Understanding the Spreading Power of All Nodes in a Network: a Continuous-Time Perspective
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Some Simple Rules for Estimating Reproduction Numbers in the Presence of Reservoir Exposure Or Imported Cases
Some simple rules for estimating reproduction numbers in the presence of reservoir exposure or imported cases Angus McLure1*, Kathryn Glass1 1 Research School of Population Health, Australian National University, 62 Mills Rd, Acton, 0200, ACT, Australia * Corresponding author: [email protected] 1 Abstract The basic reproduction number () is a threshold parameter for disease extinction or survival in isolated populations. However no human population is fully isolated from other human or animal populations. We use compartmental models to derive simple rules for the basic reproduction number for populations with local person‐to‐person transmission and exposure from some other source: either a reservoir exposure or imported cases. We introduce the idea of a reservoir‐driven or importation‐driven disease: diseases that would become extinct in the population of interest without reservoir exposure or imported cases (since 1, but nevertheless may be sufficiently transmissible that many or most infections are acquired from humans in that population. We show that in the simplest case, 1 if and only if the proportion of infections acquired from the external source exceeds the disease prevalence and explore how population heterogeneity and the interactions of multiple strains affect this rule. We apply these rules in two cases studies of Clostridium difficile infection and colonisation: C. difficile in the hospital setting accounting for imported cases, and C. difficile in the general human population accounting for exposure to animal reservoirs. We demonstrate that even the hospital‐adapted, highly‐transmissible NAP1/RT027 strain of C. difficile had a reproduction number <1 in a landmark study of hospitalised patients and therefore was sustained by colonised and infected admissions to the study hospital. -
New Foundations for Geometric Algebra1
Text published in the electronic journal Clifford Analysis, Clifford Algebras and their Applications vol. 2, No. 3 (2013) pp. 193-211 New foundations for geometric algebra1 Ramon González Calvet Institut Pere Calders, Campus Universitat Autònoma de Barcelona, 08193 Cerdanyola del Vallès, Spain E-mail : [email protected] Abstract. New foundations for geometric algebra are proposed based upon the existing isomorphisms between geometric and matrix algebras. Each geometric algebra always has a faithful real matrix representation with a periodicity of 8. On the other hand, each matrix algebra is always embedded in a geometric algebra of a convenient dimension. The geometric product is also isomorphic to the matrix product, and many vector transformations such as rotations, axial symmetries and Lorentz transformations can be written in a form isomorphic to a similarity transformation of matrices. We collect the idea Dirac applied to develop the relativistic electron equation when he took a basis of matrices for the geometric algebra instead of a basis of geometric vectors. Of course, this way of understanding the geometric algebra requires new definitions: the geometric vector space is defined as the algebraic subspace that generates the rest of the matrix algebra by addition and multiplication; isometries are simply defined as the similarity transformations of matrices as shown above, and finally the norm of any element of the geometric algebra is defined as the nth root of the determinant of its representative matrix of order n. The main idea of this proposal is an arithmetic point of view consisting of reversing the roles of matrix and geometric algebras in the sense that geometric algebra is a way of accessing, working and understanding the most fundamental conception of matrix algebra as the algebra of transformations of multiple quantities. -
Seventy-Five Years of Estimating the Force of Infection from Current Status
