Über eine Klasse von Integralgeleichungen
About: This article is published in Mathematische Zeitschrift.The article was published on 1924-12-01. It has received 11 citations till now.
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TL;DR: In this article, the authors considered the problem of finding a causal factor which appears to be adequate to account for the magnitude of the frequent epidemics of disease which visit almost every population.
Abstract: (1) One of the most striking features in the study of epidemics is the difficulty of finding a causal factor which appears to be adequate to account for the magnitude of the frequent epidemics of disease which visit almost every population. It was with a view to obtaining more insight regarding the effects of the various factors which govern the spread of contagious epidemics that the present investigation was undertaken. Reference may here be made to the work of Ross and Hudson (1915-17) in which the same problem is attacked. The problem is here carried to a further stage, and it is considered from a point of view which is in one sense more general. The problem may be summarised as follows: One (or more) infected person is introduced into a community of individuals, more or less susceptible to the disease in question. The disease spreads from the affected to the unaffected by contact infection. Each infected person runs through the course of his sickness, and finally is removed from the number of those who are sick, by recovery or by death. The chances of recovery or death vary from day to day during the course of his illness. The chances that the affected may convey infection to the unaffected are likewise dependent upon the stage of the sickness. As the epidemic spreads, the number of unaffected members of the community becomes reduced. Since the course of an epidemic is short compared with the life of an individual, the population may be considered as remaining constant, except in as far as it is modified by deaths due to the epidemic disease itself. In the course of time the epidemic may come to an end. One of the most important probems in epidemiology is to ascertain whether this termination occurs only when no susceptible individuals are left, or whether the interplay of the various factors of infectivity, recovery and mortality, may result in termination, whilst many susceptible individuals are still present in the unaffected population. It is difficult to treat this problem in its most general aspect. In the present communication discussion will be limited to the case in which all members of the community are initially equally susceptible to the disease, and it will be further assumed that complete immunity is conferred by a single infection.
8,238 citations
01 Jan 1927
TL;DR: The present communication discussion will be limited to the case in which all members of the community are initially equally susceptible to the disease, and it will be further assumed that complete immunity is conferred by a single infection.
Abstract: (1) One of the most striking features in the study of epidemics is the difficulty of finding a causal factor which appears to be adequate to account for the magnitude of the frequent epidemics of disease which visit almost every population. It was with a view to obtaining more insight regarding the effects of the various factors which govern the spread of contagious epidemics that the present investigation was undertaken. Reference may here be made to the work of Ross and Hudson (1915-17) in which the same problem is attacked. The problem is here carried to a further stage, and it is considered from a point of view which is in one sense more general. The problem may be summarised as follows: One (or more) infected person is introduced into a community of individuals, more or less susceptible to the disease in question. The disease spreads from the affected to the unaffected by contact infection. Each infected person runs through the course of his sickness, and finally is removed from the number of those who are sick, by recovery or by death. The chances of recovery or death vary from day to day during the course of his illness. The chances that the affected may convey infection to the unaffected are likewise dependent upon the stage of the sickness. As the epidemic spreads, the number of unaffected members of the community becomes reduced. Since the course of an epidemic is short compared with the life of an individual, the population may be considered as remaining constant, except in as far as it is modified by deaths due to the epidemic disease itself. In the course of time the epidemic may come to an end. One of the most important probems in epidemiology is to ascertain whether this termination occurs only when no susceptible individuals are left, or whether the interplay of the various factors of infectivity, recovery and mortality, may result in termination, whilst many susceptible individuals are still present in the unaffected population. It is difficult to treat this problem in its most general aspect. In the present communication discussion will be limited to the case in which all members of the community are initially equally susceptible to the disease, and it will be further assumed that complete immunity is conferred by a single infection.
7,769 citations
Cites background from "Über eine Klasse von Integralgeleic..."
...In equation (13) vt and d log x/dt are known functions of t and so the equation is of the type discussed by Fock (1924)....
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TL;DR: A mathematical investigation has been made of the progress of an epidemic in a homogeneous population, finding a threshold density of population is found to exist, which depends upon the infectivity, recovery and death rates peculiar to the epidemic.
1,271 citations
01 Feb 1925
TL;DR: In this paper, the authors consider the life of an individual as a succession of events, one following the other, and the method of their movement becomes a study in kinetics, and can be approached by the methods ordinarily adopted in the study of such systems.
Abstract: In the majority of the processes with which one is concerned in the study of the medical sciences, one has to deal with assemblages of individuals, be they living or be they dead, which become affected according to some characteristic. They may meet and exchange ideas, the meeting may result in the transference of some infectious disease, and so forth. The life of each individual consists of a train of such incidents, one following the other. From another point of view each member of the human community consists of an assemblage of cells. These cells react and interact amongst each other, and each individual lives a life which may be again considered as a succession of events, one following the other. If one thinks of these individuals, be they human beings or be they cells, as moving in all sorts of dimensions, reversibly or irreversibly, continuously or discontinuously, by unit stages or per saltum, then the method of their movement becomes a study in kinetics, and can be approached by the methods ordinarily adopted in the study of such systems.
1,019 citations
TL;DR: This article traced the history of the Laplace Transform up to 1880, when Poincare reinvented the transform, but did so in a more powerful context, that of properly conceived complex analysis.
Abstract: An earlier paper, to which this is a sequel, traced the history of the Laplace Transform up to 1880. In that year Poincare reinvented the transform, but did so in a more powerful context, that of properly conceived complex analysis. Rapid developments followed, culminating in Doetsch' work in which the transform took its modern shape.
26 citations