A reaction–diffusion epidemic model with free boundary incorporating an anti-disease mechanism

This paper concerns a reaction–diffusion epidemic model with free boundary including an anti-disease mechanism. The free boundary is introduced to this model and it is explored for the expanding fronts of the habitat region of infection individuals. By applying the spectral radius of the generation operator, the spatial-temporal risk index [Formula: see text] is presented for this model. The global existence of positive solutions is verified and the long-time behavior of positive solutions is considered with finite cross-diffusion coefficient. The sufficient conditions are found for the disappearance and propagation of disease in new environment. Moreover, the propagating-disappearing criteria are established. The results show that the diseases will disappear eventually and not establish themselves if [Formula: see text] or [Formula: see text], cross-diffusion coefficient is relatively big and propagating coefficient [Formula: see text] is sufficiently small, and the disease will propagate successfully if [Formula: see text] for some [Formula: see text], or [Formula: see text] with the relative big propagating coefficient and the relative small cross-diffusion coefficient.