Stability and bifurcation of disease spreading in complex networks

A general nonlinear model of disease spreading is proposed, describing the effect of the new link-adding probability p in the topological transition of the N-W small-world network model. The new nonlinear model covers both limiting cases of regular lattices and random networks, and presents a more flexible internal nonlinear interaction than a previous model. Hopf bifurcation is proved to exist during disease spreading in all typical cases of regular lattices, small-world networks, and random networks described by this model. It is shown that probability p not only determines the topological transition of the N-W small-world network model, but also dominates the stability of the local equilibria and bifurcating periodic solutions, and moreover can be further applied to stabilize a periodic spreading behaviour onto a stable equilibrium over the network.

[1]  B. Hassard,et al.  Theory and applications of Hopf bifurcation , 1981 .

[2]  X. Yang,et al.  Chaos in small-world networks. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  Guanrong Chen,et al.  Complex networks: small-world, scale-free and beyond , 2003 .

[4]  S. Milgram,et al.  Acquaintance Networks Between Racial Groups: Application of the Small World Method. , 1970 .

[5]  M. Newman,et al.  Scaling and percolation in the small-world network model. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[6]  Albert-László Barabási,et al.  Internet: Diameter of the World-Wide Web , 1999, Nature.

[7]  Xin-She Yang,et al.  Fractals in small-world networks with time-delay , 2002, 1003.4949.

[8]  Guanrong Chen.  Controlling Chaos and Bifurcations in Engineering Systems , 1999 .

[9]  M. G. Cosenza,et al.  Turbulence in small-world networks. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[10]  Xiao Fan Wang,et al.  Synchronization in Small-World Dynamical Networks , 2002, Int. J. Bifurc. Chaos.

[11]  Mauricio Barahona,et al.  Synchronization in small-world systems. , 2002, Physical review letters.

[12]  C. Moukarzel.  Spreading and shortest paths in systems with sparse long-range connections. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[13]  A. Rbnyi.  ON THE EVOLUTION OF RANDOM GRAPHS , 2001 .

[14]  Mark Newman,et al.  Models of the Small World , 2000 .

[15]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.

[16]  P. Erdos,et al.  On the evolution of random graphs , 1984 .

[17]  Guanrong Chen,et al.  Complexity and synchronization of the World trade Web , 2003 .

[18]  Gesine Reinert,et al.  Small worlds , 2001, Random Struct. Algorithms.

[19]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[20]  Albert-László Barabási,et al.  Statistical mechanics of complex networks , 2001, ArXiv.

[21]  Mark E. J. Newman,et al.  The Structure and Function of Complex Networks , 2003, SIAM Rev..

[22]  Marián Boguñá,et al.  Topology of the world trade web. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.