researchers have increasingly explored the utilization of fractional calculus. Fractional calculus offers a promising tool for highlighting the intricate characteristics of infectious disease dynamics, as it allows for the incorporation of memory-dependent and non-local effects. Traditional models often suppose an instantaneous mixing by assuming that past interactions have no influence on future outcomes. So, from this point, the notion of fractional calculus might be utilized as an alternative to these models, as it can enable one to formulate a model of long-range dependence with extra freedomof fractional-order values. This would enable a more realistic representation of the spread and evolution of the disease; see [4]. In the context of Mpox, fractional calculus can potentially enhance our understanding of various epidemiological factors such as transmission rates, recovery rates, and the impact of interventions. By incorporating fractional derivatives and integrals into the modeling framework, researchers can address the non-local nature of interactions and the presence of memory effects in disease transmission. These effects can reflect the persistence of immunity, delayed responses to interventions, or even other temporal dependencies; see [5]. Developing a fractional calculus-based model for Mpox involves considering fractional differential equations that can describe the growth of the diseased population over time. The fractional-order derivatives correspond to the memory or long-range dependence, which might be observed in the transmission of disease. Accurate estimation of the fractional-order value requires a combination of empirical evidence, fitting the model to available data, and a deep understanding of the underlying epidemiological processes [6]. The utilization of fractional calculus in modeling Mpox has the potential to yield valuable insights into the disease's dynamics and control strategies. Incorporating memory effects and non-local interactions in modeling the disease can provide a more accurate representation of its spread and evolution over time. Furthermore, such modeling can assist public health officials in evaluating and optimizing intervention strategies, estimating the impact of vaccination programs, and assessing the effectiveness of different control measures [7]. In this study, we aim to explore the application of fractional calculus in modeling Mpox transmission dynamics. By introducing fractional-order derivatives and considering memory-dependent effects, we attempt to enhance our understanding of the disease's behavior and provide valuable tools for policymakers and health care professionals in their efforts to mitigate and control Mpox outbreaks. For this purpose, a novel fractional-order version of the SEIR model is established in light of operating the Caputo operator. This model is analyzed with the help of proposing certain novel stability results, including the investigation of its invariant region, equilibria, Basic Reproductive Number (BRN), elasticity, and boundedness of solution, which is observed by using the Adomian decomposition method.