Threshold effects for two pathogens spreading on a network

Diseases spread through host populations over the networks of contacts between individuals and a number of results about this process have been derived in recent years by exploiting connections between epidemic processes and bond percolation on networks. Here we investigate the case of two pathogens in a single population, which has been the subject of recent interest among epidemiologists. We demonstrate that two pathogens competing for the same hosts can both spread through a population only for intermediate values of the bond occupation probability that lie above the classic epidemic threshold and below a second higher value, which we call the coexistence threshold, corresponding to a distinct topological phase transition in networked systems.     

Stochastic epidemics and rumours on finite random networks

In this paper, we investigate the stochastic spread of epidemics and rumours on networks. We focus on the general stochastic (SIR) epidemic model and a recently proposed rumour model on networks in Nekovee et al. (2007) [3], and on networks with different random structures, taking into account the structure of the underlying network at the level of the degree–degree correlation function. Using embedded Markov chain techniques and ignoring density correlations between neighbouring nodes, we derive a set of equations for the final size of the epidemic/rumour on a homogeneous network that can be solved numerically, and compare the resulting distribution with the solution of the corresponding mean-field deterministic model. The final size distribution is found to switch from unimodal to bimodal form (indicating the possibility of substantial spread of the epidemic/rumour) at a threshold value that is higher than that for the deterministic model. However, the difference between the two thresholds decreases with the network size, n, following a n^−1/3 behaviour. We then compare results (obtained by Monte Carlo simulation) for the full stochastic model on a homogeneous network, including density correlations at neighbouring nodes, with those for the approximating stochastic model and show that the latter reproduces the exact simulation results with great accuracy. Finally, further Monte Carlo simulations of the full stochastic model are used to explore the effects on the final size distribution of network size and structure (using homogeneous networks, simple random graphs and the Barabasi–Albert scale-free networks).     

Local and occupation time of a particle diffusing in a random medium

We consider a particle moving in a one-dimensional potential which has a symmetric deterministic part and a quenched random part. We study analytically the probability distributions of the local time (spent by the particle around its mean value) and the occupation time (spent above its mean value) within an observation time window of size t. In the absence of quenched randomness, these distributions have three typical asymptotic behaviors depending on whether the deterministic potential is unstable, stable, or flat. These asymptotic behaviors are shown to get drastically modified when the random part of the potential is switched on, leading to the loss of self-averaging and wide sample to sample fluctuations.     

Stability analysis of multi-group deterministic and stochastic epidemic models with vaccination rate

We discuss in this paper a deterministic multi-group MSIR epidemic model with a vaccination rate, the basic reproduction number R0, a key parameter in epidemiology, is a threshold which determines the persistence or extinction of the disease. By using Lyapunov function techniques, we show if R0 is greater than 1 and the deterministic model obeys some conditions, then the disease will prevail, the infective persists and the endemic state is asymptotically stable in a feasible region. If R0 is less than or equal to 1, then the infective disappear so the disease dies out. In addition, stochastic noises around the endemic equilibrium will be added to the deterministic MSIR model in order that the deterministic model is extended to a system of stochastic ordinary differential equations. In the stochastic version, we carry out a detailed analysis on the asymptotic behavior of the stochastic model. In addition, regarding the value of R0, when the stochastic system obeys some conditions and R0 is greater than 1, we deduce the stochastic system is stochastically asymptotically stable.Finally, the deterministic and stochastic model dynamics are illustrated through computer simulations.     

Phase Transition between Synchronous and Asynchronous Updating Algorithms

We update a one-dimensional chain of Ising spins of length L with algorithms which are parameterized by the probability p for a certain site to get updated in one time step. The result of the update event itself is determined by the energy change due to the local change in the configuration. In this way we interpolate between the Metropolis algorithm at zero temperature when p is of the order of 1/L and L is large, and a synchronous deterministic updating procedure for p=1. As a function of p we observe a phase transition between the stationary states to which the algorithm drives the system. These are non-absorbing stationary states with antiferromagnetic domains for p>p _c , and absorbing states with ferromagnetic domains for p≤p _c . This means that above this transition the stationary states have lost any remnants of the ferromagnetic Ising interaction. A measurement of the critical exponents shows that this transition belongs to the universality class of parity conservation.     

