Pair Approximation Models for Disease Spread

We consider a Susceptible-Infective-Recovered (SIR) model, where the mechanism for the renewal of susceptibles is demographic, on a ring with next nearest neighbour interactions, and a family of correlated pair approximations (CPA), parametrized by a measure of the relative contributions of loops and open triplets of the sites involved in the infection process. We have found that the phase diagram of the CPA, at fixed coordination number, changes qualitatively as the relative weight of the loops increases, from the phase diagram of the uncorrelated pair approximation to phase diagrams typical of one-dimensional systems. In addition, we have performed computer simulations of the same model and shown that while the CPA with a constant correlation parameter cannot describe the global behaviour of the model, a reasonable description of the endemic equilibria as well as of the phase diagram may be obtained by allowing the parameter to depend on the demographic rate.Comment: 6 pages, 3 figures, LaTeX2e+SVJour+AmSLaTeX, NEXTSigmaPhi 2005; metadata title corrected wrt paper titl

Liquid crystals boojum-colloids

Colloidal particles dispersed in a liquid crystal lead to distortions of the director field. The distortions are responsible for long-range effective colloidal interactions whose asymptotic behaviour is well understood. The short distance behaviour of the interaction, however, is sensitive to the structure and dynamics of the topological defects nucleated near the colloidal particles in the strong anchoring regime. The full non-linear theory is required in order to determine the interaction at short separations. Spherical colloidal particles with sufficiently strong planar degenerate anchoring nucleate a pair of antipodal surface topological defects, known as boojums. We use the Landau-de Gennes formalism in order to resolve the mesoscopic structure of the boojum cores and to determine the pairwise colloidal interaction. We compare the results in three (3D) and two (2D) spatial dimensions. The corresponding free energy functionals are minimized numerically using finite elements with adaptive meshes. Boojums are always point-like in 2D, but acquire a rather complex structure in 3D which depends on the combination of the anchoring potential, the radius of the colloid, the temperature and the LC elastic anisotropy. We identify three types of defect cores in 3D which we call single, double and split core boojums, and investigate the associated structural transitions. In the presence of two colloidal particles there are substantial re-arrangements of the defects at short distances, both in 3D and 2D. These re-arrangements lead to qualitative changes in the force-distance profile when compared to the asymptotic quadrupole-quadrupole interaction. In line with the experimental results, the presence of the defects prevents coalescence of the colloidal particles in 2D, but not in 3D systems.Comment: 18 pages, 21 figure