Epidemiol. Infect. (2010), 138, 802–812. f Cambridge University Press 2009 doi:10.1017/S0950268809990781 Seventy-five years of estimating the force of infection from current status data N. HENS 1,2*, M. AERTS 1,C.FAES1,Z.SHKEDY1, O. LEJEUNE2, P. VAN DAMME2 AND P. BEUTELS2 1 Interuniversity Institute of Biostatistics and Statistical Bioinformatics, Hasselt University, Diepenbeek, Belgium 2 Centre for Health Economics Research & Modelling Infectious Diseases; Centre for the Evaluation of Vaccination, Vaccine and Infectious Disease Institute, University of Antwerp, Antwerp, Belgium (Accepted 18 August 2009; first published online 21 September 2009) SUMMARY The force of infection, describing the rate at which a susceptible person acquires an infection, is a key parameter in models estimating the infectious disease burden, and the effectiveness and cost-effectiveness of infectious disease prevention. Since Muench formulated the first catalytic model to estimate the force of infection from current status data in 1934, exactly 75 years ago, several authors addressed the estimation of this parameter by more advanced statistical methods, while applying these to seroprevalence and reported incidence/case notification data. In this paper we present an historical overview, discussing the relevance of Muench’s work, and we explain the wide array of newer methods with illustrations on pre-vaccination serological survey data of two airborne infections: rubella and parvovirus B19. We also provide guidance on deciding which method(s) to apply to estimate the -
Multiscale Mobility Networks and the Spatial Spreading of Infectious Diseases
Multiscale mobility networks and the spatial spreading of infectious diseases Duygu Balcana,b, Vittoria Colizzac, Bruno Gonc¸alvesa,b, Hao Hud, Jose´ J. Ramascob, and Alessandro Vespignania,b,c,1 aCenter for Complex Networks and Systems Research, School of Informatics and Computing, Indiana University, Bloomington, IN 47408; bPervasive Technology Institute, Indiana University, Bloomington, IN 47404; cComputational Epidemiology Laboratory, Institute for Scientific Interchange Foundation, 10133 Torino, Italy; and dDepartment of Physics, Indiana University, Bloomington, IN 47406 Edited by H. Eugene Stanley, Boston University, Boston, MA, and approved October 13, 2009 (received for review June 19, 2009) Among the realistic ingredients to be considered in the computational two questions stand out: (i) Is there a most relevant mobility scale modeling of infectious diseases, human mobility represents a crucial in the definition of the global epidemic pattern? and (ii) At which challenge both on the theoretical side and in view of the limited level of resolution of the epidemic behavior does a given mobility availability of empirical data. To study the interplay between short- scale become relevant, and to what extent? scale commuting flows and long-range airline traffic in shaping the To begin addressing these questions, we use high-resolution spatiotemporal pattern of a global epidemic we (i) analyze mobility worldwide population data that allow for the definition of sub- data from 29 countries around the world and find a gravity model populations according to a Voronoi decomposition of the world able to provide a global description of commuting patterns up to 300 surface centered on the locations of International Air Transport kms and (ii) integrate in a worldwide-structured metapopulation Association (IATA)-indexed airports (www.iata.org). -
Epidemic Response to Physical Distancing Policies and Their Impact on the Outbreak Risk
Epidemic response to physical distancing policies and their impact on the outbreak risk Fabio Vanni∗†‡ David Lambert§ ‡ Luigi Palatella¶ Abstract We introduce a theoretical framework that highlights the impact of physical distancing variables such as human mobility and physical proximity on the evolution of epidemics and, crucially, on the reproduction number. In particular, in response to the coronavirus disease (CoViD-19) pandemic, countries have introduced various levels of ’lockdown’ to reduce the number of new infections. Specifically we use a collisional approach to an infection-age struc- tured model described by a renewal equation for the time homogeneous evolution of epidemics. As a result, we show how various contributions of the lockdown policies, namely physical prox- imity and human mobility, reduce the impact of SARS-CoV-2 and mitigate the risk of disease resurgence. We check our theoretical framework using real-world data on physical distanc- ing with two different data repositories, obtaining consistent results. Finally, we propose an equation for the effective reproduction number which takes into account types of interac- tions among people, which