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**Epidemiological models for heterogeneous populations: Proportionate mixing, parameter estimation, and immunization programs.**
*(English)*
Zbl 0619.92006

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. R. M. May and 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.

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. R. M. May and 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.

Reviewer: T.Postelnicu

### MSC:

92D25 | Population dynamics (general) |

### Keywords:

epidemiology; subpopulation sizes; deterministic models; long-term endemic diseases; homogeneously mixing; SIRS model; equilibria; thresholds; asymptotic behavior; homogeneous population; Fractional activity levels; contact numbers; proportionate-mixing assumption; Parameter estimations; contact-rate matrices; heterogeneous population models; spatial heterogeneity; immunization programs### Citations:

Zbl 0564.92016
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\textit{H. W. Hethcote} and \textit{J. W. van Ark}, Math. Biosci. 84, 85--118 (1987; Zbl 0619.92006)

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### References:

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