Modelling disease spread with spatio-temporal fractional derivative equations and saturated incidence rate.

The global analysis of a spatio-temporal fractional order SIR infection model with saturated incidence function is suggested and studied in this paper. The dynamics of the infection is described by three partial differential equations including a time-fractional derivative order for each one of them. The equations of our model describe the evolution of the susceptible, the infected and the recovered individuals with taking into account the spatial diffusion for each compartment. We will choose a saturated incidence rate in order to describe the nonlinear force of the infection. First, we will prove the well-posedness of our suggested model in terms of existence and uniqueness of the solution. Also in this context, the boundedness and the positivity of solutions are established. Afterward, we will give the forms of the disease-free equilibrium and the endemic one. It was demonstrated that the global stability of the each equilibrium depends mainly on the basic reproduction number. Finally, numerical simulations are performed to validate the theoretical results and to show the effect of vaccination in reducing the infection severity. It was shown that the fractional derivative order has no effect on the equilibria stability but only on the convergence speed towards the steady states. It was also observed that vaccination is amongst the good strategies in controlling the disease spread.