may help policy makers to improve remote-working organizational structure. Keywords: Renewal equation, Epidemic Risk, Lockdown, Social Distancing, Mobility, Smart Work. arXiv:2007.14620v2 [physics.soc-ph] 31 Jul 2020 1 Introduction As the coronavirus disease (CoViD-19) epidemic worsens, understanding the effectiveness of public messaging and large-scale physical distancing interventions is critical in order to manage the acute ∗Sciences Po, OFCE , France †Institute of Economics, Sant’Anna School of Advanced Studies, Pisa, Italy ‡Department of Physics, University of North Texas, USA §Department of Mathematics, University of North Texas, USA ¶Liceo Scientifico Statale “C. -
Similarity Preserving Linear Maps on Upper Triangular Matrix Algebras ୋ
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Linear Algebra and its Applications 414 (2006) 278–287 www.elsevier.com/locate/laa Similarity preserving linear maps on upper triangular matrix algebras ୋ Guoxing Ji ∗, Baowei Wu College of Mathematics and Information Science, Shaanxi Normal University, Xian 710062, PR China Received 14 June 2005; accepted 5 October 2005 Available online 1 December 2005 Submitted by P. Šemrl Abstract In this note we consider similarity preserving linear maps on the algebra of all n × n complex upper triangular matrices Tn. We give characterizations of similarity invariant subspaces in Tn and similarity preserving linear injections on Tn. Furthermore, we considered linear injections on Tn preserving similarity in Mn as well. © 2005 Elsevier Inc. All rights reserved. AMS classification: 47B49; 15A04 Keywords: Matrix; Upper triangular matrix; Similarity invariant subspace; Similarity preserving linear map 1. Introduction In the last few decades, many researchers have considered linear preserver problems on matrix or operator spaces. For example, there are many research works on linear maps which preserve spectrum (cf. [4,5,11]), rank (cf. [3]), nilpotency (cf. [10]) and similarity (cf. [2,6–8]) and so on. Many useful techniques have been developed; see [1,9] for some general techniques and background. Hiai [2] gave a characterization of the similarity preserving linear map on the algebra of all complex n × n matrices. ୋ This research was supported by the National Natural Science Foundation of China (no. 10571114), the Natural Science Basic Research Plan in Shaanxi Province of China (program no. -
Reproduction Numbers for Infections with Free-Living Pathogens Growing in the Environment
Journal of Biological Dynamics Vol. 6, No. 2, March 2012, 923–940 Reproduction numbers for infections with free-living pathogens growing in the environment Majid Bani-Yaghouba*, Raju Gautama, Zhisheng Shuaib, P. van den Driesscheb and Renata Ivaneka aDepartment of Veterinary Integrative Biosciences, College of Veterinary Medicine and Biomedical Sciences, Texas A&M University, College Station, TX 77843, USA; bDepartment of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada V8W 3R4 (Received 29 January 2012; final version received 6 May 2012) The basic reproduction number 0 for a compartmental disease model is often calculated by the next generation matrix (NGM) approach.R When the interactions within and between disease compartments are interpreted differently, the NGM approach may lead to different 0 expressions. This is demonstrated by considering a susceptible–infectious–recovered–susceptible modelR with free-living pathogen (FLP) growing in the environment.Although the environment could play different roles in the disease transmission process, leading to different 0 expressions, there is a unique type reproduction number when control R strategies are applied to the host population.All 0 expressions agree on the threshold value 1 and preserve their order of magnitude. However, using data forR salmonellosis and cholera, it is shown that the estimated 0 values are substantially different. This study highlights the utility and limitations of reproduction numbersR to accurately quantify the effects of control strategies for infections with FLPs growing in the environment. Keywords: SIRSP model; infection control; free-living pathogen; basic reproduction number; type reproduction number 1. Introduction The basic reproduction number, 0, is considered as one of the most practical tools that R Downloaded by [University of Central Florida] at 08:28 15 August 2012 mathematical thinking has brought to epidemic theory [26]. -
