A Discrete SIR Model with Spatial Distribution on a Torus for COVID-19 Analysis using Local Neighborhood Properties

A simple and effective mathematical procedure for the description of observed COVID-19 data and calculation of future projections is presented. An exponential function E(t) with a time-varying Growth Constant k(t) is used. E(t) closely approximates observed COVID-19 Daily Confirmed Cases with NRMSDs of 1 to 2%. An example of prediction of future cases is presented. The Effective Growth Rates of a discrete SIR model were estimated on the basis of k(t) for COVID-19 data for Germany, and were found to be consistent with those reported in a previous study (1). The proposed procedure, which involves less than ten basic algebraic, logarithm and exponentiation operations for each data point, is suitable for use in promoting interdisciplinary research, exchange and sharing of information.

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Fitting a Three-Phase Discrete SIR Model to New Coronavirus Cases in New York State

International Journal of Data Analytics ◽

10.4018/ijda.2021070104 ◽

2021 ◽

Vol 2 (2) ◽

pp. 59-74

Author(s):

Kris H. Green

Keyword(s):

New York ◽

New York State ◽

The United States ◽

Sir Model ◽

High Fidelity ◽

Optimal Model ◽

Training Set ◽

York State ◽

Three Phase ◽

Discrete Sir Model

CDC data on new coronavirus cases in New York State between March 4, 2020 and June 26, 2020 show three distinct phases for the spread of the virus. The authors demonstrate fitting of a simple discrete SIR model with three phases to model these data, achieving a high fidelity to the data. Optimal model fits using both R and Excel are compared, and various issues are discussed. Finally, the model for New York State is treated as a training set for extending and applying the model to the outbreak in other areas of the United States and the country as a whole.

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Time-continuous and time-discrete SIR models revisited: theory and applications

Advances in Difference Equations ◽

10.1186/s13662-020-02995-1 ◽

2020 ◽

Vol 2020 (1) ◽

Author(s):

Benjamin Wacker ◽

Jan Schlüter

Keyword(s):

Computer Science ◽

Unique Solvability ◽

Interdisciplinary Research ◽

Numerical Scheme ◽

Sir Model ◽

Implicit Time ◽

Time Varying ◽

Mathematical Epidemiology ◽

Continuous Version ◽

Discrete Sir Model

Abstract Since Kermack and McKendrick have introduced their famous epidemiological SIR model in 1927, mathematical epidemiology has grown as an interdisciplinary research discipline including knowledge from biology, computer science, or mathematics. Due to current threatening epidemics such as COVID-19, this interest is continuously rising. As our main goal, we establish an implicit time-discrete SIR (susceptible people–infectious people–recovered people) model. For this purpose, we first introduce its continuous variant with time-varying transmission and recovery rates and, as our first contribution, discuss thoroughly its properties. With respect to these results, we develop different possible time-discrete SIR models, we derive our implicit time-discrete SIR model in contrast to many other works which mainly investigate explicit time-discrete schemes and, as our main contribution, show unique solvability and further desirable properties compared to its continuous version. We thoroughly show that many of the desired properties of the time-continuous case are still valid in the time-discrete implicit case. Especially, we prove an upper error bound for our time-discrete implicit numerical scheme. Finally, we apply our proposed time-discrete SIR model to currently available data regarding the spread of COVID-19 in Germany and Iran.

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Lithography below 100 nm

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100074367 ◽

1983 ◽

Vol 41 ◽

pp. 96-99

Author(s):

L. D. Jackel

Keyword(s):

Spatial Distribution ◽

Electron Beam ◽

Electron Scattering ◽

Electron Beam Lithography ◽

Substrate Material ◽

Local Electron ◽

Electron Dose ◽

Positive Resist ◽

Exposure Process

Most production electron beam lithography systems can pattern minimum features a few tenths of a micron across. Linewidth in these systems is usually limited by the quality of the exposing beam and by electron scattering in the resist and substrate. By using a smaller spot along with exposure techniques that minimize scattering and its effects, laboratory e-beam lithography systems can now make features hundredths of a micron wide on standard substrate material. This talk will outline sane of these high- resolution e-beam lithography techniques.We first consider parameters of the exposure process that limit resolution in organic resists. For concreteness suppose that we have a “positive” resist in which exposing electrons break bonds in the resist molecules thus increasing the exposed resist's solubility in a developer. Ihe attainable resolution is obviously limited by the overall width of the exposing beam, but the spatial distribution of the beam intensity, the beam “profile” , also contributes to the resolution. Depending on the local electron dose, more or less resist bonds are broken resulting in slower or faster dissolution in the developer.

