5,113 research outputs found
Temporal Gillespie algorithm: Fast simulation of contagion processes on time-varying networks
Stochastic simulations are one of the cornerstones of the analysis of
dynamical processes on complex networks, and are often the only accessible way
to explore their behavior. The development of fast algorithms is paramount to
allow large-scale simulations. The Gillespie algorithm can be used for fast
simulation of stochastic processes, and variants of it have been applied to
simulate dynamical processes on static networks. However, its adaptation to
temporal networks remains non-trivial. We here present a temporal Gillespie
algorithm that solves this problem. Our method is applicable to general Poisson
(constant-rate) processes on temporal networks, stochastically exact, and up to
multiple orders of magnitude faster than traditional simulation schemes based
on rejection sampling. We also show how it can be extended to simulate
non-Markovian processes. The algorithm is easily applicable in practice, and as
an illustration we detail how to simulate both Poissonian and non-Markovian
models of epidemic spreading. Namely, we provide pseudocode and its
implementation in C++ for simulating the paradigmatic
Susceptible-Infected-Susceptible and Susceptible-Infected-Recovered models and
a Susceptible-Infected-Recovered model with non-constant recovery rates. For
empirical networks, the temporal Gillespie algorithm is here typically from 10
to 100 times faster than rejection sampling.Comment: Minor changes and updates to reference
Optimized Gillespie algorithms for the simulation of Markovian epidemic processes on large and heterogeneous networks
Numerical simulation of continuous-time Markovian processes is an essential
and widely applied tool in the investigation of epidemic spreading on complex
networks. Due to the high heterogeneity of the connectivity structure through
which epidemics is transmitted, efficient and accurate implementations of
generic epidemic processes are not trivial and deviations from statistically
exact prescriptions can lead to uncontrolled biases. Based on the Gillespie
algorithm (GA), in which only steps that change the state are considered, we
develop numerical recipes and describe their computer implementations for
statistically exact and computationally efficient simulations of generic
Markovian epidemic processes aiming at highly heterogeneous and large networks.
The central point of the recipes investigated here is to include phantom
processes, that do not change the states but do count for time increments. We
compare the efficiencies for the susceptible-infected-susceptible, contact
process and susceptible-infected-recovered models, that are particular cases of
a generic model considered here. We numerically confirm that the simulation
outcomes of the optimized algorithms are statistically indistinguishable from
the original GA and can be several orders of magnitude more efficient.Comment: 12 pages, 9 figure
Efficient sampling of spreading processes on complex networks using a composition and rejection algorithm
Efficient stochastic simulation algorithms are of paramount importance to the
study of spreading phenomena on complex networks. Using insights and analytical
results from network science, we discuss how the structure of contacts affects
the efficiency of current algorithms. We show that algorithms believed to
require or even operations per
update---where is the number of nodes---display instead a polynomial
scaling for networks that are either dense or sparse and heterogeneous. This
significantly affects the required computation time for simulations on large
networks. To circumvent the issue, we propose a node-based method combined with
a composition and rejection algorithm, a sampling scheme that has an
average-case complexity of per update for general
networks. This systematic approach is first set-up for Markovian dynamics, but
can also be adapted to a number of non-Markovian processes and can enhance
considerably the study of a wide range of dynamics on networks.Comment: 12 pages, 7 figure
Stochastic spreading on complex networks
Complex interacting systems are ubiquitous in nature and society. Computational modeling of these systems is, therefore, of great relevance for science and engineering. Complex networks are common representations of these systems (e.g., friendship networks or road networks). Dynamical processes (e.g., virus spreading, traffic jams) that evolve on these networks are shaped and constrained by the underlying connectivity. This thesis provides numerical methods to study stochastic spreading processes on complex networks. We consider the processes as inherently probabilistic and analyze their behavior through the lens of probability theory. While powerful theoretical frameworks (like the SIS-epidemic model and continuous-time Markov chains) already exist, their analysis is computationally challenging. A key contribution of the thesis is to ease the computational burden of these methods. Particularly, we provide novel methods for the efficient stochastic simulation of these processes. Based on different simulation studies, we investigate techniques for optimal vaccine distribution and critically address the usage of mathematical models during the Covid-19 pandemic. We also provide model-reduction techniques that translate