We examine the coevolution of N cycles and force chains as part of a broader study which is designed to quantitatively characterize the role of the laterally supporting contact network to the evolution of force chains. Here, we elucidate the rheological function of these coexisting structures, especially in the lead up to failure. In analogy to force chains, we introduce the concept of force cycles: N cycles whose contacts each bear above average force. We examine their evolution around force chains in a discrete element simulation of a dense granular material under quasistatic biaxial loading. Three-force cycles are shown to be stabilizing structures that inhibit relative particle rotations and provide strong lateral support to force chains. These exhibit distinct behavior from other cycles. Their population decreases rapidly during the initial stages of the strain-hardening regime-a trend that is suddenly interrupted and reversed upon commencement of force chain buckling prior to peak shear stress. Results suggest that the three-force cycles are called upon for reinforcements to ward off failure via shear banding. Ultimately though, the resistance to buckling proves futile; buckling wins under the combined effects of dilatation and increasing compressive load. The sudden increase in three-force cycles may thus be viewed as an indicator of imminent failure via shear bands.

We examine epidemic thresholds for disease spread using susceptible-infected-susceptible models on scalefree networks with variable infectivity. Infectivity between nodes is modeled as a piecewise linear function of the node degree ͑rather than the less realistic linear transformation considered previously͒. With this nonlinear infectivity, we derive conditions for the epidemic threshold to be positive. The effects of various immunization schemes including ring and targeted vaccination are studied and compared. We find that both targeted and ring immunization strategies compare favorably to a proportional scheme in terms of effectiveness.

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a b s t r a c tOne of the great challenges in the science of complex materials -materials capable of emergent behavior such as self-organized pattern formation -is deciphering their ''inherent" structural design principles as they deform in response to external loads. We have been exploring the efficacy of techniques from complex networks to the study of dense granular materials as a means to: (i) uncover such design principles and (ii) identify suitable metrics that quantify the evolution of structure during deformation. Herein, we characterize the developing network structure and loss of connectivity in a quasistatically deforming granular medium from the perspective of complex networks. Attention is paid to the evolution of the contact and contact force networks at the local or mesoscopic level, i.e., a particle and its immediate neighbors, as well as the macroscopic level. We explore network motifs and other topological properties at these multiple length scales, in an attempt to find that which best correlates with the constitutive properties of nonaffine deformation and dissipation, spatially and with respect to strain. Key processes or rearrangement events that cause loss of connectivity within the material domain, e.g. microbanding and force chain buckling, are investigated. Network statistics of these processes, previously shown to be major sources of energy dissipation and nonaffine deformation, are then tied to corresponding trends observed in the evolving macroscopic network. It is shown that consideration of the unweighted contact network alone is insufficient to tie dissipation to loss of material connectivity.

We describe a stochastic small-world network model of transmission of the SARS virus. Unlike the standard Susceptible-Infected-Removed models of disease transmission, our model exhibits both geographically localised outbreaks and "super-spreaders". Moreover, the combination of localised and long range links allows for more accurate modelling of partial isolation and various public health policies. From this model, we derive an expression for the probability of a widespread outbreak and a condition to ensure that the epidemic is controlled. Moreover, multiple simulations are used to make predictions of the likelihood of various eventual scenarios for fixed initial conditions. The main conclusions of this study are: (i) "super-spreaders" may occur even if the infectiousness of all infected individuals is constant; (ii) consistent with previous reports, extended exposure time beyond 3-5 days (i.e. significant nosocomial transmission) was the key factor in the severity of the SARS outbreak in Hong Kong; and, (iii) the spread of SARS can be effectively controlled by either limiting long range links (imposing a partial quarantine) or enforcing rapid hospitalisation and isolation of symptomatic individuals. In addition, the epidemiological data currently available for Hong Kong is far superior to that of the Chinese mainland. 1 Two characteristic features were observed during the SARS outbreak in Hong Kong in 2003 (see Fig. 1 E-mail address: ensmall@polyu.edu.hk (M. Small). 1 During the epidemic, mainland authorities classified information on SARS infections as a state secret. Moreover, bureaucracy caused much of the available information to be concealed. Despite this, subsequent official investigation indicates that the infection rate for the Chinese mainland was significantly overreported. The reliability of data from China is therefore uncertain. a large number of cases; and persistent transmission within the community. Two widely cited SSEs were observed early in the epidemic and have been the subject of much attention: at the Amoy Gardens housing estate and at the Prince of Wales hospital. Epidemiological studies [5,1] have found that in Hong Kong:• the fatality rate was approximately 17% (compared to 11% globally); • the mean incubation period was 6.4 days (range 2-10 days) [6]; • the duration between onset of symptoms and hospitalisation was 3-5 days; and, • the mean number of individuals infected by each case during the initial phase of the epidemic (excluding SSEs) was 2.7 [4]. Standard deterministic SIR (susceptible-infected-removed) models of the spread of infectious diseases [7] make several important assumptions. An alternative approach [8], particularly popular for the study of sexually transmitted diseases [9][10][11], is to build an explicit network and model 0167-2789/$ -see front matter c