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      The Failure of R 0

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          Abstract

          The basic reproductive ratio, R 0, is one of the fundamental concepts in mathematical biology. It is a threshold parameter, intended to quantify the spread of disease by estimating the average number of secondary infections in a wholly susceptible population, giving an indication of the invasion strength of an epidemic: if R 0 < 1, the disease dies out, whereas if R 0 > 1, the disease persists. R 0 has been widely used as a measure of disease strength to estimate the effectiveness of control measures and to form the backbone of disease-management policy. However, in almost every aspect that matters, R 0 is flawed. Diseases can persist with R 0 < 1, while diseases with R 0 > 1 can die out. We show that the same model of malaria gives many different values of R 0, depending on the method used, with the sole common property that they have a threshold at 1. We also survey estimated values of R 0 for a variety of diseases, and examine some of the alternatives that have been proposed. If R 0 is to be used, it must be accompanied by caveats about the method of calculation, underlying model assumptions and evidence that it is actually a threshold. Otherwise, the concept is meaningless.

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          Experimental Infection of North American Birds with the New York 1999 Strain of West Nile Virus

          To evaluate transmission dynamics, we exposed 25 bird species to West Nile virus (WNV) by infectious mosquito bite. We monitored viremia titers, clinical outcome, WNV shedding (cloacal and oral), seroconversion, virus persistence in organs, and susceptibility to oral and contact transmission. Passeriform and charadriiform birds were more reservoir competent (a derivation of viremia data) than other species tested. The five most competent species were passerines: Blue Jay (Cyanocitta cristata), Common Grackle (Quiscalus quiscula), House Finch (Carpodacus mexicanus), American Crow (Corvus brachyrhynchos), and House Sparrow (Passer domesticus). Death occurred in eight species. Cloacal shedding of WNV was observed in 17 of 24 species, and oral shedding in 12 of 14 species. We observed contact transmission among four species and oral in five species. Persistent WNV infections were found in tissues of 16 surviving birds. Our observations shed light on transmission ecology of WNV and will benefit surveillance and control programs.
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            Epidemics with two levels of mixing

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              Comparative estimation of the reproduction number for pandemic influenza from daily case notification data.

              The reproduction number, R, defined as the average number of secondary cases generated by a primary case, is a crucial quantity for identifying the intensity of interventions required to control an epidemic. Current estimates of the reproduction number for seasonal influenza show wide variation and, in particular, uncertainty bounds for R for the pandemic strain from 1918 to 1919 have been obtained only in a few recent studies and are yet to be fully clarified. Here, we estimate R using daily case notifications during the autumn wave of the influenza pandemic (Spanish flu) in the city of San Francisco, California, from 1918 to 1919. In order to elucidate the effects from adopting different estimation approaches, four different methods are used: estimation of R using the early exponential-growth rate (Method 1), a simple susceptible-exposed-infectious-recovered (SEIR) model (Method 2), a more complex SEIR-type model that accounts for asymptomatic and hospitalized cases (Method 3), and a stochastic susceptible-infectious-removed (SIR) with Bayesian estimation (Method 4) that determines the effective reproduction number Rt at a given time t. The first three methods fit the initial exponential-growth phase of the epidemic, which was explicitly determined by the goodness-of-fit test. Moreover, Method 3 was also fitted to the whole epidemic curve. Whereas the values of R obtained using the first three methods based on the initial growth phase were estimated to be 2.98 (95% confidence interval (CI): 2.73, 3.25), 2.38 (2.16, 2.60) and 2.20 (1.55, 2.84), the third method with the entire epidemic curve yielded a value of 3.53 (3.45, 3.62). This larger value could be an overestimate since the goodness-of-fit to the initial exponential phase worsened when we fitted the model to the entire epidemic curve, and because the model is established as an autonomous system without time-varying assumptions. These estimates were shown to be robust to parameter uncertainties, but the theoretical exponential-growth approximation (Method 1) shows wide uncertainty. Method 4 provided a maximum-likelihood effective reproduction number 2.10 (1.21, 2.95) using the first 17 epidemic days, which is consistent with estimates obtained from the other methods and an estimate of 2.36 (2.07, 2.65) for the entire autumn wave. We conclude that the reproduction number for pandemic influenza (Spanish flu) at the city level can be robustly assessed to lie in the range of 2.0-3.0, in broad agreement with previous estimates using distinct data.
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                Author and article information

                Journal
                Comput Math Methods Med
                CMMM
                Computational and Mathematical Methods in Medicine
                Hindawi Publishing Corporation
                1748-670X
                1748-6718
                2011
                16 August 2011
                : 2011
                : 527610
                Affiliations
                1Department of Mathematics, Pennsylvania State University, University Park, State College, PA 16802, USA
                2School of Mathematics and Statistics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH, UK
                3Department of Mathematics and Faculty of Medicine, The University of Ottawa, 585 King Edward Avenue, Ottawa ON, Canada K1N 6N5
                Author notes
                *Robert J. Smith?: rsmith43@ 123456uottawa.ca

                Academic Editor: Haitao Chu

                Article
                10.1155/2011/527610
                3157160
                21860658
                78600bd7-4765-47fd-98c8-d9672e554345
                Copyright © 2011 Jing Li et al.

                This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

                History
                : 30 January 2011
                : 18 May 2011
                : 18 May 2011
                Categories
                Review Article

                Applied mathematics
                Applied mathematics

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