In this paper, we conduct mathematical and numerical analyses for COVID-19. To predict the trend of COVID-19, we propose a time-dependent SIR model that tracks the transmission and recovering rate at time t. Using the data provided by China authority, we show our one-day prediction errors are almost less than 3%. The turning point and the total number of confirmed cases in China are predicted under our model. To analyze the impact of the undetectable infections on the spread of disease, we extend our model by considering two types of infected persons: detectable and undetectable infected persons. Whether there is an outbreak is characterized by the spectral radius of a 2 Â 2 matrix. If R 0 > 1, then the spectral radius of that matrix is greater than 1, and there is an outbreak. We plot the phase transition diagram of an outbreak and show that there are several countries on the verge of COVID-19 outbreaks on Mar. 2, 2020. To illustrate the effectiveness of social distancing, we analyze the independent cascade model for disease propagation in a configuration random network. We show two approaches of social distancing that can lead to a reduction of the effective reproduction number R e .
In this paper, we develop a formal framework for what a good community should look like and how strong a community is (community strength). One of the key innovations is to incorporate the concept of relative centrality into structural analysis of networks. In our framework, relative centrality is a measure that measures how important a set of nodes in a network is with respect to another set of nodes, and it is a generalization of centrality. Building on top of relative centrality, the community strength for a set of nodes is measured by the difference between its relative centrality with respect to itself and its centrality. A community is then a set of nodes with a nonnegative community strength. We show that our community strength is related to conductance that is commonly used for measuring the strength of a small community. We define the modularity for a partition of a network as the average community strength for a randomly selected node. Such a definition generalizes the original Newman's modularity and recovers the stability in as special cases. For the local community detection problem, we also develop efficient agglomerative algorithms that guarantee the community strength of the detected local community.
Clustering is one of the most fundamental problems in data analysis and it has been studied extensively in the literature. Though many clustering algorithms have been proposed, clustering theories that justify the use of these clustering algorithms are still unsatisfactory. In particular, one of the fundamental challenges is to address the following question: What is a cluster in a set of data points? In this paper, we make an attempt to address such a question by considering a set of data points associated with a distance measure (metric). We first propose a new cohesion measure in terms of the distance measure. Using the cohesion measure, we define a cluster as a set of points that are cohesive to themselves. For such a definition, we show there are various equivalent statements that have intuitive explanations. We then consider the second question: How do we find clusters and good partitions of clusters under such a definition? For such a question, we propose a hierarchical agglomerative algorithm and a partitional algorithm. Unlike standard hierarchical agglomerative algorithms, our hierarchical agglomerative algorithm has a specific stopping criterion and it stops with a partition of clusters. Our partitional algorithm, called the K-sets algorithm in the paper, appears to be a new iterative algorithm. Unlike the Lloyd iteration that needs two-step minimization, our K-sets algorithm only takes one-step minimization. One of the most interesting findings of our paper is the duality result between a distance measure and a cohesion measure. Such a duality result leads to a dual K-sets algorithm for clustering a set of data points with a cohesion measure. The dual K-sets algorithm converges in the same way as a sequential version of the classical kernel K-means algorithm. The key difference is that a cohesion measure does not need to be positive semi-definite
In comparison with individual testing, group testing is more efficient in reducing the number of tests and potentially leading to tremendous cost reduction. There are two key elements in a group testing technique: (i) the pooling matrix that directs samples to be pooled into groups, and (ii) the decoding algorithm that uses the group test results to reconstruct the status of each sample. In this paper, we propose a new family of pooling matrices from packing the pencil of lines (PPoL) in a finite projective plane. We compare their performance with various pooling matrices proposed in the literature, including 2Dpooling, P-BEST, and Tapestry, using the two-stage definite defectives (DD) decoding algorithm. By conducting extensive simulations for a range of prevalence rates up to 5%, our numerical results show that there is no pooling matrix with the lowest relative cost in the whole range of the prevalence rates. To optimize the performance, one should choose the right pooling matrix, depending on the prevalence rate. The family of PPoL matrices can dynamically adjust their construction parameters according to the prevalence rates and could be a better alternative than using a fixed pooling matrix.
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