Global dynamics of an SIR epidemic model with nonlocal diffusion. (October 2018) - Kuniya, Toshikazu; Wang, Jinliang
Abstract: In this paper, we are concerned with the global asymptotic stability of each equilibrium of an SIR epidemic model with nonlocal diffusion. Under the assumption of Lipschitz continuity of parameters, the eigenvalue problem associated with the linearized system around the disease-free equilibrium has a principal eigenvalue corresponding to a strictly positive eigenfunction. By setting the eigenfunction as the integral kernel of a Lyapunov function, we prove the global asymptotic stability of the disease-free equilibrium when the basic reproduction number R 0 is less than one. We also prove the uniform persistence of the system when R 0 > 1 by using the persistent theory for dynamical systems. Furthermore, in a special case where the dispersal rate for susceptible individuals is equal to zero, we prove the existence, uniqueness and global asymptotic stability of the endemic equilibrium when R 0 > 1 by constructing a suitable Lyapunov function.