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 rF0(t)and RF0(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 rF0(0)1 and the initial number of the infected birds is small,the avian influenza vanishes in the bird world.Furthermore,if rF0(0)1 and RF0(0)1,the avian influenza vanishes in the bird and human worlds.In the case that rF0(0)1 and RF0(0)1,spreading of the mutant avian influenza in the human world is possible.It is also shown that if rF0(t0)1 for any t00,the avian influenza spreads in the bird world.