Second look at the spread of epidemics on networks

In an important paper, M.E.J. Newman claimed that a general network-based stochastic Susceptible-Infectious-Removed (SIR) epidemic model is isomorphic to a bond percolation model, where the bonds are the edges of the contact network and the bond occupation probability is equal to the marginal probability of transmission from an infected node to a susceptible neighbor. In this paper, we show that this isomorphism is incorrect and define a semi-directed random network we call the epidemic percolation network that is exactly isomorphic to the SIR epidemic model in any finite population. In the limit of a large population, (i) the distribution of (self-limited) outbreak sizes is identical to the size distribution of (small) out-components, (ii) the epidemic threshold corresponds to the phase transition where a giant strongly-connected component appears, (iii) the probability of a large epidemic is equal to the probability that an initial infection occurs in the giant in-component, and (iv) the relative final size of an epidemic is equal to the proportion of the network contained in the giant out-component. For the SIR model considered by Newman, we show that the epidemic percolation network predicts the same mean outbreak size below the epidemic threshold, the same epidemic threshold, and the same final size of an epidemic as the bond percolation model. However, the bond percolation model fails to predict the correct outbreak size distribution and probability of an epidemic when there is a nondegenerate infectious period distribution. We confirm our findings by comparing predictions from percolation networks and bond percolation models to the results of simulations. In an appendix, we show that an isomorphism to an epidemic percolation network can be defined for any time-homogeneous stochastic SIR model.Comment: 29 pages, 5 figure

Extinction in Lotka-Volterra model

Competitive birth-death processes often exhibit an oscillatory behavior. We investigate a particular case where the oscillation cycles are marginally stable on the mean-field level. An iconic example of such a system is the Lotka-Volterra model of predator-prey competition. Fluctuation effects due to discreteness of the populations destroy the mean-field stability and eventually drive the system toward extinction of one or both species. We show that the corresponding extinction time scales as a certain power-law of the population sizes. This behavior should be contrasted with the extinction of models stable in the mean-field approximation. In the latter case the extinction time scales exponentially with size.Comment: 11 pages, 17 figure

Effects of aging and links removal on epidemic dynamics in scale-free networks

We study the combined effects of aging and links removal on epidemic dynamics in the Barab\'{a}si-Albert scale-free networks. The epidemic is described by a susceptible-infected-refractory (SIR) model. The aging effect of a node introduced at time $t_{i}$ is described by an aging factor of the form $(t-t_{i})^{-\beta}$ in the probability of being connected to newly added nodes in a growing network under the preferential attachment scheme based on popularity of the existing nodes. SIR dynamics is studied in networks with a fraction $1-p$ of the links removed. Extensive numerical simulations reveal that there exists a threshold $p_{c}$ such that for $p \geq p_{c}$, epidemic breaks out in the network. For $p < p_{c}$, only a local spread results. The dependence of $p_{c}$ on $\beta$ is studied in detail. The function $p_{c}(\beta)$ separates the space formed by $\beta$ and $p$ into regions corresponding to local and global spreads, respectively.Comment: 8 pages, 3 figures, revtex, corrected Ref.[11

Synthetic curved DNA sequences can act as transcriptional activators in Escherichia coli.

Can a transcriptional activator known to bend DNA be functionally replaced by a sequence-directed bend in Escherichia coli? To investigate this question, a partially truncated promoter was used, deleted of its -35 region and of its CRP binding site, leaving only two Pribnow boxes as functional elements. Synthetic and naturally occurring curved DNA sequences introduced upstream from these elements could restore transcription at either one of the two natural starts. Some of these hybrid promoters turned out to be more efficient than the CRP activated wild-type gal promoter in vivo. Control experiments performed with very similar sequences devoid of any curvature produced weak promoters only. Minimal changes in the location of the centre of curvature or perturbation in the amount of curvature strongly affected the level of expression. No significant stimulation of transcription could be detected in vitro. Furthermore, both gal P1 and P2 starts could be activated in vivo but also in vitro via a properly positioned CRP binding site. This partial analogy suggests that bending induced by the cAMP-CRP complex upon binding to its site may be biologically relevant to the mechanism of transcriptional activation

The role of clustering and gridlike ordering in epidemic spreading

The spreading of an epidemic is determined by the connectiviy patterns which underlie the population. While it has been noted that a virus spreads more easily on a network in which global distances are small, it remains a great challenge to find approaches that unravel the precise role of local interconnectedness. Such topological properties enter very naturally in the framework of our two-timestep description, also providing a novel approach to tract a probabilistic system. The method is elaborated for SIS-type epidemic processes, leading to a quantitative interpretation of the role of loops up to length 4 in the onset of an epidemic.Comment: Submitted to Phys. Rev. E; 15 pages, 11 figures, 5 table

Epidemic variability in complex networks

We study numerically the variability of the outbreak of diseases on complex networks. We use a SI model to simulate the disease spreading at short times, in homogeneous and in scale-free networks. In both cases, we study the effect of initial conditions on the epidemic's dynamics and its variability. The results display a time regime during which the prevalence exhibits a large sensitivity to noise. We also investigate the dependence of the infection time on nodes' degree and distance to the seed. In particular, we show that the infection time of hubs have large fluctuations which limit their reliability as early-detection stations. Finally, we discuss the effect of the multiplicity of shortest paths between two nodes on the infection time. Furthermore, we demonstrate that the existence of even longer paths reduces the average infection time. These different results could be of use for the design of time-dependent containment strategies

Stochastic Description of Aircraft Simulation Models and Numerical Approaches

Der Artikel befasst sich mit der Unsicherheitsquantifizierung eines Flugzeugsimulationsmodells. Mathematisch gesehen handelt es sich bei dem Flugzeugmodell um ein System von Differentialgleichungen zweiter Ordnung. Dabei hängt das System von Eingangsparametern ab, die der Masse, der Aerodynamik und der Struktur des Flugzeugs zugeordnet sind. Die aerodynamischen Eingangsparameter sind als Zufallsvariablen und -prozesse modelliert, deren Wahrscheinlichkeitsverteilungen nach dem Maximum-Entropie-Prinzip und verfügbaren experimentellen Daten ausgewählt werden. Für das Flugdynamikmodell wird die Unsicherheitsentwicklung der Flugzustandsverläufe mittels sogenannter nicht-intrusiver numerischer Verfahren abgeschätzt. Die Verfahren sind beispielsweise direkte Integrationsverfahren, stochastische Kollokationsverfahren und Pseudo-Galerkin Verfahren. Diese numerischen Verfahren basieren auf Stichproben von Flugzustandsverläufen, welche durch das Lösen der zugehörigen deterministischen gewöhnlichen Differentialgleichungen erhalten werden.This paper is concerned with the uncertainty quantification of an aircraft simulation model. Mathematically speaking the aircraft model represents a system of second order differential equations dependent on a set of input parameters related to the mass, aerodynamics and the structure of the aircraft. The input aerodynamic parameters are modelled as random variables and processes whose probability distributions are chosen according to the maximum entropy principle and available experimental data. For a flight dynamics model the evolution of uncertainties in the aircraft state trajectory is estimated with the help of so-called non-intrusive numerical approaches, examples of which are the direct integration method, the stochastic collocation approach and the pseudo-Galerkin method. These numerical methods rely on a set of samples of aircraft state trajectories simply obtained by solving the corresponding systems of deterministic ordinary differential equations