scholarly journals An SIR epidemic on a weighted network

2019 ◽  
Vol 7 (4) ◽  
pp. 556-580
Author(s):  
Kristoffer Spricer ◽  
Tom Britton
Keyword(s):  
Weighted Graph ◽  
Configuration Model ◽  
Final Size ◽  
Sir Epidemic ◽  
Wide Range ◽  
Major Outbreak ◽  
Standard Normal ◽  
Model Graph ◽  
Special Case ◽  
Unweighted Graph

AbstractWe introduce a weighted configuration model graph, where edge weights correspond to the probability of infection in an epidemic on the graph. On these graphs, we study the development of a Susceptible–Infectious–Recovered epidemic using both Reed–Frost and Markovian settings. For the special case of having two different edge types, we determine the basic reproduction numberR0, the probability of a major outbreak, and the relative final size of a major outbreak. Results are compared with those for a calibrated unweighted graph. The degree distributions are based on both theoretical constructs and empirical network data. In addition, bivariate standard normal copulas are used to model the dependence between the degrees of the two edge types, allowing for modeling the correlation between edge types over a wide range. Among the results are that the weighted graph produces much richer results than the unweighted graph. Also, while R0 always increases with increasing correlation between the two degrees, this is not necessarily true for the probability of a major outbreak nor for the relative final size of a major outbreak. When using copulas we see that these can produce results that are similar to those of the empirical degree distributions, indicating that in some cases a copula is a viable alternative to using the full empirical data.

scholarly journals An epidemic model with exposure-dependent severities

2005 ◽  
Vol 42 (04) ◽  
pp. 932-949 ◽  
Author(s):  
Frank Ball ◽  
Tom Britton
Keyword(s):  
Branching Process ◽  
Large Population ◽  
Final Outcome ◽  
Model Parameters ◽  
Final Size ◽  
Strong Law ◽  
Sir Epidemic ◽  
Major Outbreak ◽  
Infectious State ◽  
Process Approximation

We consider a stochastic model for the spread of a susceptible–infective–removed (SIR) epidemic among a closed, finite population, in which there are two types of severity of infectious individuals, namely mild and severe. The type of severity depends on the amount of infectious exposure an individual receives, in that infectives are always initially mild but may become severe if additionally exposed. Large-population properties of the model are derived. In particular, a coupling argument is used to provide a rigorous branching process approximation to the early stages of an epidemic, and an embedding argument is used to derive a strong law and an associated central limit theorem for the final outcome of an epidemic in the event of a major outbreak. The basic reproduction number, which determines whether or not a major outbreak can occur given few initial infectives, depends only on parameters of the mild infectious state, whereas the final outcome in the event of a major outbreak depends also on parameters of the severe state. Moreover, the limiting final size proportions need not even be continuous in the model parameters.

scholarly journals An Sir Epidemic Model on a Population with Random Network and Household Structure, and Several Types of Individuals

2012 ◽  
Vol 44 (01) ◽  
pp. 63-86 ◽  
Author(s):  
Frank Ball ◽  
David Sirl
Keyword(s):  
Random Graph ◽  
Epidemic Model ◽  
Strong Approximation ◽  
Random Network ◽  
Configuration Model ◽  
Sir Epidemic Model ◽  
Social Contacts ◽  
Sir Epidemic ◽  
Major Outbreak ◽  
Strong Approximation Theorem

We consider a stochastic SIR (susceptible → infective → removed) epidemic model with several types of individuals. Infectious individuals can make infectious contacts on two levels, within their own ‘household’ and with their neighbours in a random graph representing additional social contacts. This random graph is an extension of the well-known configuration model to allow for several types of individuals. We give a strong approximation theorem which leads to a threshold theorem for the epidemic model and a method for calculating the probability of a major outbreak given few initial infectives. A multitype analogue of a theorem of Ball, Sirl and Trapman (2009) heuristically motivates a method for calculating the expected size of such a major outbreak. We also consider vaccination and give some short numerical illustrations of our results.

scholarly journals An epidemic model with exposure-dependent severities

2005 ◽  
Vol 42 (4) ◽  
pp. 932-949 ◽  
Author(s):  
Frank Ball ◽  
Tom Britton
Keyword(s):  
Branching Process ◽  
Large Population ◽  
Final Outcome ◽  
Model Parameters ◽  
Final Size ◽  
Strong Law ◽  
Sir Epidemic ◽  
Major Outbreak ◽  
Infectious State ◽  
Process Approximation

We consider a stochastic model for the spread of a susceptible–infective–removed (SIR) epidemic among a closed, finite population, in which there are two types of severity of infectious individuals, namely mild and severe. The type of severity depends on the amount of infectious exposure an individual receives, in that infectives are always initially mild but may become severe if additionally exposed. Large-population properties of the model are derived. In particular, a coupling argument is used to provide a rigorous branching process approximation to the early stages of an epidemic, and an embedding argument is used to derive a strong law and an associated central limit theorem for the final outcome of an epidemic in the event of a major outbreak. The basic reproduction number, which determines whether or not a major outbreak can occur given few initial infectives, depends only on parameters of the mild infectious state, whereas the final outcome in the event of a major outbreak depends also on parameters of the severe state. Moreover, the limiting final size proportions need not even be continuous in the model parameters.

