Author
Herbert W. Hethcote
Bio: Herbert W. Hethcote is an academic researcher from University of Iowa. The author has contributed to research in topics: Population & Epidemic model. The author has an hindex of 49, co-authored 81 publications receiving 14898 citations.
Papers published on a yearly basis
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TL;DR: Threshold theorems involving the basic reproduction number, the contact number, and the replacement number $R$ are reviewed for classic SIR epidemic and endemic models and results with new expressions for $R_{0}$ are obtained for MSEIR and SEIR endemic models with either continuous age or age groups.
Abstract: Many models for the spread of infectious diseases in populations have been analyzed mathematically and applied to specific diseases. Threshold theorems involving the basic reproduction number $R_{0}$, the contact number $\sigma$, and the replacement number $R$ are reviewed for the classic SIR epidemic and endemic models. Similar results with new expressions for $R_{0}$ are obtained for MSEIR and SEIR endemic models with either continuous age or age groups. Values of $R_{0}$ and $\sigma$ are estimated for various diseases including measles in Niger and pertussis in the United States. Previous models with age structure, heterogeneity, and spatial structure are surveyed.
5,915 citations
TL;DR: Epidemiological models with nonlinear incidence rates λIpSq show a much wider range of dynamical behaviors than do those with bilinear incidence ratesλIS, and these behaviors are determined mainly by p and λ, and secondarily by q.
Abstract: Epidemiological models with nonlinear incidence rates λIpSqshow a much wider range of dynamical behaviors than do those with bilinear incidence rates λIS. These behaviors are determined mainly by p and λ, and secondarily by q. For such models, there may exist multiple attractive basins in phase space; thus whether or not the disease will eventually die out may depend not only upon the parameters, but also upon the initial conditions. In some cases, periodic solutions may appear by Hopf bifurcation at critical parameter values.
747 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
TL;DR: Theoretical models and epidemiologic data showed that the prevalence of gonorrhea adjusts rapidly to changes in social behavior, medical treatment, and control programs, and prevalence oscillates seasonally around an equilibrium state determined by the current social and medical conditions.
Abstract: Calculations revealed that approximately a third of the reported cases of gonorrhea in women during 1973-1975 were discoveries of the screening program. Theoretical models and epidemiologic data showed that the prevalence of gonorrhea adjusts rapidly to changes in social behavior, medical treatment, and control programs, that prevalence oscillates seasonally around an equilibrium state determined by the current social and medical conditions, and that this equilibrium moves as epidemiologic conditions change. The incidence of gonorrhea is theoretically limited by saturation in a sexually active core population, and this core causes gonorrhea to remain endemic.
404 citations
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TL;DR: There is, I think, something ethereal about i —the square root of minus one, which seems an odd beast at that time—an intruder hovering on the edge of reality.
Abstract: There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.
Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …
33,785 citations
TL;DR: Developments in this field are reviewed, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.
Abstract: Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.
17,647 citations
TL;DR: The major concepts and results recently achieved in the study of the structure and dynamics of complex networks are reviewed, and the relevant applications of these ideas in many different disciplines are summarized, ranging from nonlinear science to biology, from statistical mechanics to medicine and engineering.
9,441 citations
TL;DR: A precise definition of the basic reproduction number, R0, is presented for a general compartmental disease transmission model based on a system of ordinary differential equations and it is shown that, if R0<1, then the disease free equilibrium is locally asymptotically stable; whereas if R 0>1,Then it is unstable.
Abstract: A precise definition of the basic reproduction number, Ro, is presented for a general compartmental disease transmission model based on a system of ordinary dierential equations. It is shown that, if Ro 1, then it is unstable. Thus,Ro is a threshold parameter for the model. An analysis of the local centre manifold yields a simple criterion for the existence and stability of super- and sub-threshold endemic equilibria for Ro near one. This criterion, together with the definition of Ro, is illustrated by treatment, multigroup, staged progression, multistrain and vectorhost models and can be applied to more complex models. The results are significant for disease control.
7,106 citations
TL;DR: Threshold theorems involving the basic reproduction number, the contact number, and the replacement number $R$ are reviewed for classic SIR epidemic and endemic models and results with new expressions for $R_{0}$ are obtained for MSEIR and SEIR endemic models with either continuous age or age groups.
Abstract: Many models for the spread of infectious diseases in populations have been analyzed mathematically and applied to specific diseases. Threshold theorems involving the basic reproduction number $R_{0}$, the contact number $\sigma$, and the replacement number $R$ are reviewed for the classic SIR epidemic and endemic models. Similar results with new expressions for $R_{0}$ are obtained for MSEIR and SEIR endemic models with either continuous age or age groups. Values of $R_{0}$ and $\sigma$ are estimated for various diseases including measles in Niger and pertussis in the United States. Previous models with age structure, heterogeneity, and spatial structure are surveyed.
5,915 citations