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Register a new userDesigning Attractive Models via Automated Identification of Chaotic and Oscillatory Dynamical Regimes
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Added by
Michael Stumpf
Publication date
2011
fields
Mathematical Statistics
and research's language is
English
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Chaos and oscillations continue to capture the interest of both the scientific and public domains. Yet despite the importance of these qualitative features, most attempts at constructing mathematical models of such phenomena have taken an indirect, quantitative approach, e.g. by fitting models to a finite number of data-points. Here we develop a qualitative inference framework that allows us to both reverse engineer and design systems exhibiting these and other dynamical behaviours by directly specifying the desired characteristics of the underlying dynamical attractor. This change in perspective from quantitative to qualitative dynamics, provides fundamental and new insights into the properties of dynamical systems.
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Equipment sharing among people who inject drugs (PWID) is a key risk factor in infection by hepatitis C virus (HCV). Both the effectiveness and cost-effectiveness of interventions aimed at reducing HCV transmission in this population (such as opioid substitution therapy, needle exchange programs or improved treatment) are difficult to evaluate using field surveys. Ethical issues and complicated access to the PWID population make it difficult to gather epidemiological data. In this context, mathematical modelling of HCV transmission is a useful alternative for comparing the cost and effectiveness of various interventions. Several models have been developed in the past few years. They are often based on strong hypotheses concerning the population structure. This review presents compartmental and individual-based models in order to underline their strengths and limits in the context of HCV infection among PWID. The final section discusses the main results of the papers.
Scaling regions -- intervals on a graph where the dependent variable depends linearly on the independent variable -- abound in dynamical systems, notably in calculations of invariants like the correlation dimension or a Lyapunov exponent. In these applications, scaling regions are generally selected by hand, a process that is subjective and often challenging due to noise, algorithmic effects, and confirmation bias. In this paper, we propose an automated technique for extracting and characterizing such regions. Starting with a two-dimensional plot -- e.g., the values of the correlation integral, calculated using the Grassberger-Procaccia algorithm over a range of scales -- we create an ensemble of intervals by considering all possible combinations of endpoints, generating a distribution of slopes from least-squares fits weighted by the length of the fitting line and the inverse square of the fit error. The mode of this distribution gives an estimate of the slope of the scaling region (if it exists). The endpoints of the intervals that correspond to the mode provide an estimate for the extent of that region. When there is no scaling region, the distributions will be wide and the resulting error estimates for the slope will be large. We demonstrate this method for computations of dimension and Lyapunov exponent for several dynamical systems, and show that it can be useful in selecting values for the parameters in time-delay reconstructions.
In this paper, based on the Akaike information criterion, root mean square error and robustness coefficient, a rational evaluation of various epidemic models/methods, including seven empirical functions, four statistical inference methods and five dynamical models, on their forecasting abilities is carried out. With respect to the outbreak data of COVID-19 epidemics in China, we find that before the inflection point, all models fail to make a reliable prediction. The Logistic function consistently underestimates the final epidemic size, while the Gompertzs function makes an overestimation in all cases. Towards statistical inference methods, the methods of sequential Bayesian and time-dependent reproduction number are more accurate at the late stage of an epidemic. And the transition-like behavior of exponential growth method from underestimation to overestimation with respect to the inflection point might be useful for constructing a more reliable forecast. Compared to ODE-based SIR, SEIR and SEIR-AHQ models, the SEIR-QD and SEIR-PO models generally show a better performance on studying the COVID-19 epidemics, whose success we believe could be attributed to a proper trade-off between model complexity and fitting accuracy. Our findings not only are crucial for the forecast of COVID-19 epidemics, but also may apply to other infectious diseases.
Parkinsons disease (PD) is a common neurodegenerative disease with a high degree of heterogeneity in its clinical features, rate of progression, and change of variables over time. In this work, we present a novel data-driven, network-based Trajectory Profile Clustering (TPC) algorithm for 1) identification of PD subtypes and 2) early prediction of disease progression in individual patients. Our subtype identification is based not only on PD variables, but also on their complex patterns of progression, providing a useful tool for the analysis of large heterogenous, longitudinal data. Specifically, we cluster patients based on the similarity of their trajectories through a time series of bipartite networks connecting patients to demographic, clinical, and genetic variables. We apply this approach to demographic and clinical data from the Parkinsons Progression Markers Initiative (PPMI) dataset and identify 3 patient clusters, consistent with 3 distinct PD subtypes, each with a characteristic variable progression profile. Additionally, TPC predicts an individual patients subtype and future disease trajectory, based on baseline assessments. Application of our approach resulted in 74% accurate subtype prediction in year 5 in a test/validation cohort. Furthermore, we show that genetic variability can be integrated seamlessly in our TPC approach. In summary, using PD as a model for chronic progressive diseases, we show that TPC leverages high-dimensional longitudinal datasets for subtype identification and early prediction of individual disease subtype. We anticipate this approach will be broadly applicable to multidimensional longitudinal datasets in diverse chronic diseases.
An update is given on the current status of solar and stellar dynamos. At present, it is still unclear why stellar cycle frequencies increase with rotation frequency in such a way that their ratio increases with stellar activity. The small-scale dynamo is expected to operate in spite of a small value of the magnetic Prandtl number in stars. Whether or not the global magnetic activity in stars is a shallow or deeply rooted phenomenon is another open question. Progress in demonstrating the presence and importance of magnetic helicity fluxes in dynamos is briefly reviewed, and finally the role of nonlocality is emphasized in modeling stellar dynamos using the mean-field approach. On the other hand, direct numerical simulations have now come to the point where the models show solar-like equatorward migration that can be compared with observations and that need to be understood theoretically.
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