Periodicity, synchronization and persistence in pre-vaccination measles.

We investigate the relationship between periodicity, synchronization and persistence of measles through simulations of geographical spread on the British Isles. We show that the establishment of areas of biennial periodicity depends on the interplay between human mobility and local population size and that locations undergoing biennial cycles tend to be, on average, synchronized in phase. We show however that occurrences of opposition of phase are actually quite common and correspond to stable dynamics. We also show that persistence is strictly related to circulation of the disease in the highly populated area of London and that this ensures survival of the disease even when human mobility drops to extremely low levels.

Impact of human mobility on the periodicities and mechanisms underlying measles dynamics.

Three main mechanisms determining the dynamics of measles have been described in the literature: invasion in disease-free lands leading to import-dependent outbreaks, switching between annual and biennial attractors driven by seasonality, and amplification of stochastic fluctuations close to the endemic equilibrium. Here, we study the importance of the three mechanisms using a detailed geographical description of human mobility. We perform individual-based simulations of an SIR model using a gridded description of human settlements on top of which we implement human mobility according to the radiation model. Parallel computation permits detailed simulations of large areas. Focusing our research on the British Isles, we show that human mobility has an impact on the periodicity of measles outbreaks. Depending on the level of mobility, we observe at the global level multi-annual, annual or biennial cycles. The periodicity observed globally, however, differs from the local epidemic cycles: different locations show different mechanisms at work depending on both population size and mobility. As a result, the periodicities observed locally depend on the interplay between the local population size and human mobility.

System-size dependence of the free energy of crystalline solids.

We investigate the system-size dependence of the Helmholtz free energy of crystalline solids from computer simulation. We employ a standard thermodynamic integration technique along a reversible path that links the crystalline solid with a noninteracting Einstein crystal with the same structure. The key contribution to the free energy is computed by using the so-called expanded-ensemble technique and the results are compared with those obtained from conventional integration of the derivative of the free energy along the path using Gaussian-Legendre quadrature. We find that both methods yield fully consistent results. The free energy is found to exhibit a strong dependence with system size, in agreement with the behavior found by Polson et al. [J. Chem. Phys. 112, 5339 (2000)] but at variance with the dependence reported more recently by Chang and Sandler [J. Chem. Phys. 118, 8390 (2003)]. This has been tested for the face-centered cubic (fcc) and hexagonal close-packed phases of a crystal of hard spheres at a density close to the melting point. We also investigate any possible dependence of the free energy of the solid phase with the shape of the simulation box. We find that this contribution may not be as important as previous investigations suggest. The present results seem to indicate that there is a non-negligible contribution to the free energy arising from the orientation of the closed-packed crystal layers with respect to the simulation cell. This contribution is particularly noticeable for small system sizes and is believed to be an effect of the periodic boundary conditions used in the simulations. The results presented here corroborate the stability of the fcc phase of the hard-sphere solid close to melting.