scholarly journals An Sir Epidemic Model on a Population with Random Network and Household Structure, and Several Types of Individuals

2012 ◽  
Vol 44 (1) ◽  
pp. 63-86 ◽  
Author(s):  
Frank Ball ◽  
David Sirl
Keyword(s):  
Random Graph ◽  
Epidemic Model ◽  
Strong Approximation ◽  
Random Network ◽  
Configuration Model ◽  
Sir Epidemic Model ◽  
Social Contacts ◽  
Sir Epidemic ◽  
Major Outbreak ◽  
Strong Approximation Theorem

We consider a stochastic SIR (susceptible → infective → removed) epidemic model with several types of individuals. Infectious individuals can make infectious contacts on two levels, within their own ‘household’ and with their neighbours in a random graph representing additional social contacts. This random graph is an extension of the well-known configuration model to allow for several types of individuals. We give a strong approximation theorem which leads to a threshold theorem for the epidemic model and a method for calculating the probability of a major outbreak given few initial infectives. A multitype analogue of a theorem of Ball, Sirl and Trapman (2009) heuristically motivates a method for calculating the expected size of such a major outbreak. We also consider vaccination and give some short numerical illustrations of our results.

An Evaluation of Material Weight and Contained Fluid Volume for Eccentric Intersecting Cylinders of Arbitrary Size

1989 ◽  
Vol 111 (3) ◽  
pp. 342-347
Author(s):  
Y. J. Chao ◽  
M. A. Sutton
Keyword(s):  
Vessel Diameter ◽  
Fluid Volume ◽  
Wide Range ◽  
Single Integral ◽  
Material Volume ◽  
Thin Walled ◽  
Special Case ◽  
Range Of Values ◽  
Engineering Personnel ◽  
Quantities Of Interest

Engineering personnel in industries which use pressurized containment vessels having attached nozzles are required not only to design portions of the lifting mechanism, but also to estimate the fluid volume which the vessel and nozzles will contain; most designers use simplified formulas for computing the quantities of interest. Typically, these formulas are valid approximations when the nozzle diameter is much smaller than the vessel diameter. The enclosed work develops three single-integral expressions which can be programmed and numerically integrated to obtain accurate estimates for both the material volume and also the containment volume present in a pair of eccentrically, or concentrically, intersecting thin-walled cylinders of arbitrary diameters. A table of such values is presented for a wide range of values of the standard nozzle pipe diameter and vessel diameter, for the special case of a concentric nozzle. In addition, an example is presented which compares the numerically integrated values for both the material volume and the containment volume to simplified upper and lower-bound estimates.

A Neoclassical Growth Model for Population Dynamics in a Homogeneous Habitat

2001 ◽  
Author(s):  
Peter Vadasz ◽  
Alisa S. Vadasz
Keyword(s):  
Experimental Data ◽  
Growth Model ◽  
Logistic Growth ◽  
Lag Phase ◽  
Growth Stages ◽  
Initial Growth ◽  
Neoclassical Model ◽  
Wide Range ◽  
Logistic Growth Model ◽  
Special Case

Abstract A neoclassical model is proposed for the growth of cell and other populations in a homogeneous habitat. The model extends on the Logistic Growth Model (LGM) in a non-trivial way in order to address the cases where the Logistic Growth Model (LGM) fails short in recovering qualitative as well as quantitative features that appear in experimental data. These features include in some cases overshooting and oscillations, in others the existence of a “Lag Phase” at the initial growth stages, as well as an inflection point in the “In curve” of the population size. The proposed neoclassical model recovers also the Logistic Growth Curve as a special case. Comparisons of the solutions obtained from the proposed neoclassical model with experimental data confirm its quantitative validity, as well as its ability to recover a wide range of qualitative features captured in experiments.

Other indicators of severity

2012 ◽  
Author(s):  
Odo Diekmann ◽  
Hans Heesterbeek ◽  
Tom Britton
Keyword(s):  
Growth Rate ◽  
Structured Populations ◽  
Infectious Period ◽  
Final Size ◽  
Transmission Parameters ◽  
Contact Rates ◽  
Major Outbreak ◽  
Characteristic Features ◽  
Deterministic Setting ◽  
The Basic Reproduction Number

This chapter is devoted to the initial real-time growth rate r, the probability of a major outbreak, the final size, and the endemic level, in structured populations, with special attention for computational simplifications in the case of separable mixing. Chapter 7 studied the basic reproduction number R₀ for epidemic models in populations manifesting various forms of heterogeneity. It was illustrated that R₀ depends on the transmission parameters, contact rates, the infectious period and on the community structure. The importance of R₀ lies in the fact that an epidemic can, and will in the deterministic setting, take off only if R₀ > 1, a characteristic referred to as supercritical. In a community having births or immigration of susceptibles, this also means that the disease can become endemic. If the parameters and community are such that R₀ < 1 (or R₀ = 1), we are in the subcritical (critical) regime and an epidemic outbreak cannot occur. The chapter examines important supplementary characteristic features and shows how they depend on the different parameters of the model.

