Global analysis of a time fractional order spatio-temporal SIR model

Abstract

We deal in this paper with a diffusive SIR epidemic model described by reaction–diffusion equations involving a fractional derivative. The existence and uniqueness of the solution are shown, next to the boundedness of the solution. Further, it has been shown that the global behavior of the solution is governed by the value of R_0, which is known in epidemiology by the basic reproduction number. Indeed, using the Lyapunov direct method it has been proved that the disease will extinct for R_0 <1 for any value of the diffusion constants. For R_0>1, the disease will persist and the unique positive equilibrium is globally stable. Some numerical illustrations have been used to confirm our theoretical results.