Spatial spreading of a logistic SI epidemic model with partially degenerate diffusion and double free boundaries

In this paper, we investigate a logistic SI epidemic model with partially degenerate diffusion and double free boundaries to describe the spatial spreading of disease. The existence, uniqueness, and estimates of the global solution are discussed firstly. Then we prove a spreading-vanishing dichotomy. Namely the infective class either successfully spreads to infinity as t → ∞, or vanishes in a finite area. Besides, the long time behavior of the solution and criteria for spreading and vanishing are also obtained. Especially, we find the Basic Reproduction Number R₀ is not the unique factor which determine whether or not an infectious disease can spread through a population: when R₀ ≤ 1, vanishing always happens and the disease will die out; when R₀ > 1, whether or not to vanish depends on the size of the initial habitat and the rate of expansion. This phenomenon reveals the role of free boundaries in the epidemic. In the end, we give some numerical results as supplements to the theoretical results.