Universal crossing probabilities and incipient spanning clusters in directed percolation

Shape-dependent universal crossing probabilities are studied, via Monte Carlo simulations, for bond and site directed percolation on the square lattice in the diagonal direction, at the percolation threshold. In a dynamical interpretation, the crossing probability is the probability that, on a system with size L, an epidemic spreading without immunization remains active at time t. Since the system is strongly anisotropic, the shape dependence in space-time enters through the effective aspect ratio r _eff = ct/L ^z, where c is a non-universal constant and z the anisotropy exponent. A particular attention is paid to the influence of the initial state on the universal behaviour of the crossing probability. Using anisotropic finite-size scaling and generalizing a simple argument given by Aizenman for isotropic percolation, we also obtain the behaviour of the probability to find n incipient spanning clusters on a finite system at time t. The numerical results are in good agreement with the conjecture. Received 10 February 2003 Published online 20 June 2003 RID="a" ID="a"e-mail: turban@lpm.u-nancy.fr RID="b" ID="b"UMR CNRS 7556     

Finite-Time Current Probabilities in the Asymmetric Exclusion Process on a Ring

We calculate the time-dependent probability distribution of current through a selected bond in the totally asymmetric exclusion process with periodic boundary conditions. We derive a general formula for the probability that the integrated current exceeds a given value N at the moment of time t. The formula is written in a form of a contour integral of a determinant of a Toeplitz matrix. Transforming the determinant expression, we obtain a generalization of the known formula derived by Johansson for the infinite one-dimensional lattice. To check the general formula, we consider the specific case corresponding to the probability of a minimal non-zero current. For this case we get an explicit analytical expression and analyze its asymptotics.     

On Random Flights with Non-uniformly Distributed Directions

This paper deals with a new class of random flights in ℝ ^d , d≥2, characterized by non-uniform probability distributions on the multidimensional sphere. These random motions differ from similar models appeared in literature where the directions are taken according to the uniform law. The family of angular probability distributions introduced in this paper depends on a parameter ν≥0, which gives the anisotropy of the motion. Furthermore, we assume that the number of changes of direction performed by the random flight is fixed. The time lengths between two consecutive changes of orientation have joint probability distribution given by a Dirichlet density function.     

Description of a class of Markov processes “equivalent” toK-shifts

It results from recent works of Prigogine and collaborators that one can construct a nonunitary operator which realizes an “equivalence” between the positive actions of a reversible dynamical system and an irreversible Markov process going to equilibrium. We consider here this construction and we prove that (a) forK-shifts the transition probability of the associated Markov process is concentrated in the stable manifold of the transformed point by the shift with a point mass concentrated on the deterministic trajectory; and (b) for Bernoulli shifts the measures which go to equilibrium are the same for the deterministic system and the Markov process.     

Time distribution and loss of scaling in granular flow

Two cellular automata models with directed mass flow and internal time scales are studied by numerical simulations. Relaxation rules are a combination of probabilistic critical height (probability of toppling p) and deterministic critical slope processes with internal correlation time t_c equal to the avalanche lifetime, in model A, and ,in model B. In both cases nonuniversal scaling properties of avalanche distributions are found for , where is related to directed percolation threshold in d=3. Distributions of avalanche durations for are studied in detail, exhibiting multifractal scaling behavior in model A, and finite size scaling behavior in model B, and scaling exponents are determined as a function of p. At a phase transition to noncritical steady state occurs. Due to difference in the relaxation mechanisms, avalanche statistics at approaches the parity conserving universality class in model A, and the mean-field universality class in model B. We also estimate roughness exponent at the transition. Received: 29 May 1998 / Revised: 8 September 1998 / Accepted: 10 September 1998     

A Maximum Entropy Method for Particle Filtering

Standard ensemble or particle filtering schemes do not properly represent states of low priori probability when the number of available samples is too small, as is often the case in practical applications. We introduce here a set of parametric resampling methods to solve this problem. Motivated by a general H-theorem for relative entropy, we construct parametric models for the filter distributions as maximum-entropy/minimum-information models consistent with moments of the particle ensemble. When the prior distributions are modeled as mixtures of Gaussians, our method naturally generalizes the ensemble Kalman filter to systems with highly non-Gaussian statistics. We apply the new particle filters presented here to two simple test cases: a one-dimensional diffusion process in a double-well potential and the three-dimensional chaotic dynamical system of Lorenz.     