Similarity to Symmetric Matrices Over Finite Fields
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector FINITE FIELDS AND THEIR APPLICATIONS 4, 261Ð274 (1998) ARTICLE NO. FF980216 Similarity to Symmetric Matrices over Finite Fields Joel V. Brawley Clemson University, Clemson, South Carolina 29631 and Timothy C. Teitlo¤ Kennesaw State University, Kennesaw, Georgia 30144 E-mail: tteitlo¤@ksumail.kennesaw.edu Communicated by Zhe-Xian Wan Received October 22, 1996; revised March 31, 1998 It has been known for some time that every polynomial with coe¦cients from a finite field is the minimum polynomial of a symmetric matrix with entries from the same field. What have remained unknown, however, are the possible sizes for the symmetric matrices with a specified minimum polynomial and, in particular, the least possible size. In this paper we answer these questions using only the prime factoriz- ation of the given polynomial. Closely related is the question of whether or not a given matrix over a finite field is similar to a symmetric matrix over that field. Although partial results on that question have been published before, this paper contains a complete characterization. ( 1998 Academic Press 1. INTRODUCTION This paper seeks to solve two problems: PROBLEM 1. Determine if a given polynomial with coe¦cients from Fq, the finite field of q elements, is the minimum polynomial of a sym- metric matrix, and if so, determine the possible sizes of such symmetric matrices. PROBLEM 2. Characterize those matrices over Fq that are similar over Fq to symmetric matrices. 261 1071-5797/98 $25.00 Copyright ( 1998 by Academic Press All rights of reproduction in any form reserved. -
Judgement of Matrix Similarity
International Journal of Trend in Research and Development, Volume 8(5), ISSN: 2394-9333 www.ijtrd.com Judgement of Matrix Similarity Hailing Li School of Mathematics and Statistics, Shandong University of Technology, Zibo, China Abstract: Matrix similarity is very important in matrix theory. characteristic matrix of B. In this paper, several methods of judging matrix similarity are given. II. JUDGEMENT OF SIMILARITY OF MATRICES Keywords: Similarity; Transitivity; Matrix equivalence; Matrix Type 1 Using the Definition 1 to determine that the abstract rank two matrices are similar. I. INTRODUCTION Example 1 Let be a 2-order matrix, be a Matrix similarity theory has important applications not only in various branches of mathematics, such as differential 2-order invertible matrix, if , please equation, probability and statistics, computational mathematics, but also in other fields of science and technology. There are prove A many similar properties and conclusions of matrix, so it is necessary to study and sum up the similar properties and Prove judgement of matrix. Definition 1 Let A and B be n-order matrices, if there is an invertible matrix P, such that , then A is similar to B, and denoted by A~B. Definition 2 Let A,be an n-order matrix, the matrix is called the characteristic matrix of A; its determinant that is, by the definition, it can obtain is called the characteristic polynomial of A; is called the characteristic equation of A; the that root that satisfies the characteristic equation is called an eigenvalue of A; the nonzero solution vectors of the system Type 2 For choice questions, we can using Property 3 of linear equations are called the characteristic matrices to determine that two matrices are eigenvectors of the eigenvalue . -
Global Convergence of COVID-19 Basic Reproduction Number and Estimation from Early-Time SIR Dynamics
PLOS ONE RESEARCH ARTICLE Global convergence of COVID-19 basic reproduction number and estimation from early-time SIR dynamics 1,2 1☯ 3,4☯ 5☯ Gabriel G. KatulID *, Assaad MradID , Sara BonettiID , Gabriele ManoliID , Anthony 6☯ J. ParolariID 1 Nicholas School of the Environment, Duke University, Durham, NC, United States of America, 2 Department of Civil and Environmental Engineering, Duke University, Durham, NC, United States of America, a1111111111 3 Department of Environmental Systems Science, ETH ZuÈrich, ZuÈrich, Switzerland, 4 Bartlett School of a1111111111 Environment, Energy and Resources, University College London, London, United Kingdom, 5 Department of a1111111111 Civil, Environmental and Geomatic Engineering, University College London, London, United Kingdom, 6 Department of