Mathematics最新文献

This article is concerned with the mathematical modeling of cancer virotherapy, emphasizing the impact of Allee effects on tumor cell growth. We propose a modeling framework that describes the complex interaction between tumor cells and oncolytic viruses. The efficacy of this therapy against cancer is mathematically investigated. The analysis involves linear and logistic growth scenarios coupled with different Allee effects, including weak, strong, and hyper Allee forms. Critical points are identified, and their existence and stability are analyzed using dynamical system theories and bifurcation techniques. Also, bifurcation diagrams and numerical simulations are utilized to verify and extend analytical results. It is observed that Allee effects significantly influence the stability of the system and the conditions necessary for tumor control and eradication.

Falls are a major cause of injury among older adults. The Physio-fEedback Exercise pRogram (PEER) combines physio-feedback, cognitive reframing, and guided exercises to reduce fall risk. However, its impact on physical activity (PA) over time is underexplored. Functional time-series analysis offers insight into behavior patterns and sustainability. This preliminary study assessed PEER's effectiveness in improving PA levels immediately and over time. A total of 64 community-dwelling older adults were cluster-randomized into PEER $(N=33)$ or control groups $(N=31)$ . Participants wore Fitbit trackers, generating time-series data on activity. The PEER group completed an 8-week program, while the control group received CDC fall prevention pamphlets. PA data were analyzed using smoothing spline analysis of variance (SSANOVA), chosen for its flexibility in modeling complex, non-linear relationships in time-series data and its ability to handle skewed distributions and repeated measures. Unlike traditional parametric models, SSANOVA decomposes temporal trends into interpretable components, capturing both smooth trends and abrupt changes, such as those occurring on group workout days. This capability ensures robust and nuanced analysis of intervention effects. Results showed PEER participants significantly increased evenly and had very active minutes and reduced sedentary behavior during the intervention. No significant effect was found for light active minutes. Specifically, during the intervention period, PEER participants engaged in an average of 6.7% fewer sedentary minutes per day, 13.8% additional fairly active minutes per day, and 2.8% additional very active minutes per day compared to the control group. While the reduction in sedentary minutes and increase in fairly active minutes were not statistically significant, the increase in very active minutes was significant. However, our functional time-series analysis revealed these improvements diminished over the 15-week follow-up, indicating challenges in maintaining PA. In conclusion, PEER boosts PA and reduces sedentary behavior short-term, but strategies are needed to sustain these benefits. In conclusion, PEER boosts PA and reduces sedentary behavior short-term, but strategies are needed to sustain these benefits. Public health policies should emphasize technology-driven fall risk assessments, community-based prevention programs, and initiatives that promote physical activity, home safety, and chronic condition management.

Persistent topological Laplacians constitute a new class of tools in topological data analysis (TDA). They are motivated by the necessity to address challenges encountered in persistent homology when handling complex data. These Laplacians combine multiscale analysis with topological techniques to characterize the topological and geometrical features of functions and data. Their kernels fully retrieve the topological invariants of corresponding persistent homology, while their non-harmonic spectra provide supplementary information. Persistent topological Laplacians have demonstrated superior performance over persistent homology in the analysis of large-scale protein engineering datasets. In this survey, we offer a pedagogical review of persistent topological Laplacians formulated in various mathematical settings, including simplicial complexes, path complexes, flag complexes, digraphs, hypergraphs, hyperdigraphs, cellular sheaves, and $N$ -chain complexes.

Spatial, temporal, and space-time scan statistics can be used for geographical surveillance, identifying temporal and spatial patterns, and detecting outliers. While statistical cluster analysis is a valuable tool for identifying patterns, optimizing resource allocation, and supporting decision-making, accurately predicting future spatial clusters remains a significant challenge. Given the known relative risks of spatial clusters over the past $k$ time intervals, the main objective of the present study is to predict the relative risks for the subsequent interval, $k+1$ . Building on our prior research, we propose a predictive Markov chain model with an embedded corrector component. This corrector utilizes either multiple linear regression or exponential smoothing method, selecting the one that minimizes the relative distance between observed and predicted values in the $k$ -th interval. To test the proposed method, we first calculated the relative risks of statistically significant spatial clusters of COVID-19 mortality in the U.S. over seven time intervals from May 2020 to March 2023. Then, for each time interval, we selected the top 25 clusters with the highest relative risks and iteratively predicted the relative risks of clusters from intervals three to seven. The predictive accuracies ranged from moderate to high, indicating the potential applicability of this method for predictive disease analytics and future pandemic preparedness.

Hierarchical clustering analysis (HCA) is a widely used unsupervised learning method. Limitations of HCA, however, include imposing an artificial hierarchy onto non-hierarchical data and fixed two-way mergers at every level. To address this, the current work describes a novel rootlets hierarchical principal component analysis (hPCA). This method extends typical hPCA using multivariate statistics to construct adaptive multiway mergers and Riemannian geometry to visualize nested dependencies. The rootlets hPCA algorithm and its projection onto the Poincaré disk are presented as examples of this extended framework. The algorithm constructs high-dimensional mergers using a single parameter, interpreted as a $p$ -value. It decomposes a similarity matrix from $GL(m,R)$ using a sequence of rotations from $SO\left(k\right)$ , $k\ll m$ . Analysis shows that the rootlets algorithm limits the number of distinct eigenvalues for any merger. Nested clusters of arbitrary size but equal correlations are constructed and merged using their leading principal components. The visualization method then maps elements of $SO\left(k\right)$ onto a low-dimensional hyperbolic manifold, the Poincaré disk. Rootlets hPCA was validated using simulated datasets with known hierarchical structure, and a neuroimaging dataset with an unknown hierarchy. Experiments demonstrate that rootlets hPCA accurately reconstructs known hierarchies and, unlike HCA, does not impose a hierarchy on data.

In systems and network neuroscience, many common practices in brain connectomic analysis are often not properly scrutinized. One such practice is mapping a predetermined set of sub-circuits, like functional networks (FNs), onto subjects' functional connectomes (FCs) without adequately assessing the information-theoretic appropriateness of the partition. Another practice that goes unchallenged is thresholding weighted FCs to remove spurious connections without justifying the chosen threshold. This paper leverages recent theoretical advances in Stochastic Block Models (SBMs) to formally define and quantify the information-theoretic fitness (e.g., prominence) of a predetermined set of FNs when mapped to individual FCs under different fMRI task conditions. Our framework allows for evaluating any combination of FC granularity, FN partition, and thresholding strategy, thereby optimizing these choices to preserve the important topological features of the human brain connectomes. By applying to the Human Connectome Project with Schaefer parcellations at multiple levels of granularity, the framework showed that the common thresholding value of 0.25 was indeed information-theoretically valid for group-average FCs, despite its previous lack of justification. Our results pave the way for the proper use of FNs and thresholding methods, and provide insights for future research in individualized parcellations.