264 MITOCHONDRIAL DNA COPY NUMBER IN OOCYTES OF PREPUBERTAL AND CYCLIC GILTS

2011 ◽  
Vol 23 (1) ◽  
pp. 230
Author(s):  
P. Pawlak ◽  
E. Pers-Kamczyc ◽  
D. Lechniak-Cieslak
Keyword(s):  
Mitochondrial Dna ◽  
Copy Number ◽  
Blastocyst Stage ◽  
Developmental Competence ◽  
Published Data ◽  
Mtdna Copy Number ◽  
Final Size ◽  
Domestic Species ◽  
Mitochondrial Dna Copy Number ◽  
Wide Range

In many domestic species (pig, cow, sheep), oocytes from prepubertal females show impaired quality when compared with those from adult animals. Incomplete cytoplasmic maturation is thought to be the main factor responsible for reduced developmental competence of embryos derived from prepubertal oocytes. The status of ooplasm maturation is also reflected by the copy number of mitochondrial DNA (mtDNA). Because replication of mtDNA ceases when oocytes reach their final size and occurs again at the blastocyst stage, the mtDNA copy number is a proved marker of oocyte quality in the pig (El Shourbagy et al. 2006 Reproduction 131, 233–245). The number of mtDNA copies in the grown oocyte is crucial to sustain the first embryonic divisions. To increase the rate of good-quality blastocysts, oocytes of domestic animals have been evaluated by the brilliant cresyl blue test (BCB). According to El Shourbagy et al. (2006), more competent BCB+ oocytes possess higher copy number of mtDNA (on average 222 446) than do their BCB– counterparts (115 352). However, there are no published data on the variation in mtDNA copy number in oocytes derived from ovaries of prepubertal (NCL) and cyclic (CL) gilts. Ovaries of NCL and CL gilts were collected in a local slaughterhouse. Cumulus–oocyte complexes (COC) were aspirated from nonatretic follicles 2 to 6 mm in diameter and evaluated morphologically. Only COC with a proper morphology were subjected to the BCB test. A group of non-BCB-treated COC served as control. Four groups of COC were collected: BCB+ (CL, NCL) and control (CL, NCL). Follicular cells attached to oocytes were removed by pipetting, and completely denuded gametes were individually frozen in liquid nitrogen. Analysis of the mtDNA copy number included isolation of the total DNA followed by amplification of the Cytochrome b (CYTB) gene by real-time PCR (one copy per one mitochondrial genome). Differences in mtDNA copy number among experimental groups were evaluated by Student’s t-test. To date, 30 BCB+ oocytes have been analysed individually (15 CL and 15 NCL). The analysed parameter varied in a wide range from 79 852 to 522 712 copies in CL oocytes and from 52 270 to 287 852 copies in NCL oocytes. Oocytes from cyclic gilts contained significantly more mtDNA copies (on average 267 524) than did gametes of prepubertal females (179 339; P < 0.05). The data on the mtDNA copy number in the control oocytes are currently under investigation. The preliminary results indicate that impaired oocytes quality of prepubertal gilts may be also attributed to the reduced copy number of mtDNA. This project was sponsored by MSHE Poland (grant no. 451/N-COST/2009/0).

Equilibrium probability distributions for low density highway traffic

1966 ◽  
Vol 3 (1) ◽  
pp. 247-260 ◽  
Author(s):  
G. F. Newell
Keyword(s):  
Poisson Process ◽  
Probability Distributions ◽  
Initial Distribution ◽  
Order Effects ◽  
Highway Traffic ◽  
Wide Range ◽  
Headway Distribution ◽  
Equilibrium Distributions ◽  
Initial Distributions ◽  
Special Case

If on a long homogeneous highway there is no interaction between cars, then, under a wide range of conditions, an initial distribution of cars will in the course of time tend toward that of a Poisson process with statistically independent velocities for the cars in any finite interval of highway. Here we will generalize this known property to obtain the following. Suppose cars do interact in such a way as to delay a car when it passes another, but the density of cars is so low that we can neglect simultaneous interactions between three or more cars. There will again be equilibrium distributions of cars to which general classes of initial distributions will converge. These equilibrium distributions are superpositions of two statistically independent processes, one a Poisson process of single free cars with statistically independent velocities, and the other a Poisson process of interacting pairs of cars with various velocities. In the limit of zero interaction, the density of pairs vanishes leaving only the Poisson process of single cars as a special case. To the same order of approximation, including the first order effects of interactions, the headway distribution between consecutive cars will still have exponential tail outside the range of interaction.

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