Determinism,noise, and spurious estimations in a generalised model of population growth

We study a generalised model of population growth in which the state variable is population growth rate instead of population size. Stochastic parametric perturbations, modelling phenotypic variability, lead to a Langevin system with two sources of multiplicative noise. The stationary probability distributions have two characteristic power-law scales. Numerical simulations show that noise suppresses the explosion of the growth rate which occurs in the deterministic counterpart. Instead, in different parameter regimes populations will grow with “anomalous” stochastic rates and (i) stabilise at “random carrying capacities”, or (ii) go extinct in random times. Using logistic fits to reconstruct the simulated data, we find that even highly significant estimations do not recover or reflect information about the deterministic part of the process. Therefore, the logistic interpretation is not biologically meaningful. These results have implications for distinct model-aided calculations in biological situations because these kinds of estimations could lead to spurious conclusions.     

Stochastic aspects of pattern formation during the catalytic oxidation of CO on Pd(111) surfaces

We present experimental results on rare transitions between two states due to intrinsic noise between two states in a bistable surface reaction, namely the catalytic oxidation of CO on Pd(111) surfaces. The mean time scales involved are typically of order 10⁴ s and the probability distribution shows two peaks over a large part of the bistable regime of this surface reaction. We use measurements of the resulting CO₂ rate as well as photoelectron emission microscopy (PEEM) to characterize these rare transitions. From our dynamic data we can extract probability distributions for the CO₂ rate. We use x-t plots from PEEM measurements to describe the transitions, which are-as we demonstrate-characterized by one wall moving through the field of view in PEEM measurements. The resulting probability distributions for the CO₂ rate are shown to depend strongly on the value, Y, of the CO fraction in the reactant flux inside the bistable regime. We find that the probability distribution is strongly asymmetric indicating that the two basins of attraction are rather different in depth and width. This is also concluded from the PEEM measurements, which show in one case a rather sharp and narrow domain wall going one way, while it is rather wide and diffuse for the motion in the opposite direction. To have two basins of attraction in the bistable regime, which are rather different in nature is reminiscent of other bistable systems such as, for example, optical bistability, although the time scales involved in the present system are entirely different.     

Finite Time Corrections in KPZ Growth Models

We consider some models in the Kardar-Parisi-Zhang universality class, namely the polynuclear growth model and the totally/partially asymmetric simple exclusion process. For these models, in the limit of large time t, universality of fluctuations has been previously obtained. In this paper we consider the convergence to the limiting distributions and determine the (non-universal) first order corrections, which turn out to be a non-random shift of order t ^−1/3 (of order 1 in microscopic units). Subtracting this deterministic correction, the convergence is then of order t ^−2/3. We also determine the strength of asymmetry in the exclusion process for which the shift is zero. Finally, we discuss to what extend the discreteness of the model has an effect on the fitting functions.     

Effects of the Low Frequencies of Noise on On–Off Intermittency

A bifurcating system subject to multiplicative noise can exhibit on–off intermittency close to the instability threshold. For a canonical system, we discuss the dependence of this intermittency on the Power Spectrum Density (PSD) of the noise. Our study is based on the calculation of the Probability Density Function (PDF) of the unstable variable. We derive analytical results for some particular types of noises and interpret them in the framework of on-off intermittency. Besides, we perform a cumulant expansion (N. G. van Kampen, 24, 171 (1976).) for a random noise with arbitrary power spectrum density and we show that the intermittent regime is controlled by the ratio between the departure from the threshold and the value of the PSD of the noise at zero frequency. Our results are in agreement with numerical simulations performed with two types of random perturbations: colored Gaussian noise and deterministic fluctuations of a chaotic variable. Extensions of this study to another, more complex, system are presented and the underlying mechanisms are discussed. PACS Number: 05.40.-a, 05.45.-a, 91.25.-r     

