Dynamics for an Sir Epidemic Model with Nonlocal Diffusion and Free Boundaries

This paper is concerned with the spatial propagation of an SIR epidemic model with nonlocal diffusion and free boundaries describing the evolution of a disease. This model can be viewed as a nonlocal version of the free boundary problem studied by Kim et al. (An SIR epidemic model with free boundary. Nonlinear Anal RWA, 2013, 14: 1992–2001). We first prove that this problem has a unique solution defined for all time, and then we give sufficient conditions for the disease vanishing and spreading. Our result shows that the disease will not spread if the basic reproduction number R0 < 1, or the initial infected area h0, expanding ability μ, and the initial datum S0 are all small enough when $$1 < {R_0} < 1 + {d \over {{\mu _2} + \alpha }}$$ . Furthermore, we show that if $$1 < {R_0} < 1 + {d \over {{\mu _2} + \alpha }}$$ , the disease will spread when h0 is large enough or h0 is small but μ is large enough. It is expected that the disease will always spread when $${R_0} \ge 1 + {d \over {{\mu _2} + \alpha }}$$ , which is different from the local model.