Epidemiological models for heterogeneous populations: Proportionate mixing, parameter estimation, and immunization programs
type: Item
instance of: scholarly article
title: Epidemiological models for heterogeneous populations: Proportionate mixing, parameter estimation, and immunization programs
zbMATH Open document ID: 0619.92006
DOI: 10.1016/0025-5564(87)90044-7
author: Herbert W. Hethcote
author: James W. van Ark
published in: Mathematical Biosciences
publication date: 1987-01-01T00:00:00Z
review text: In this paper, deterministic models are presented for epidemics which occur quickly and for long-term endemic diseases where birth and death must be considered. Initially the models have assumed that the population being considered is uniform and homogeneously mixing. But it is appropriate to consider a population divided into subpopulations which differ from each other. Subpopulations can be determined also on the basis of social, cultural, economic, demographic, and geographic factors. In the model here, each subpopulation in the heterogeneous population is further subdivided into susceptible, infective, and removed classes. The model is called a SIRS model, since susceptible individuals become infectious, then removed with immunity, and then susceptible again if the immunity is temporary. Complete assumptions and several formulations are given in Section 2, and the equilibria, thresholds, and asymptotic behavior are also described for the model. Methods are presented in Sections 4 and 5 for estimating the contact number for diseases in one homogeneous population from either epidemic or endemic data. Fractional activity levels and contact numbers are defined for the subpopulations in Section 6, and it is shown that the threshold is an average contact number under the proportionate-mixing assumption. Parameter estimations are given for an endemic disease in a heterogeneous population (Section 7), and for an epidemic in such a population (Section 8). Methods which have been used by other authors to formulate contact-rate matrices in heterogeneous population models are briefly described in Section 9. \textit{R. M. May} and \textit{R. M. Anderson}'s paper [ibid. 72, 83-111 (1984; Zbl 0564.92016)] on spatial heterogeneity is presented in Section 10. In Section 11, a model to compare the three immunization programs in a spatially heterogeneous population is developed, and the ''city and villages'' example ends this interesting paper.
Mathematics Subject Classification ID: 92D25
zbMATH DE Number: 4006044
zbMATH Keywords: epidemiology
zbMATH Keywords: subpopulation sizes
zbMATH Keywords: deterministic models
zbMATH Keywords: long-term endemic diseases
zbMATH Keywords: homogeneously mixing
zbMATH Keywords: SIRS model
zbMATH Keywords: equilibria
zbMATH Keywords: thresholds
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: homogeneous population
zbMATH Keywords: Fractional activity levels
zbMATH Keywords: contact numbers
zbMATH Keywords: proportionate-mixing assumption
zbMATH Keywords: Parameter estimations
zbMATH Keywords: contact-rate matrices
zbMATH Keywords: heterogeneous population models
zbMATH Keywords: spatial heterogeneity
zbMATH Keywords: immunization programs
reviewed by: T. Postelnicu
MaRDI profile type: Publication
full work available at URL: 0025-5564(87)90044-7
OpenAlex ID: W2079458878
cites work: Q5724331
cites work: On estimating the contagiousness of a disease transmitted from person to person
cites work: Q3943082
cites work: Q4085555
cites work: Q3713901
cites work: Qualitative analyses of communicable disease models
cites work: An immunization model for a heterogeneous population
cites work: Q3934187
cites work: Stability of the endemic equilibrium in epidemic models with subpopulations
cites work: Q3331035
cites work: On the identification of superspreaders for infectious disease
cites work: A deterministic model for gonorrhea in a nonhomogeneous population
cites work: The generalized discrete-time epidemic model with immunity: A synthesis
cites work: Household and Community Transmission Parameters from Final Distributions of Infections in Households
cites work: Spatial heterogeneity and the design of immunization programs
cites work: Q3752200
cites work: Q4151515
cites work: Heterogeneity in disease-transmission modeling
cites work: Endemic disease in environments with spatially heterogeneous host populations
cites work: A mathematical model for the global spread of influenza
cites work: Q3752201
cites work: Eradication of infectious diseases in heterogeneous populations
cites work: An age-dependent epidemic model with application to measles
description: scientific article; zbMATH DE number 4006044