complicated models into simpler ones that can be solved without resorting to simulations. Lastly, we show how to infer the underlying contact data from node-level observations.Komplexe, interagierende Systeme sind in Natur und Gesellschaft allgegenwärtig. Die computergestützte Modellierung dieser Systeme ist daher von immenser Bedeutung für Wissenschaft und Technik. Netzwerke sind eine gängige Art, diese Systeme zu repräsentieren (z. B. Freundschaftsnetzwerke, Straßennetze). Dynamische Prozesse (z. B. Epidemien, Staus), die sich auf diesen Netzwerken ausbreiten, werden durch die spezifische Konnektivität geformt. In dieser Arbeit werden numerische Methoden zur Untersuchung stochastischer Ausbreitungsprozesse in komplexen Netzwerken entwickelt. Wir betrachten die Prozesse als inhärent probabilistisch und analysieren ihr Verhalten nach wahrscheinlichkeitstheoretischen Fragestellungen. Zwar gibt es bereits theoretische Grundlagen und Paradigmen (wie das SIS-Epidemiemodell und zeitkontinuierliche Markov-Ketten), aber ihre Analyse ist rechnerisch aufwändig. Ein wesentlicher Beitrag dieser Arbeit besteht darin, die Rechenlast dieser Methoden zu verringern. Wir erforschen Methoden zur effizienten Simulation dieser Prozesse. Anhand von Simulationsstudien untersuchen wir außerdem Techniken für optimale Impfstoffverteilung und setzen uns kritisch mit der Verwendung mathematischer Modelle bei der Covid-19-Pandemie auseinander. Des Weiteren führen wir Modellreduktionen ein, mit denen komplizierte Modelle in einfachere umgewandelt werden können. Abschließend zeigen wir, wie man von Beobachtungen einzelner Knoten auf die zugrunde liegende Netzwerkstruktur schließt
Simulation and inference algorithms for stochastic biochemical reaction networks: from basic concepts to state-of-the-art
Stochasticity is a key characteristic of intracellular processes such as gene
regulation and chemical signalling. Therefore, characterising stochastic
effects in biochemical systems is essential to understand the complex dynamics
of living things. Mathematical idealisations of biochemically reacting systems
must be able to capture stochastic phenomena. While robust theory exists to
describe such stochastic models, the computational challenges in exploring
these models can be a significant burden in practice since realistic models are
analytically intractable. Determining the expected behaviour and variability of
a stochastic biochemical reaction network requires many probabilistic
simulations of its evolution. Using a biochemical reaction network model to
assist in the interpretation of time course data from a biological experiment
is an even greater challenge due to the intractability of the likelihood
function for determining observation probabilities. These computational
challenges have been subjects of active research for over four decades. In this
review, we present an accessible discussion of the major historical
developments and state-of-the-art computational techniques relevant to
simulation and inference problems for stochastic biochemical reaction network
models. Detailed algorithms for particularly important methods are described
and complemented with MATLAB implementations. As a result, this review provides
a practical and accessible introduction to computational methods for stochastic
models within the life sciences community
Modeling competition between two pharmaceutical drugs using innovation diffusion models
The study of competition among brands in a common category is an interesting
strategic issue for involved firms. Sales monitoring and prediction of
competitors' performance represent relevant tools for management. In the
pharmaceutical market, the diffusion of product knowledge plays a special role,
different from the role it plays in other competing fields. This latent feature
naturally affects the evolution of drugs' performances in terms of the number
of packages sold. In this paper, we propose an innovation diffusion model that
takes the spread of knowledge into account. We are motivated by the need of
modeling competition of two antidiabetic drugs in the Italian market.Comment: Published at http://dx.doi.org/10.1214/15-AOAS868 in the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Some Remarks about the Complexity of Epidemics Management
Recent outbreaks of Ebola, H1N1 and other infectious diseases have shown that
the assumptions underlying the established theory of epidemics management are
too idealistic. For an improvement of procedures and organizations involved in
fighting epidemics, extended models of epidemics management are required. The
necessary extensions consist in a representation of the management loop and the
potential frictions influencing the loop. The effects of the non-deterministic
frictions can be taken into account by including the measures of robustness and
risk in the assessment of management options. Thus, besides of the increased
structural complexity resulting from the model extensions, the computational
complexity of the task of epidemics management - interpreted as an optimization
problem - is increased as well. This is a serious obstacle for analyzing the
model and may require an additional pre-processing enabling a simplification of
the analysis process. The paper closes with an outlook discussing some
forthcoming problems
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