Civil, Construction, and Environmental Engineering, Marquette University, Milwaukee, a1111111111 Wisconsin, United States of America a1111111111 ☯ These authors contributed equally to this work. * [email protected] OPEN ACCESS Abstract Citation: Katul GG, Mrad A, Bonetti S, Manoli G, Parolari AJ (2020) Global convergence of COVID- The SIR (`susceptible-infectious-recovered') formulation is used to uncover the generic 19 basic reproduction number and estimation from spread mechanisms observed by COVID-19 dynamics globally, especially in the early early-time SIR dynamics. PLoS ONE 15(9): phases of infectious spread. During this early period, potential controls were not effectively e0239800. https://doi.org/10.1371/journal. pone.0239800 put in place or enforced in many countries. Hence, the early phases of COVID-19 spread in countries where controls were weak offer a unique perspective on the ensemble-behavior of Editor: Maria Vittoria Barbarossa, Frankfurt Institute for Advanced Studies, GERMANY COVID-19 basic reproduction number Ro inferred from SIR formulation. -
Into to Epidemic Modeling
INTRO TO EPIDEMIC MODELING Introduction to epidemic modeling is usually made through one of the first epidemic models proposed by Kermack and McKendrick in 1927, a model known as the SIR epidemic model When a disease spreads in a population it splits the population into nonintersecting classes. In one of the simplest scenarios there are 3 classes: • The class of individuals who are healthy but can contract the disease. These people are called susceptible individuals or susceptibles. The size of this class is usually denoted by S. • The class of individuals who have contracted the disease and are now sick with it, called infected individuals. In this model it is assumed that infected individuals are also infectious. The size of the class of infectious/infected individuals is denoted by I. • The class of individuals who have recovered and cannot contract the disease again are called removed/recovered individuals. The class of recovered individuals is usually denoted by R. The number of individuals in each of these classes changes with time, that is, S(t), I(t), and R(t). The total population size N is the sum of these three classes 푁 푡 = 푆 푡 + 퐼 푡 + 푅(푡) Assumptions: (1) Infected individuals are also infectious; (2) The total population size remains constant; (3) The population is closed (no immigration/emigration); (4) No births/deaths; (5)All recovered individuals have complete immunity Epidemiological models consist of systems of ODEs which describe the dynamics in each class. To derive the differential equations, we consider how the classes change in time. When a susceptible individual enters into a contact with an infectious individual, that susceptible individual becomes infected and moves from the susceptible class into the infected class. -
Matrix Similarity
Matrix similarity Isak Johansson and Jonas Bederoff Eriksson Supervisor: Tilman Bauer Department of Mathematics 2016 1 Abstract This thesis will deal with similar matrices, also referred to as matrix conju- gation. The first problem we will attack is whether or not two given matrices are similar over some field. To solve this problem we will introduce the Ratio- nal Canonical Form, RCF. From this normal form, also called the Frobenius normal form, we can determine whether or not the given matrices are sim- ilar over any field. We can also, given some field F , see whether they are similar over F or not. To be able to understand and prove the existence and uniqueness of the RCF we will introduce some additional module theory. The theory in this part will build up to finally prove the theorems regarding the RCF that can be used to solve our problem. The next problem we will investigate is regarding simultaneous conjugation, i.e. conjugation by the same matrix on a pair of matrices. When are two pairs of similar matrices simultaneously conjugated? Can we find any necessary or even sufficient conditions on the matrices? We will address this more complicated issue with the theory assembled in the first part. 2 Contents 1 Introduction 4 2 Modules 5 2.1 Structure theorem for finitely generated modules over a P.I.D, existence . 8 2.2 Structure theorem for finitely generated modules over a P.I.D, uniqueness . 10 3 Similar matrices and normal forms 12 3.1 Linear transformations . 12 3.2 Rational Canonical Form .