The spreading frontiers of avian-human influenza described by the free boundary

In this paper, a reaction-diffusion system is proposed to investigate avian-human influenza. Two free boundaries are introduced to describe the spreading frontiers of the avian influenza. The basic reproduction numbers r (0)(F) (t) and R (0)(F)(t) are defined for the bird with the avian influenza and for the human with the mutant avian influenza of the free boundary problem, respectively. Properties of these two time-dependent basic reproduction numbers are obtained. Sufficient conditions both for spreading and for vanishing of the avian influenza are given. It is shown that if r (0)(F) (0) < 1 and the initial number of the infected birds is small, the avian influenza vanishes in the bird world. Furthermore, if r (0)(F) (0) < 1 and R (0)(F)(0) < 1, the avian influenza vanishes in the bird and human worlds. In the case that r (0)(F) (0) < 1 and R (0)(F)(0) > 1, spreading of the mutant avian influenza in the human world is possible. It is also shown that if r (0)(F) (t (0)) ⩾ 1 for any t (0) ⩾ 0, the avian influenza spreads in the bird world.