A weighted configuration model and inhomogeneous epidemics

A random graph model with prescribed degree distribution and degree dependent edge weights is introduced. Each vertex is independently equipped with a random number of half-edges and each half-edge is assigned an integer valued weight according to a distribution that is allowed to depend on the degree of its vertex. Half-edges with the same weight are then paired randomly to create edges. An expression for the threshold for the appearance of a giant component in the resulting graph is derived using results on multi-type branching processes. The same technique also gives an expression for the basic reproduction number for an epidemic on the graph where the probability that a certain edge is used for transmission is a function of the edge weight. It is demonstrated that, if vertices with large degree tend to have large (small) weights on their edges and if the transmission probability increases with the edge weight, then it is easier (harder) for the epidemic to take off compared to a randomized epidemic with the same degree and weight distribution. A recipe for calculating the probability of a large outbreak in the epidemic and the size of such an outbreak is also given. Finally, the model is fitted to three empirical weighted networks of importance for the spread of contagious diseases and it is shown that $R_0$ can be substantially over- or underestimated if the correlation between degree and weight is not taken into account

The military and administrative reforms of the emperor Gallienus

The military and administrative reforms of Gallienus affected the tactical organisation of the army, the composition and structure of the officer corps, and the social composition of the provincial administration. Gallienus' tactical reforms created a powerful body of cavalry, which represented the culmination to a century of increasing mobility and depth in defence. This "battlecavalry" was not the direct ancestor of the later comitatenses. but its tactical value and elite status paved the way for a "dual" system of defence involving frontier garrisons and centralised field-armies. His reforms of the officer corps involved the replacement of senatorial officers by equestrians, and the introduction of the protectorate. He hoped thereby to bind military commanders more closely to himself. The administrative reforms involved increasing the number of equestrian governors. Though senatorial governorships continued to occur, senators were henceforth without any real military power. Whether Gallienus ever issued an "edict" to this effect is uncertain, as is the motive behind the reform. Most scholars see in it an attempt to improve the quality of military leadership, but there is little evidence for this. It is preferable to agree with Aurelius Victor, that Gallienus feared the senators, and so deprived them of their commands. Together, Gallienus' creation of the "battlecavalry" and his reforms of the officer corps and provincial administration created the pre-conditions for the recovery of the Roman empire under the Illyrian emperors, and laid the foundations to the "Military Monarchy" of the later Roman empire

Room temperature spin relaxation in GaAs/AlGaAs multiple quantum wells

We have explored the dependence of electron spin relaxation in undoped GaAs/AlGaAs quantum wells on well width (confinement energy) at 300 K. For wide wells, the relaxation rate tends to the intrinsic bulk value due to the D'yakonov-Perel (DP) mechanism with momentum scattering by phonons. In narrower wells, there is a strong dependence of relaxation rate on well width, as expected for the DP mechanism, but also considerable variation between samples from different sources, which we attribute to differences in sample interface morphology. (C) 1998 American Institute of Physics. [S0003-6951(98)02541-8].</p

High-velocity feature of the class I methanol maser in G309.38-0.13

The Australia Telescope Compact Array (ATCA) has been used to map class I methanol masers at 36 and 44 GHz in G309.38-0.13. Maser spots are found at nine locations in an area of 50''x30'', with both transitions reliably detected at only two locations. The brightest spot is associated with shocked gas traced by 4.5 micron emission. The data allowed us to make a serendipitous discovery of a high-velocity 36-GHz spectral feature, which is blue-shifted by about 30 km/s from the peak velocity at this frequency, but spatially located close to (within a few arcseconds of) the brightest maser spot. We interpret this as indicating an outflow parallel to the line of sight. Such a high velocity spread of maser features, which has not been previously reported in the class I methanol masers associated with a single molecular cloud, suggests that the outflow most likely interacts with a moving parcel of gas.Comment: 6 pages, 2 figures, accepted by MNRAS Letter

Characterizing the Initial Phase of Epidemic Growth on some Empirical Networks

A key parameter in models for the spread of infectious diseases is the basic reproduction number $R_0$, which is the expected number of secondary cases a typical infected primary case infects during its infectious period in a large mostly susceptible population. In order for this quantity to be meaningful, the initial expected growth of the number of infectious individuals in the large-population limit should be exponential. We investigate to what extent this assumption is valid by performing repeated simulations of epidemics on selected empirical networks, viewing each epidemic as a random process in discrete time. The initial phase of each epidemic is analyzed by fitting the number of infected people at each time step to a generalised growth model, allowing for estimating the shape of the growth. For reference, similar investigations are done on some elementary graphs such as integer lattices in different dimensions and configuration model graphs, for which the early epidemic behaviour is known. We find that for the empirical networks tested in this paper, exponential growth characterizes the early stages of the epidemic, except when the network is restricted by a strong low-dimensional spacial constraint, such as is the case for the two-dimensional square lattice. However, on finite integer lattices of sufficiently high dimension, the early development of epidemics shows exponential growth.Comment: To be included in the conference proceedings for SPAS 2017 (International Conference on Stochastic Processes and Algebraic Structures), October 4-6, 201

Intensity enhancement of O VI ultraviolet emission lines in solar spectra due to opacity

Opacity is a property of many plasmas, and it is normally expected that if an emission line in a plasma becomes optically thick, its intensity ratio to that of another transition that remains optically thin should decrease. However, radiative transfer calculations undertaken both by ourselves and others predict that under certain conditions the intensity ratio of an optically thick to thin line can show an increase over the optically thin value, indicating an enhancement in the former. These conditions include the geometry of the emitting plasma and its orientation to the observer. A similar effect can take place between lines of differing optical depth. Previous observational studies have focused on stellar point sources, and here we investigate the spatially-resolved solar atmosphere using measurements of the I(1032 A)/I(1038 A) intensity ratio of O VI in several regions obtained with the Solar Ultraviolet Measurements of Emitted Radiation (SUMER) instrument on board the Solar and Heliospheric Observatory (SoHO) satellite. We find several I(1032 A)/I(1038 A) ratios observed on the disk to be significantly larger than the optically thin value of 2.0, providing the first detection (to our knowledge) of intensity enhancement in the ratio arising from opacity effects in the solar atmosphere. Agreement between observation and theory is excellent, and confirms that the O VI emission originates from a slab-like geometry in the solar atmosphere, rather than from cylindrical structures.Comment: 17 pages, 4 figures, ApJ Letters, in pres