Global dynamic of spatio-temporal fractional order SEIR model

The global analysis of a spatio-temporal fractional order SEIR infection epidemic model is studied and analyzed in this paper.  The dynamics of the infection is described by four partial differential equations with a fractional derivative order and with diffusion.  The equations of our model describe the evolution of the susceptible, the exposed, the infected and the recovered  individuals with taking into account the spatial diffusion for each compartment.  At first, we will prove the existence and uniqueness of the solution using the results of the fixed point theorem, and the equilibrium points are established and presented according to ${\cal R}_{0}$.  Next, the bornitude and the positivity of the solutions of the proposed model are established.  Using the Lyapunov direct method it has been proved that the global stability of the each equilibrium depends mainly on the basic reproduction number ${\cal R}_{0}$.  Finally, numerical simulations are performed to validate the theoretical results.