On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations

  title={On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations},
  author={Odo Diekmann and J. A. P. Heesterbeek and Johan A. J. Metz},
  journal={Journal of Mathematical Biology},
The expected number of secondary cases produced by a typical infected individual during its entire period of infectiousness in a completely susceptible population is mathematically defined as the dominant eigenvalue of a positive linear operator. It is shown that in certain special cases one can easily compute or estimate this eigenvalue. Several examples involving various structuring variables like age, sexual disposition and activity are presented. 
On the biological interpretation of a definition for the parameter R0 in periodic population models
The present paper shows that in demography and epidemiology, the basic reproduction number R0 is the asymptotic ratio of total births in two successive generations of the family tree. Expand
Estimating the basic reproduction number at the beginning of an outbreak under incomplete data
This work compares different methods of estimating the basic reproduction number, R0, focusing on the early stages of an epidemic, and considering weekly reports of new infecteds, and examines the sensitivity of the estimators to the model structure. Expand
The estimation of the basic reproduction number for infectious diseases
  • K. Dietz
  • Biology, Medicine
  • Statistical methods in medical research
  • 1993
The present paper gives a survey about the various estimation methods available for the basic reproduction number Ro, which may vary considerably for different infectious diseases but also for the same disease in different populations. Expand
A non-parametric method for determining epidemiological reproduction numbers
  • F. Pijpers
  • Medicine
  • Journal of mathematical biology
  • 2021
A non-parametric inverse method for extracting the full transfer function of infection, of which the reproduction number is the integral is described, demonstrated by applying it to the timeline of hospitalisation admissions for covid-19 in the Netherlands up to May 20 2020. Expand
Disease Control in Age Structure Population
We combine the Leslie model and its derivatives with the classical compartmental SIRS models to build a model of transmission of infected diseases, in a population of hosts, whether opened or closedExpand
The basic reproduction ratio for sexually transmitted diseases: I. Theoretical considerations.
It is shown how one can calculate the basic reproduction ratio R0 for infectious disease models where an arbitrary but finite number of disease states are recognized and where the phenomena of pairExpand
On the Exact Measure of Disease Spread in Stochastic Epidemic Models
This paper is specifically concerned with the quantification of the spread of a disease modeled by a Markov chain and presents two alternative measures, namely, the exact reproduction number, Re0, and the population transmission number, Rp, that overcome this difficulty and provide valuable insight. Expand
In this short note we give threshold quantities that determine the stability of the infection-free steady state for periodic deterministic systems that describe the spread of infectious diseases inExpand
Perspectives on the basic reproductive ratio
An overview of common methods of formulating R0 and surrogate threshold parameters from deterministic, non-structured models and the recent use of R0 in assessing emerging diseases, such as severe acute respiratory syndrome and avian influenza, a number of recent livestock diseases, and vector-borne diseases malaria, dengue and West Nile virus are surveyed. Expand
The type-reproduction number T in models for infectious disease control.
The type-reproduction number T for an infectious disease is defined, and it is shown that this not only has the required threshold behaviour, but also correctly determines the critical control effort for heterogeneous populations. Expand


Heterogeneity in disease-transmission modeling
The dynamic behavior of a family of disease models for a heterogeneous population and asymptotic patterns of disease prevalence are related to contact and disease duration parameters for gonorrhea are discussed. Expand
Proportionate mixing models for age-dependent infection transmission
We present explicit formulas for the transmission potential of an immunizing infection where the contact rates and the vaccination rates depend on the chronological age of an individual, and theExpand
Epidemiological models for heterogeneous populations: proportionate mixing, parameter estimation, and immunization programs
Abstract Deterministic models are presented for epidemics which occur quickly and for long-term endemic diseases where births and deaths must be considered. Contact-rate matrices are formulated inExpand
Persistence of an infectious disease in a subdivided population.
The transmission dynamics of a communicable disease in a subdivided population where the spread among groups follows the proportionate mixing model while the within-group transmission can correspondExpand
Modeling and analyzing HIV transmission: the effect of contact patterns
The most important finding is that the pattern of contacts between the different groups has a major effect on the spread of HIV, an effect inadequately recognized or studied heretofore. Expand
Using mathematical models to understand the AIDS epidemic
This work uses a simplified approach to investigate the effects of variation in incubation periods and infectivity specific to the AIDS virus, and compares a model of random partner choices with a model in which partners both come from similar behavior groups. Expand
The growth and composition of branching populations
A single-type general branching population develops by individuals reproducing according to i.i.d. point processes on R +, interpreted as the individuals' ages. Such a population can be measured orExpand
Macdonald's model and the transmission of bilharzia.
  • A. Barbour
  • Biology, Medicine
  • Transactions of the Royal Society of Tropical Medicine and Hygiene
  • 1978
It is shown that the threshold parameter governing whether or not an endemic cycle can be established is closely related to the proportion of infected snails in a community, and that this proportion is normally observed to be rather smaller than is compatible with the model. Expand
Models for Vector-Borne Parasitic Diseases
There are several reasons for concentrating on the class of vector-borne diseases from a modelling point of view: 1) They still figure among the health problems of highest priority in mostExpand
The transmission dynamics of human immunodeficiency virus (HIV).
  • R. May, R. Anderson
  • Medicine
  • Philosophical transactions of the Royal Society of London. Series B, Biological sciences
  • 1988
The paper first reviews data on HIV infections and AIDS disease among homosexual men, heterosexuals, intravenous drug abusers and children born to infected mothers, in both developed and developing countries, and develops models for the transmission dynamics of HIV. Expand