Dynamics of nonlocal and local SIR diffusive epidemic model with free boundaries

Abstract

This paper is concerned with nonlocal and local diffusive SIR epidemic model with free boundaries including convolution, which is natural extension of reaction diffusion systems with free boundary problems and local diffusions. The existence of unique global solution for this model is considered. Dichotomy of the spreading and vanishing is established. A spreading barrier line is found to determine whether the spreading of disease will fail finally. The spreading of disease will fail when it cannot spread across the spreading barrier line l∗, while it will be successful when it transcends over this barrier line. The results show that if the basic reproduction number R0<1, the spreading of disease will fail eventually, and if R0>1+d2μ2+α, the spreading of disease will get success finally. We also find that the spreading coefficients play important role in the spreading achievement. When 1+βθμ1<R0<1+d2μ2+α, the spreading coefficient decides whether the spreading of disease will be successful. It is shown that the spreading will be successful when the spreading coefficient is relatively big, while the spreading will fail if the spreading coefficient is relatively small.