Analyzing the dynamics of fractional spatio-temporal $\mathrm{S}\mathrm{E}\mathrm{I}\mathrm{R}$ epidemic model

In this work, we present a comprehensive analysis of the spatio-temporal $\mathrm{S}\mathrm{E}\mathrm{I}\mathrm{R}$ epidemic model of fractional order. The infection dynamics in the proposed fractional order model (FOM) are described by a system of partial differential equations (PDEs) within a time-fractional order and diffusion operator in one-dimensional space, considering that the total population is split into four compartments: Susceptible, exposed, infected, and recovered individuals denoted as $\mathrm{S}$, $\mathrm{E}$, $\mathrm{I}$ and $\mathrm{R}$, respectively. Our contributions commence by establishing the existence and uniqueness of positively bounded solutions for the proposed FOM. Moreover, we determined all equilibrium points (EPs) and investigated their local stability based on the basic reproduction number (BRN) ${\mathcal{R}}_0$, which is calculated by the next-generation matrix (NGM) method. Additionally, we demonstrated global stability using an appropriate Lyapunov function with fractional LaSalle's invariance principle (LIP). Sensitivity analysis of the FOM parameters was discussed to identify the most critical parameters by which the volume of disease propagation can be measured. The theoretical findings were corroborated by numerical simulations of solutions that are displayed in 3D and 2D graphs. Graphical simulations highlight the effect of vaccination on infection severity. Changing the fractional order $\alpha$ in the proposed FOM has an influence on the speed of convergence to the steady state as a result of the memory effect. Furthermore, vaccination emerges as an effective strategy for disease control.