Reproduction numbers and the expanding fronts for a diffusion–advection SIS model in heterogeneous time-periodic environment

• A free boundary problem is proposed to describe the expanding of disease in a heterogeneous time-periodic environment.
• The impact of small advection on the spreading of disease is considered.
• The basic reproduction number introduced here depends on the time.
• Numerical simulations are presented to illustrate the effect of small advection, the diffusion rate and the expanding capability.

This paper deals with a simplified SIS model, which describes the transmission of infectious disease in time-periodic heterogeneous environment. To grasp the impact of spatial heterogeneity of environment, temporal periodicity and small advection intensity on the persistence and eradication of the disease, the left and right free boundaries are introduced to represent the expanding fronts. The basic reproduction numbers R0D and R0F(τ), which depend on spatial heterogeneity, temporal periodicity, spatial diffusion and advection, are introduced. A spreading–vanishing dichotomy is established and sufficient conditions for the spreading and vanishing of the disease are given. The asymptotic spreading speeds for the left and right fronts are also obtained, and numerical simulations are presented to illustrate the influences of the advection intensity, dispersal rate and expanding capability on the moving fronts.