Contributions to the Mathematical Theory of Epidemics. II. The Problem of Endemicity
01 Oct 1932-Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences (The Royal Society)-Vol. 138, Iss: 834, pp 55-83
TL;DR: Under special cases investigated when either immigration or birth is operative in the supply of fresh individuals, as well as in the general case, only one steady state of disease is possible and it has been shown that in all cases, except one, the steady states are stable ones.
Abstract: In a previous communication an attempt was made to investigate mathematically the course of an epidemic in a closed population of susceptible individuals. In order to simplify the problem certain definite assumptions were made, namely, that all individuals were equally susceptible, and that death resulted, or complete immunity was conferred, as the result of an attack. The infectivity of the individual and his chances of death or recovery were represented by arbitrary functions, and the chance of a new infection occurring was assumed to be proportional to the product of the infected and susceptible members of the population. In spite of the introduction of the arbitrary functions, it was shown that in general a critical density of population existed, such that if the actual density was less than this, no epidemic could occur, but if it exceeded this by n an epidemic would appear on the introduction of a focus of infection, and further that if n was small relative to the population density, the size of the epidemic would be 2 n per unit area. It was shown that these conclusions could be readily extended to the case of a metaxenous disease, that is, one in which transmission takes place through an intermediate host. It is the purpose of the present paper to consider the effect of the continuous introduction of fresh susceptible individuals into the population. It appeared desirable to investigate this point, since it might make it possible to interpret certain aspects of the incidence of disease not only in human communities where there is usually an influx of fresh susceptible individuals either by immigration or by birth, but also in the animal experiments carried out by Topley and others—where fresh animals were introduced at a constant rate into the cages in which cases of disease were already present—from which certain definite results were obtained.
Citations
More filters
TL;DR: In this article, the Kermack-McKendrick deterministic model is generalized, introducing an interaction term in which the dependence upon the number of infectives occurs via a nonlinear bounded function.
Abstract: In this paper the Kermack-McKendrick deterministic model is generalized, introducing an interaction term in which the dependence upon the number of infectives occurs via a nonlinear bounded function which may take into account saturation phenomena for large numbers of infectives. An extension of the well-known threshold theorem is obtained, after a stability analysis of the equilibrium points of the system. A numerical example is carried out in detail.
999 citations
TL;DR: Deterministic communicable disease models which are initial value problems for a system of ordinary differential equations are considered, where births and deaths occur at equal rates with all newborns being susceptible.
Abstract: Deterministic communicable disease models which are initial value problems for a system of ordinary differential equations are considered, where births and deaths occur at equal rates with all newborns being susceptible. Asymptotic stability regions are determined for the equilibrium points for models involving temporary immunity, disease-related fatalities, carriers, migration, dissimilar interacting groups, and transmission by vectors. Epidemiological interpretations of all results are given.
682 citations
28 Apr 2014
TL;DR: Social Media Mining introduces the unique problems arising from social media data and presents fundamental concepts, emerging issues, and effective algorithms for network analysis and data mining.
Abstract: The growth of social media over the last decade has revolutionized the way individuals interact and industries conduct business. Individuals produce data at an unprecedented rate by interacting, sharing, and consuming content through social media. Understanding and processing this new type of data to glean actionable patterns presents challenges and opportunities for interdisciplinary research, novel algorithms, and tool development. Social Media Mining integrates social media, social network analysis, and data mining to provide a convenient and coherent platform for students, practitioners, researchers, and project managers to understand the basics and potentials of social media mining. It introduces the unique problems arising from social media data and presents fundamental concepts, emerging issues, and effective algorithms for network analysis and data mining. Suitable for use in advanced undergraduate and beginning graduate courses as well as professional short courses, the text contains exercises of different degrees of difficulty that improve understanding and help apply concepts, principles, and methods in various scenarios of social media mining.
550 citations
Cites methods from "Contributions to the Mathematical T..."
...The SIR model, first introduced by Kermack, and McKendrick [148], adds more detail to the standard SI model....
[...]
TL;DR: The Ross-Macdonald theory has since played a central role in development of research on mosquito-borne pathogen transmission and the development of strategies for mosquitoes-borne disease prevention.
Abstract: Ronald Ross and George Macdonald are credited with developing a mathematical model of mosquito-borne pathogen transmission. A systematic historical review suggests that several mathematicians and scientists contributed to development of the Ross-Macdonald model over a period of 70 years. Ross developed two different mathematical models, Macdonald a third, and various “Ross-Macdonald” mathematical models exist. Ross-Macdonald models are best defined by a consensus set of assumptions. The mathematical model is just one part of a theory for the dynamics and control of mosquito-transmitted pathogens that also includes epidemiological and entomological concepts and metrics for measuring transmission. All the basic elements of the theory had fallen into place by the end of the Global Malaria Eradication Programme (GMEP, 1955–1969) with the concept of vectorial capacity, methods for measuring key components of transmission by mosquitoes, and a quantitative theory of vector control. The Ross-Macdonald theory has since played a central role in development of research on mosquito-borne pathogen transmission and the development of strategies for mosquito-borne disease prevention.
481 citations
01 Jan 2008
TL;DR: Compartmental models for disease transmission are described and analyzed, showing how to calculate the basic reproduction number and the final size of the epidemic and studying age of infection models which give a unifying framework for more complicated compartmental models.
Abstract: We describe and analyze compartmental models for disease transmission. We begin with models for epidemics, showing how to calculate the basic reproduction number and the final size of the epidemic. We also study models with multiple compartments, including treatment or isolation of infectives. We then consider models including births and deaths in which there may be an endemic equilibrium and study the asymptotic stability of equilibria. We conclude by studying age of infection models which give a unifying framework for more complicated compartmental models.
437 citations