Spatial dynamics and optimization method for a rumor propagation model in both homogeneous and heterogeneous environment.

Considering the influence of environmental capacity and forgetting on rumor spreading, we improve the traditional SIR (susceptible-infected-removed) rumor propagation model and give two dynamic models of rumor propagation in heterogeneous environment and homogeneous environment, respectively. The main purpose of this paper is to make a dynamic analysis of rumor propagation models. In the spatial heterogeneous environment, we have analyzed the uniform persistence of the rumor propagation model and the asymptotic behavior of the positive equilibrium point when the diffusion rate of rumor-susceptible tends to zero. In the spatial homogeneous environment, we discuss the stability of rumor propagation model. Further, optimal control and the necessary optimality conditions are obtained by using the maximum principle. Finally, we study the Hopf bifurcation phenomenon through inducing time delay in the reaction-diffusion model. In addition, the existence of Hopf bifurcation is verified and the influence of diffusion coefficients is studied by numerical simulations.

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.