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SEM evaluation of the TEM cold-stage double-film specimen

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100111471 ◽

1984 ◽

Vol 42 ◽

pp. 272-273

Author(s):

Jayesh Bellare

Keyword(s):

Spatial Distribution ◽

Sample Preparation ◽

Sample Thickness ◽

Specimen Preparation ◽

Preparation Technique ◽

Liquid Sample ◽

Thin Specimen ◽

Sample Preparation Technique ◽

Cold Stage ◽

Film Specimen

Seeing is believing, but only after the sample preparation technique has received a systematic study and a full record is made of the treatment the sample gets.For microstructured liquids and suspensions, fast-freeze thermal fixation and cold-stage microscopy is perhaps the least artifact-laden technique. In the double-film specimen preparation technique, a layer of liquid sample is trapped between 100- and 400-mesh polymer (polyimide, PI) coated grids. Blotting against filter paper drains excess liquid and provides a thin specimen, which is fast-frozen by plunging into liquid nitrogen. This frozen sandwich (Fig. 1) is mounted in a cooling holder and viewed in TEM.Though extremely promising for visualization of liquid microstructures, this double-film technique suffers from a) ireproducibility and nonuniformity of sample thickness, b) low yield of imageable grid squares and c) nonuniform spatial distribution of particulates, which results in fewer being imaged.

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Macroprobe Mode for the Study of Segregation in Steels

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100134909 ◽

1990 ◽

Vol 48 (2) ◽

pp. 260-261

Author(s):

Auclair Gilles ◽

Benoit Danièle

Keyword(s):

Mechanical Properties ◽

Spatial Distribution ◽

High Performance ◽

Volume Fraction ◽

Sheet Steel ◽

Acquisition Time ◽

Chemical Information ◽

X Ray ◽

Accurate Quantitative Analysis ◽

Optical Micrographs

During these last 10 years, high performance correction procedures have been developed for classical EPMA, and it is nowadays possible to obtain accurate quantitative analysis even for soft X-ray radiations. It is also possible to perform EPMA by adapting this accurate quantitative procedures to unusual applications such as the measurement of the segregation on wide areas in as-cast and sheet steel products.The main objection for analysis of segregation in steel by means of a line-scan mode is that it requires a very heavy sampling plan to make sure that the most significant points are analyzed. Moreover only local chemical information is obtained whereas mechanical properties are also dependant on the volume fraction and the spatial distribution of highly segregated zones. For these reasons we have chosen to systematically acquire X-ray calibrated mappings which give pictures similar to optical micrographs. Although mapping requires lengthy acquisition time there is a corresponding increase in the information given by image anlysis.

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Ultrastructural analysis of the spatial distribution of MRNA

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100123167 ◽

1992 ◽

Vol 50 (1) ◽

pp. 552-553

Author(s):

Gary Bassell ◽

Robert H. Singer

Keyword(s):

Electron Microscopy ◽

Spatial Distribution ◽

In Situ Hybridization ◽

High Resolution ◽

Nucleic Acids ◽

Colloidal Gold ◽

Dna Probe ◽

Nucleotide Analogs ◽

Gold Particles

We have been investigating the spatial distribution of nucleic acids intracellularly using in situ hybridization. The use of non-isotopic nucleotide analogs incorporated into the DNA probe allows the detection of the probe at its site of hybridization within the cell. This approach therefore is compatible with the high resolution available by electron microscopy. Biotinated or digoxigenated probe can be detected by antibodies conjugated to colloidal gold. Because mRNA serves as a template for the probe fragments, the colloidal gold particles are detected as arrays which allow it to be unequivocally distinguished from background.

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Membrane microdomains: lessons from microscopy

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100140257 ◽

1995 ◽

Vol 53 ◽

pp. 776-777

Author(s):

J.M. Robinson ◽

J.M Oliver

Keyword(s):

Spatial Distribution ◽

Plasma Membrane ◽

Epithelial Cells ◽

Plasma Membranes ◽

Cell Types ◽

Imaging Modalities ◽

Membrane Microdomains ◽

Membrane Domains ◽

Lateral Heterogeneity ◽

Functional Importance

Specialized regions of plasma membranes displaying lateral heterogeneity are the focus of this Symposium. Specialized membrane domains are known for certain cell types such as differentiated epithelial cells where lateral heterogeneity in lipids and proteins exists between the apical and basolateral portions of the plasma membrane. Lateral heterogeneity and the presence of microdomains in membranes that are uniform in appearance have been more difficult to establish. Nonetheless a number of studies have provided evidence for membrane microdomains and indicated a functional importance for these structures.This symposium will focus on the use of various imaging modalities and related approaches to define membrane microdomains in a number of cell types. The importance of existing as well as emerging imaging technologies for use in the elucidation of membrane microdomains will be highlighted. The organization of membrane microdomains in terms of dimensions and spatial distribution is of considerable interest and will be addressed in this Symposium.

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