Towards a Theory of Transition Paths

We construct a statistical theory of reactive trajectories between two pre-specified sets A and B, i.e. the portionsof the path of a Markov process during which the path makes a transition from A to B. This problem is relevant e.g. in the context of metastability, in which case the two sets A and B are metastable sets, though the formalism we propose is independent of any such assumptions on A and B. We show that various probability distributions on the reactive trajectories can be expressed in terms of the equilibrium distribution of the process and the so-called committor functions which give the probability that the process reaches first B before reaching A, either backward or forward in time. Using these objects, we obtain (i) the distribution of reactive trajectories, which gives the proportion of time reactive trajectories spend in sets outside of A and B; (ii) the hitting point distribution of the reactive trajectories on a surface, which measures where the reactive trajectories hit the surface when they cross it; (iii) the last hitting point distribution of the reactive trajectories on the surface; (iv) the probability current of reactive trajectories, the integral of which on a surface gives the net average flux of reactive trajectories across this surface; (v) the average frequency of reactive trajectories, which gives the average number of transitions between A and B per unit of time; and (vi) the traffic distribution of reactive trajectories, which gives some information about the regions the reactive trajectories visit regardless of the time they spend in these regions.     

Microscopic derivation of a Markovian Master equation in a deterministic model of chemical reaction

We consider a (deterministic, conservative) one-dimensional system of colored hard points, changing color each time they hit one another with a relative velocity above a threshold. In the limit of rare reactions, theN-particle color distribution follows a Markovian birth-and-death process. Using the reaction rate as an intrinsic time scale, we also obtain the reaction-diffusion equation for a test particle in this hydrodynamic limit. Explicit results are given for a discrete and a Maxwellian velocity distribution.     

LRS Bianchi Type II Bulk Viscous Fluid Universe with?Decaying Vacuum Energy Density Λ

The present study deals with spatially homogeneous and locally rotationally symmetric (LRS) Bianchi type II cosmological models with bulk viscous fluid distribution of matter and decaying vacuum energy density Λ. To get the deterministic models of the universe, we assume that the expansion (θ) in the model is proportional to the shear (σ). This leads to condition R=mS ⁿ , where R and S are metric potentials, m and n are constants. We have obtained two types of models of the universe for two different values of n. The vacuum energy density Λ for both models is found to be a decreasing function of time and it approaches a small positive value at late time which is supported by recent results from the observations of (SN Ia). Some physical and geometric behaviour of these models are also discussed.     

Fluctuation Theorems for Entropy Production and Heat Dissipation in Periodically Driven Markov Chains

Asymptotic fluctuation theorems are statements of a Gallavotti-Cohen symmetry in the rate function of either the time-averaged entropy production or heat dissipation of a process. Such theorems have been proved for various general classes of continuous-time deterministic and stochastic processes, but always under the assumption that the forces driving the system are time independent, and often relying on the existence of a limiting ergodic distribution. In this paper we extend the asymptotic fluctuation theorem for the first time to inhomogeneous continuous-time processes without a stationary distribution, considering specifically a finite state Markov chain driven by periodic transition rates. We find that for both entropy production and heat dissipation, the usual Gallavotti-Cohen symmetry of the rate function is generalized to an analogous relation between the rate functions of the original process and its corresponding backward process, in which the trajectory and the driving protocol have been time-reversed. The effect is that spontaneous positive fluctuations in the long time average of each quantity in the forward process are exponentially more likely than spontaneous negative fluctuations in the backward process, and vice-versa, revealing that the distributions of fluctuations in universes in which time moves forward and backward are related. As an additional result, the asymptotic time-averaged entropy production is obtained as the integral of a periodic entropy production rate that generalizes the constant rate pertaining to homogeneous dynamics.