Stability analysis of reaction–diffusion fractional-order SEIR model with vaccination and saturated incidence rate

Abstract

In this work, we present a mathematical analysis of a reaction–diffusion susceptible–exposed–infected–recovered $\left(\mathrm{SEIR}\right)$ model with a time fractional order derivative. The model describes the transmission of infectious diseases among the four $\mathrm{SEIR}$ compartments using fractional-order differential equations. We take into account the spatial diffusion of each variable. To represent the non-linear force of infection, a saturated incidence function is taken into consideration. First, we prove the well-posedness of the suggested model by demonstrating existence, uniqueness and boundedness of solutions. Next, we give the basic reproduction number $R_0$ and the problem steady states. The global stability of both equilibria is fulfilled using the Lyapunov method. Finally, the numerical simulations are conducted to validate the theoretical findings and highlight the impact of vaccination on reducing the severity of infection, as well as the impact of the fractional derivative order on equilibria stability. It has been demonstrated that the fractional derivative order has no impact on the stability of the equilibria, but it has a big effect on the speed of the convergence towards the steady states. In addition, it is observed that when the diffusion parameters are increased, the peak of infected and exposed individuals also gets maximized.