Vanishing and spreading conditions for a free-boundary epidemic model with subclinical infections and vaccination.

This paper presents a free-boundary epidemic model with subclinical infections and vaccination.We prove the existence and uniqueness of solutions to the model.Moreover, sufficient conditions for the disease vanishing and spreading are given.The disease will vanish if the basic reproduction number $ R_0 < 1 $, that the corresponding ODE model defines without spatial expansion. However, the disease will spread to the whole area if $ R^F_0(t_0) > 1 $ for some $ t_0 > 0 $ when it is introduced spatial heterogeneity. $ R^F_0(0) < R_0 $ implies that the spillovers from hotspots to areas with no confirmed cases will reduce the outbreak threshold and increase the difficulty of prevention and control in the whole region. Under the condition $ R^F_0(0) < 1 < R_0 $, if the free boundary condition of infectives $ h(t) < \infty $, $ t \to \infty $, then the disease is vanishing, which indicates that $ R^F_0(0) < 1 $ can also control the disease if the scope of hotspots expansion is limited. Furthermore, the numerical simulations illustrate that the routine vaccination would decrease the basic reproduction number and then change the disease from spreading to vanishing.