The Final Size of an Epidemic and Its Relation to the Basic Reproduction Number

Andreasen, Viggo

doi:10.1007/s11538-010-9623-3

Cited by 110 publications

(135 citation statements)

References 42 publications

(48 reference statements)

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“…In doing so, one departure from the classic relationship between R 0 and attack rate that we made was to rescale R 0 by an exponent α ∈ (0, 1] to allow for better correspondence with observed attack rates. Although there is no theoretical justification for this or any other particular scaling relationship, it is consistent with theoretical expectations 21 that attack rates should be lower in populations with equal R 0 values but more heterogeneous contact patterns, which are typical for transmission by Ae. aegypti 22 .…”

supporting

confidence: 73%

Section: Methodsmentioning

confidence: 89%

“…Mathematical analyses 21 in this case show that a seemingly infinite complexity of relationships between R 0 and final epidemic size are possible in a heterogeneous system. As a general rule, however, final epidemic size in a system with contact heterogeneity and proportional mixing is expected to be strictly less than the final epidemic size in an otherwise equivalent system with homogeneous contacts 21 . How the ratio of these final epidemic sizes scales as a function of R 0 depends entirely on the details of a given system and would therefore be extremely difficult to generalize without copious data on mosquito-human contact and further investigation, which is beyond the scope of our study.…”

Section: Methodsmentioning

confidence: 89%

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Model-based projections of Zika virus infections in childbearing women in the Americas

et al. 2016

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Zika virus is a mosquito-borne pathogen that is rapidly spreading across the Americas. Due to associations between Zika virus infection and a range of fetal maladies 1,2 , the epidemic trajectory of this viral infection poses a significant concern for the nearly 15 million children born in the Americas each year. Ascertaining the portion of this population that is truly at risk is an important priority. One recent estimate 3 suggested that 5.42 million childbearing women live in areas of the Americas that are suitable for Zika occurrence. To improve on that estimate, which did not take into account the protective effects of herd immunity, we developed a new approach that combines classic results from epidemiological theory with seroprevalence data and highly spatially resolved data about drivers of transmission to make location-specific projections of epidemic attack rates. Our results suggest that 1.65 (1.45-2.06) million childbearing women and 93.4 (81.6-117.1) million people in total could become infected before the first wave of the epidemic concludes. Based on current estimates of rates of adverse fetal outcomes among infected women 2,4,5 , these results suggest that tens of thousands of pregnancies could be negatively impacted by the first wave of the epidemic. These projections constitute a revised upper limit of populations at risk in the current Zika epidemic, and our approach offers a new way to make rapid assessments of the threat posed by emerging infectious diseases more generally.On 1 February 2016, the World Health Organization (WHO) designated the ongoing Zika virus epidemic in the Americas as a Public Health Emergency of International Concern (PHEIC), defined as an 'extraordinary event' that 'potentially require[s] a coordinated international response' 6 . This declaration acknowledges the high potential for Zika to establish across the Americas given that its dominant vector, the Aedes aegypti mosquito, is endophilic and occupies an exceptionally broad geographical range 7 . Concern underlying this rare WHO declaration also stems from an association between Zika virus infection in pregnant women and a range of adverse fetal outcomes 2 , most notably congenital microcephaly 1 . As of 30 June 2016, there were 1,674 confirmed cases of microcephaly associated with Zika virus infection in five countries 8 , and there is widespread concern that these numbers could increase further as the virus continues to spread across the Americas 9 .A number of uncertainties surround the future of the Zika epidemic in the Americas, particularly questions about how many women may be at risk of having children with congenital microcephaly and other adverse outcomes associated with Zika virus infection 10 . Of women who become infected with Zika virus during a vulnerable stage of their pregnancy, evidence is emerging that 1-13% may go on to develop congenital microcephaly 2,4,5 .However, the number of women who become infected with Zika virus during that timeframe is difficult to ascertain. One recent study 3 estimat...

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supporting

confidence: 73%

Section: Methodsmentioning

confidence: 89%

Section: Methodsmentioning

confidence: 89%

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Model-based projections of Zika virus infections in childbearing women in the Americas

et al. 2016

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“…Moreover, the final size formula is related to the basic reproduction number. This relation is extended to more general models including a subdivided population with general mixing patterns among subgroups [2]. In that paper it is proved that the final size equation determines a unique value for the final epidemic size.…”

Section: Introductionmentioning

confidence: 94%

Solvability of implicit final size equations for SIR epidemic models

Bidari¹,

Chen²,

Peters³

et al. 2016

Mathematical Biosciences

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Final epidemic size relations play a central role in mathematical epidemiology. These can be written in the form of an implicit equation which is not analytically solvable in most of the cases. While final size relations were derived for several complex models, including multiple infective stages and models in which the durations of stages are arbitrarily distributed, the solvability of those implicit equations have been less studied. In this paper the SIR homogeneous mean-field and pairwise models and the heterogeneous mean-field model are studied. It is proved that the implicit equation for the final epidemic size has a unique solution, and that through writing the implicit equation as a fixed point equation in a suitable form, the iteration of the fixed point equation converges to the unique solution. The Markovian SIR epidemic model on finite networks is also studied by using the generation-based approach. Explicit analytic formulas are derived for the final size distribution for line and star graphs of arbitrary size. Iterative formulas for the final size distribution enable us to study the accuracy of mean-field approximations for the complete graph.

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“…Complex network is a method used to study the physical process or data, which is widely used in economics [14], sociology [15], medical science [16] and physics [17]. Recently, complex network has also been gradually employed to analyze the physical and mechanical properties of granular matter.…”

Section: Introductionmentioning

confidence: 99%

Effects of Inter-Particle Frictional Coefficients on Evolution of Contact Networks in Landslide Process

Lian¹,

Liu²,

Tian³

et al. 2017

ENG

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During the process of landslide, its dynamic mechanism is important to understand and predict these kinds of natural hazard. In this paper, a new method, based on concepts of complex networks, has been proposed to investigate the evolution of contact networks in mesoscale during the sliding process of slope. A slope model was established using the discrete element method (DEM), and influences of inter-particle frictional coefficients with four different values on dynamic landslides were studied. Both macroscopic analysis on slope landslide and mesoanalysis on structure evolution of contact networks, including the average degree, clustering coefficient and N-cycle, were done during the process of landslide. The analysis results demonstrate that: 1) with increasing inter-particle frictional coefficients, the displacement of slope decreases and the stable angle of slope post-failure increases, which is smaller than the peak internal frictional angle; 2) the average degree decreases with the increase of inter-particle frictional coefficient. When the displacement at the toe of the slope is smaller, the average degree there changes more greatly with increasing inter-particle frictional coefficient; 3) during the initial stage of landslide, the clustering coefficient reduces sharply, which may leads to easily slide of slope. As the landslide going on, however, the clustering coefficient increases denoting increasing stability with increasing inter-particle frictional coefficients. When the inter-particle frictional coefficient is smaller than 0.3, its variation can affect the clustering coefficient and stable inclination of slope post-failure greatly; and 4) the number of 3-cycle increases, but 4-cycle and 5-cycle decrease with increasing inter-particle frictional coefficients.

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The Final Size of an Epidemic and Its Relation to the Basic Reproduction Number

Cited by 110 publications

References 42 publications

Model-based projections of Zika virus infections in childbearing women in the Americas

Model-based projections of Zika virus infections in childbearing women in the Americas

Solvability of implicit final size equations for SIR epidemic models

Effects of Inter-Particle Frictional Coefficients on Evolution of Contact Networks in Landslide Process

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