Analysis of Petri Net Models through Stochastic Differential Equations

It is well known, mainly because of the work of Kurtz, that density dependent Markov chains can be approximated by sets of ordinary differential equations (ODEs) when their indexing parameter grows very large. This approximation cannot capture the stochastic nature of the process and, consequently, it can provide an erroneous view of the behavior of the Markov chain if the indexing parameter is not sufficiently high. Important phenomena that cannot be revealed include non-negligible variance and bi-modal population distributions. A less-known approximation proposed by Kurtz applies stochastic differential equations (SDEs) and provides information about the stochastic nature of the process. In this paper we apply and extend this diffusion approximation to study stochastic Petri nets. We identify a class of nets whose underlying stochastic process is a density dependent Markov chain whose indexing parameter is a multiplicative constant which identifies the population level expressed by the initial marking and we provide means to automatically construct the associated set of SDEs. Since the diffusion approximation of Kurtz considers the process only up to the time when it first exits an open interval, we extend the approximation by a machinery that mimics the behavior of the Markov chain at the boundary and allows thus to apply the approach to a wider set of problems. The resulting process is of the jump-diffusion type. We illustrate by examples that the jump-diffusion approximation which extends to bounded domains can be much more informative than that based on ODEs as it can provide accurate quantity distributions even when they are multi-modal and even for relatively small population levels. Moreover, we show that the method is faster than simulating the original Markov chain

Emergent spatial correlations in stochastically evolving populations

We study the spatial pattern formation and emerging long range correlations in a model of three species coevolving in space and time according to stochastic contact rules. Analytical results for the pair correlation functions, based on a truncation approximation and supported by computer simulations, reveal emergent strategies of survival for minority agents based on selection of patterns. Minority agents exhibit defensive clustering and cooperative behavior close to phase transitions.Comment: 11 pages, 4 figures, Adobe PDF forma

Migration paths saturations in meta-epidemic systems

In this paper we consider a simple two-patch model in which a population affected by a disease can freely move. We assume that the capacity of the interconnected paths is limited, and thereby influencing the migration rates. Possible habitat disruptions due to human activities or natural events are accounted for. The demographic assumptions prevent the ecosystem to be wiped out, and the disease remains endemic in both populated patches at a stable equilibrium, but possibly also with an oscillatory behavior in the case of unidirectional migrations. Interestingly, if infected cannot migrate, it is possible that one patch becomes disease-free. This fact could be exploited to keep disease-free at least part of the population

Classification of image distortions in terms of Petrov types

An observer surrounded by sufficiently small spherical light sources at a fixed distance will see a pattern of elliptical images distributed over the sky, owing to the distortion effect (shearing effect) of the spacetime geometry upon light bundles. In lowest non-trivial order with respect to the distance, this pattern is completely determined by the conformal curvature tensor (Weyl tensor) at the observation event. In this paper we derive formulas that allow to calculate these distortion patterns in terms of the Newman-Penrose formalism. Then we represent the distortion patterns graphically for all Petrov types, and we discuss their dependence on the velocity of the observer.Comment: 22 pages, 8 eps-figures; revised version, parts of Introduction and Conclusions rewritte

Activity driven modeling of time varying networks

Network modeling plays a critical role in identifying statistical regularities and structural principles common to many systems. The large majority of recent modeling approaches are connectivity driven. The structural patterns of the network are at the basis of the mechanisms ruling the network formation. Connectivity driven models necessarily provide a time-aggregated representation that may fail to describe the instantaneous and fluctuating dynamics of many networks. We address this challenge by defining the activity potential, a time invariant function characterizing the agents' interactions and constructing an activity driven model capable of encoding the instantaneous time description of the network dynamics. The model provides an explanation of structural features such as the presence of hubs, which simply originate from the heterogeneous activity of agents. Within this framework, highly dynamical networks can be described analytically, allowing a quantitative discussion of the biases induced by the time-aggregated representations in the analysis of dynamical processes.Comment: 10 pages, 4 figure

Epidemic Enhancement in Partially Immune Populations

We observe that a pathogen introduce/pmcdata/journal/plosone/2-2007/1/ingest/pmcmod/sgml/pone.0000165.xmld into a population containing individuals with acquired immunity can result in an epidemic longer in duration and/or larger in size than if the pathogen were introduced into a naive population. We call this phenomenon “epidemic enhancement,” and use simple dynamical models to show that it is a realistic scenario within the parameter ranges of many common infectious diseases. This finding implies that repeated pathogen introduction or intermediate levels of vaccine coverage can lead to pathogen persistence in populations where extinction would otherwise be expected

Cost analysis of a vaccination startegy for respiratory syncytial virus (RSV) in a network model

[EN] In this paper an age-structured mathematical model for respiratory syncytial virus (RSV) is proposed where children younger than one year old, who are the most affected by this illness, are specially considered. Real data of hospitalized children in the Spanish region of Valencia are used in order to determine some seasonal parameters of the model. Once the parameters are determined, we propose a complete stochastic network model to study the seasonal evolution of the respiratory syncytial virus (RSV) epidemics. In this model every susceptible individual can acquire the disease after a random encounter with any infected individual in the social network. The edges of a complete graph connecting every pair of individuals in the network simulate these encounters and a season dependent probability, beta(t), determines whether the healthy susceptible individual becomes infected or not. We show that the prediction of this model is compatible with the above mentioned age-structured model based upon differential equations, but sharper peaks are obtained in the case of the network. Then, on the network model, we propose the vaccination of children at 2 months, 4 months and 1 year old, and we study the cost of this vaccination strategy, which is emerging as the most plausible one to be applied when the vaccine hits the market. It is worth to note that this vaccination strategy is simulated in the network model because to implement it in the continuous model is very difficult and increases its complexity. (C) 2010 Elsevier Ltd. All rights reserved.Acedo Rodríguez, L.; Moraño Fernández, JA.; Diez-Domingo, J. (2010). Cost analysis of a vaccination startegy for respiratory syncytial virus (RSV) in a network model. Mathematical and Computer Modelling. 52(7):1016-1022. doi:10.1016/j.mcm.2010.02.041S1016102252